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Deep Learning · Algorithms

Sifeng SHANG, JIAYI ZHOU, Chenyu Lin, Minxian Li, Kaiyang Zhou

As the size of large language models grows exponentially, GPU memory has become a bottleneck for adapting these models to downstream tasks. In this paper, we aim to push the limits of memory-efficient training by minimizing memory usage on model weights, gradients, and optimizer states, within a unified framework. Our idea is to eliminate both gradients and optimizer states using zeroth-order optimization, which approximates gradients by perturbing weights during forward passes to identify gradient directions. To minimize memory usage on weights, we employ model quantization, e.g., converting from bfloat16 to int4. However, directly applying zeroth-order optimization to quantized weights is infeasible due to the precision gap between discrete weights and continuous gradients, which would otherwise require de-quantization and re-quantization. To overcome this challenge, we propose Quantized Zeroth-order Optimization (QZO), a simple yet effective approach that perturbs the continuous quantization scale for gradient estimation and uses a directional derivative clipping method to stabilize training. QZO is orthogonal to both scalar-based and codebook-based post-training quantization methods. Compared to full-parameter fine-tuning in 16 bits, QZO can reduce the total memory cost by more than 18$\times$ for 4-bit LLMs, and enables fine-tuning Llama-2-13B within a single 24GB GPU.

Parham Rezaei, Filip Kovačević, Francesco Locatello, Marco Mondelli

Despite the progress in the development of generative models, their usefulness in creating synthetic data that improve prediction performance of classifiers has been put into question. Besides heuristic principles such as ''synthetic data should be close to the real data distribution'', it is actually not clear which specific properties affect the generalization error. Our paper addresses this question through the lens of high-dimensional regression. Theoretically, we show that, for linear models, the *covariance shift* between the target distribution and the distribution of the synthetic data affects the generalization error but, surprisingly, the mean shift does not. Furthermore, in some regimes, we prove that matching the covariance of the target distribution is optimal. Remarkably, the theoretical insights for linear models carry over to deep neural networks and generative models. We empirically demonstrate that the *covariance matching* procedure (matching the covariance of the synthetic data with that of the data coming from the target distribution) performs well against several recent approaches for synthetic data selection, across various training paradigms, datasets and generative models used for augmentation.

Chaitanya Kapoor, Alex Williams, Meenakshi Khosla

Representational similarity metrics typically force all units to be matched, making them susceptible to noise and outliers common in neural representations. We extend the soft-matching distance to a partial optimal transport setting that allows some neurons to remain unmatched, yielding rotation-sensitive but robust correspondences. This partial soft-matching distance provides theoretical advantages---relaxing strict mass conservation while maintaining interpretable transport costs---and practical benefits through efficient neuron ranking in terms of cross-network alignment without costly iterative recomputation. In simulations, it preserves correct matches under outliers and reliably selects the correct model in noise-corrupted identification tasks. On fMRI data, it automatically excludes low-reliability voxels and produces voxel rankings by alignment quality that closely match computationally expensive brute-force approaches. It achieves higher alignment precision across homologous brain areas than standard soft-matching, which is forced to match all units regardless of quality. In deep networks, highly matched units exhibit similar maximally exciting images, while unmatched units show divergent patterns. This ability to partition by match quality enables focused analyses, \emph{e.g.,} testing whether networks have privileged axes even within their most aligned subpopulations. Overall, partial soft-matching provides a principled and practical method for representational comparison under partial correspondence.

Theory · Learning Theory

Julian D. Schiller, Malte Heinrich, Victor Lopez, Matthias Müller

Training recurrent neural networks (RNNs) with standard backpropagation through time (BPTT) can be challenging, especially in the presence of long input sequences. A practical alternative to reduce computational and memory overhead is to perform BPTT repeatedly over shorter segments of the training data set, corresponding to truncated BPTT. In this paper, we examine the training of RNNs when using such a truncated learning approach for time series tasks. Specifically, we establish theoretical bounds on the accuracy and performance loss when optimizing over subsequences instead of the full data sequence. This reveals that the burn-in phase of the RNN is an important tuning knob in its training, with significant impact on the performance guarantees. We validate our theoretical results through experiments on standard benchmarks from the fields of system identification and time series forecasting. In all experiments, we observe a strong influence of the burn-in phase on the training process, and proper tuning can lead to a reduction of the prediction error on the training and test data of more than 60% in some cases.

Deep Learning · Generative Models and Autoencoders

Roman Tarasov, Petr Mokrov, Milena Gazdieva, Evgeny Burnaev, Aleksandr Korotin

Neural network-based optimal transport (OT) is a recent and fruitful direction in the generative modeling community. It finds its applications in various fields such as domain translation, image super-resolution, computational biology and others. Among the existing OT approaches, of considerable interest are adversarial minimax solvers based on semi-dual formulations of OT problems. While promising, these methods lack theoretical investigation from a statistical learning perspective. Our work fills this gap by establishing upper bounds on the generalization error of an approximate OT map recovered by the minimax quadratic OT solver. Importantly, the bounds we derive depend solely on some standard statistical and mathematical properties of the considered functional classes (neural nets). While our analysis focuses on the quadratic OT, we believe that similar bounds could be derived for general OT case, paving the promising direction for future research. Our experimental illustrations are available online https://github.com/milenagazdieva/StatOT.

Deep Learning · Graph Neural Networks

Atamert Rahma, Chinmay Datar, Ana Cukarska, Felix Dietrich

Learning dynamical systems that respect physical symmetries and constraints remains a fundamental challenge in data-driven modeling. Integrating physical laws with graph neural networks facilitates principled modeling of complex N-body dynamics and yields accurate and permutation-invariant models. However, training graph neural networks with iterative, gradient-descent-based optimization algorithms (e.g., Adam, RMSProp, LBFGS) often leads to slow training, especially for large, complex systems. In comparison to 15 different optimizers, we demonstrate that Hamiltonian Graph Networks (HGN) can be trained 150-600× faster - but with comparable accuracy - by replacing iterative optimization with random feature-based parameter construction. We show robust performance in diverse simulations, including N-body mass-spring and molecular dynamics systems in up to $3$ dimensions and 10,000 particles with different geometries, while retaining essential physical invariances with respect to permutation, rotation, and translation. Our proposed approach is benchmarked using a NeurIPS 2022 Datasets and Benchmarks Track publication to further demonstrate its versatility. We reveal that even when trained on minimal 8-node systems, the model can generalize in a zero-shot manner to systems as large as 4096 nodes without retraining. Our work challenges the dominance of iterative gradient-descent-based optimization algorithms for training neural network models for physical systems.

Chinmay Datar, Taniya Kapoor, Abhishek Chandra, Qing Sun, Erik L. Bolager, Iryna Burak, Anna Veselovska, Massimo Fornasier, Felix Dietrich

Solving time-dependent Partial Differential Equations (PDEs) is one of the most critical problems in computational science. While Physics-Informed Neural Networks (PINNs) offer a promising framework for approximating PDE solutions, their accuracy and training speed are limited by two core barriers: gradient-descent-based iterative optimization over complex loss landscapes and non-causal treatment of time as an extra spatial dimension. We present Frozen-PINN, a novel PINN based on the principle of space-time separation that leverages random features instead of training with gradient descent, and incorporates temporal causality by construction. On nine PDE benchmarks, including challenges like extreme advection speeds, shocks, and high-dimensionality, Frozen-PINNs achieve superior training efficiency and accuracy over state-of-the-art PINNs, often by several orders of magnitude. Our work addresses longstanding training and accuracy bottlenecks of PINNs, delivering quickly trainable, highly accurate, and inherently causal PDE solvers, a combination that prior methods could not realize. Our approach challenges the reliance of PINNs on stochastic gradient-descent-based methods and specialized hardware, leading to a paradigm shift in PINN training and providing a challenging benchmark for the community.

Optimization · Learning for Optimization

Benjamin Thérien, Charles-Étienne Joseph, Boris Knyazev, Edouard Oyallon, Irina Rish, Eugene Belilovsky

Learned optimizers (LOs) have the potential to significantly reduce the wall-clock training time of neural networks. However, they can struggle to optimize unseen tasks (*meta-generalize*), especially when training networks wider than those seen during meta-training. To address this, we derive the Maximal Update Parametrization ($\mu$P) for two state-of-the-art learned optimizer architectures and propose a simple meta-training recipe for $\mu$-parameterized LOs ($\mu$LOs). Our empirical evaluation demonstrates that LOs meta-trained with our recipe substantially improve meta-generalization to wider unseen tasks when compared to LOs trained under standard parametrization (SP) using the same compute budget. We also empirically observe that $\mu$LOs exhibit unexpectedly improved meta-generalization to deeper networks ($5\times$ meta-training) and surprising generalization to much longer training horizons ($25\times$ meta-training) when compared to SP LOs.

Optimization · Non-Convex

Pranav Vaidhyanathan, Lucas Schorling, Natalia Ares, Michael Osborne

We introduce Velocity-Regularized Adam (VRAdam), a physics-inspired optimizer for training deep neural networks that draws on ideas from quartic terms for kinetic energy with its stabilizing effects on various system dynamics. Previous algorithms, including the ubiquitous Adam, operate at the so-called adaptive edge of stability regime during training, leading to rapid oscillations and slowed convergence of loss. However, VRAdam adds a higher order penalty on the learning rate based on the velocity such that the algorithm automatically slows down whenever weight updates become large. In practice, we observe that the effective dynamic learning rate shrinks in high-velocity regimes, and damping oscillations. By combining this velocity‑based regularizer for global damping with Adam’s per‑parameter scaling, we create a powerful hybrid optimizer. For this optimizer, we provide rigorous theoretical analysis of operation at the edge of stability from a physical and control perspective for the momentum. Furthermore, we derive convergence bounds with the rate $\mathcal{O}(\ln(N)/\sqrt{N})$ for a stochastic non‑convex objective under mild assumptions. We demonstrate that VRAdam exceeds the performance against standard optimizers including AdamW. We benchmark various tasks such as image classification, language modeling, and generative modeling using diverse architectures and training methodologies including Convolutional Neural Networks (CNNs), Transformers, and GFlowNets.

Reinforcement Learning · Deep RL

Zihao Wu, Hongyao Tang, Yi Ma, Jiashun Liu, YAN ZHENG, Jianye Hao

Deep reinforcement learning (RL) suffers from plasticity loss severely due to the nature of non-stationarity, which impairs the ability to adapt to new data and learn continually. Unfortunately, our understanding of how plasticity loss arises, dissipates, and can be dissolved remains limited to empirical findings, leaving the theoretical end underexplored. To address this gap, we study the plasticity loss problem from the theoretical perspective of network optimization. By formally characterizing the two culprit factors in online RL process: the non-stationarity of data distributions and the non-stationarity of targets induced by bootstrapping, our theory attributes the loss of plasticity to two mechanisms: the rank collapse of the Neural Tangent Kernel (NTK) Gram matrix and the Θ(1/k) decay of gradient magnitude. The first mechanism echoes prior empirical findings from the theoretical perspective and sheds light on the effects of existing methods, e.g., network reset, neuron recycle, and noise injection. Against this backdrop, we focus primarily on the second mechanism and aim to alleviate plasticity loss by addressing the gradient attenuation issue, which is orthogonal to existing methods. We propose Sample Weight Decay (SWD) --- a lightweight method to restore gradient magnitude, as a general remedy to plasticity loss for deep RL methods based on experience replay. In experiments, we evaluate the efficacy of SWD upon TD3, SAC with SimBa architecture in MuJoCo and DeepMind Control Suite tasks. The results demonstrate that SWD effectively alleviates plasticity loss and consistently improves learning performance across various configurations of deep RL algorithms, UTD, network architectures, and environments, achieving SOTA performance on challenging DMC Humanoid tasks.

Isaac Han, Sangyeon Park, Seungwon Oh, Donghu Kim, Hojoon Lee, KyungJoong Kim

Deep neural networks trained on nonstationary data must balance stability (i.e., retaining prior knowledge) and plasticity (i.e., adapting to new tasks). Standard reinitialization methods, which reinitialize weights toward their original values, are widely used but difficult to tune: conservative reinitializations fail to restore plasticity, while aggressive ones erase useful knowledge. We propose FIRE, a principled reinitialization method that explicitly balances the stability–plasticity tradeoff. FIRE quantifies stability through Squared Frobenius Error (SFE), measuring proximity to past weights, and plasticity through Deviation from Isometry (DfI), reflecting weight isotropy. The reinitialization point is obtained by solving a constrained optimization problem, minimizing SFE subject to DfI being zero, which is efficiently approximated by Newton–Schulz iteration. FIRE is evaluated on continual visual learning (CIFAR-10 with ResNet-18), language modeling (OpenWebText with GPT-0.1B), and reinforcement learning (HumanoidBench with SAC and Atari games with DQN). Across all domains, FIRE consistently outperforms both naive training without intervention and standard reinitialization methods, demonstrating effective balancing of the stability–plasticity tradeoff.

Deep Learning · Theory

Bohan Zhang, Zihao Wang, Hengyu Fu, Jason Lee

In deep learning, a central issue is to understand how neural networks efficiently learn high-dimensional features. To this end, we explore the gradient descent learning of a general Gaussian Multi-index model $f(\boldsymbol{x})=g(\boldsymbol{U}\boldsymbol{x})$ with hidden subspace $\boldsymbol{U}\in \mathbb{R}^{r\times d}$, which is the canonical setup to study representation learning. We prove that under generic non-degenerate assumptions on the link function, a standard two-layer neural network trained via layer-wise gradient descent can agnostically learn the target with $o_d(1)$ test error using $\widetilde{\mathcal{O}}(d)$ samples and $\widetilde{\mathcal{O}}(d^2)$ time. The sample and time complexity both align with the information-theoretic limit up to leading order and are therefore optimal. During the first stage of gradient descent learning, the proof proceeds via showing that the inner weights can perform a power-iteration process. This process implicitly mimics a spectral start for the whole span of the hidden subspace and eventually eliminates finite-sample noise and recovers this span. It surprisingly indicates that optimal results can only be achieved if the first layer is trained for more than $\mathcal{O}(1)$ steps. This work demonstrates the ability of neural networks to effectively learn hierarchical functions with respect to both sample and time efficiency.

Deep Learning · Graph Neural Networks

Shao-En Lin, Ming-Yi Hong, Miao-Chen Chiang, Chih-Yu Wang, Che Lin

Heterogeneous Information Networks (HINs) provide a powerful framework for modeling multi-typed entities and relations, typically defined under a fixed schema. Yet, most research assumes this structure is given, overlooking the fact that alternative designs can emphasize different aspects of the data and substantially influence downstream performance. As a theoretical foundation for such designs, we introduce the principle of entity-attribute duality: attributes can be atomized as entities with their associated relations, while entities can, in turn, serve as attributes of others. This principle motivates atomic HIN, a canonical representation that makes all modeling choices explicit and achieves maximal expressiveness. Building on this foundation, we propose a systematic framework for task-specific schema refinement. Within this framework, we demonstrate that widely used benchmarks correspond to heuristic refinements of the atomic HIN—often far from optimal. Across eight datasets, refinement alone enables a simplified Relational GCN (sRGCN) to achieve state-of-the-art performance on node- and link-level tasks, with further gains from advanced HGNNs. These results highlight schema design as a key dimension in heterogeneous graph modeling. By releasing the atomic HINs, searched schemas, and refinement framework, we enable principled benchmarking and open the way for future work on schema-aware learning, automated structure discovery, and next-generation HGNNs. Our code is available at: https://github.com/ntuidssplab/AtomHIN.

Optimization · Learning for Optimization

Yile Chen, Zeyi Wen, Jian Chen, Jin Huang

Optimizers are critical for training deep neural networks. Existing training processes rely on a single static optimizer (e.g., SGD) or a simple hybrid of two optimizers, which miss the opportunity to exploit evolving dynamics in different training states, degrading model quality and convergence. In this paper, we propose a novel dynamic optimizer switching method called **D**ynamic **O**ptimizer **I**nterleaving **T**raining (DOIT) method, which builds surrogate models to predict different optimizers' performance from current parameter states. DOIT uses an acquisition function that combines the results from surrogate models with transferability assessments and process information to select a suitable optimizer for the subsequent training. Experiments on various models and tasks (e.g., image and text classification, machine translation, and object detection) show that DOIT effectively enhances the training, achieving faster convergence (i.e., 2\% to 10\% faster) and higher accuracy (i.e., 1\% to 3\% improvement). Additional independent experiments and case studies further validate DOIT's effectiveness.

Wangduo Xie, Matthew Blaschko

Computed Tomography (CT) plays a vital role in inspecting the internal structures of industrial objects. Furthermore, achieving high-quality CT reconstruction from sparse views is essential for reducing production costs. While classic implicit neural networks have shown promising results for sparse reconstruction, they are unable to leverage shape priors of objects. Motivated by the observation that numerous industrial objects exhibit rectangular structures, we propose a novel \textbf{N}eural \textbf{A}daptive \textbf{B}inning (\textbf{NAB}) method that effectively integrates rectangular priors into the reconstruction process. Specifically, our approach first maps coordinate space into a binned vector space. This mapping relies on an innovative binning mechanism based on differences between shifted hyperbolic tangent functions, with our extension enabling rotations around the input-plane normal vector. The resulting representations are then processed by a neural network to predict CT attenuation coefficients. This design enables end-to-end optimization of the encoding parameters---including position, size, steepness, and rotation---via gradient flow from the projection data, thus enhancing reconstruction accuracy. By adjusting the smoothness of the binning function, NAB can generalize to objects with more complex geometries. This research provides a new perspective on integrating shape priors into neural network-based reconstruction. Extensive experiments demonstrate that NAB achieves superior performance on two industrial datasets. It also maintains robust on medical datasets when the binning function is extended to more general expression. The code is available at \url{https://github.com/Wangduo-Xie/NAB_CT_reconstruction}.

Optimization · Learning for Optimization

Maria Chzhen, Priya Donti

Enforcing complex (e.g., nonconvex) operational constraints is a critical challenge in real-world learning and control systems. However, existing methods struggle to efficiently enforce general classes of constraints. To address this, we propose a novel data-driven amortized approach that uses a trained autoencoder as an approximate projector to provide fast corrections to infeasible predictions. Specifically, we train an autoencoder using an adversarial objective to learn a structured, convex latent representation of the feasible set. This enables rapid correction of neural network outputs by projecting their associated latent representations onto a simple convex shape before decoding into the original feasible set. We test our approach on a diverse suite of constrained optimization and reinforcement learning problems with challenging nonconvex constraints. Results show that our method effectively enforces constraints at a low computational cost, offering a practical alternative to expensive feasibility correction techniques based on traditional solvers.

Deep Learning · Generative Models and Autoencoders

Jacob Bamberger, Iolo Jones, Dennis Duncan, Michael Bronstein, Pierre Vandergheynst, Adam Gosztolai

Deep generative models often face a fundamental tradeoff: high sample quality can come at the cost of memorisation, where the model reproduces training data rather than generalising across the underlying data geometry. We introduce Carré du champ flow matching (CDC-FM), a generalisation of flow matching (FM), that improves the quality-generalisation tradeoff by regularising the probability path with a geometry-aware noise. Our method replaces the homogeneous, isotropic noise in FM with a spatially varying, anisotropic Gaussian noise whose covariance captures the local geometry of the latent data manifold. We prove that this geometric noise can be optimally estimated from the data and is scalable to large data. Further, we provide an extensive experimental evaluation on diverse datasets (synthetic manifolds, point clouds, single-cell genomics, animal motion capture, and images) as well as various neural network architectures (MLPs, CNNs, and transformers). We demonstrate that CDC-FM consistently offers a better quality-generalisation tradeoff, even when used as a latent space generation model. We observe significant improvements over standard FM in data-scarce regimes and in highly non-uniformly sampled datasets, which are often encountered in AI for science applications. Our work provides a mathematical framework for studying the interplay between data geometry, generalisation and memorisation in generative models, as well as a robust and scalable algorithm that can be readily integrated into existing flow matching pipelines.

Computer Vision · 3D Rendering & Reconstruction

Xinhang Liu, Yuxi Xiao, Donny Y. Chen, Jiashi Feng, Yu-Wing Tai, Chi-Keung Tang, Bingyi Kang

Building 4D video representations to model underlying spacetime constitutes a crucial step toward understanding dynamic scenes, yet there is no consensus on the paradigm: current approaches resort to additional estimators such as depth, flow, or tracking, or to heavy per-scene optimization, making them brittle and hard to generalize. In a video, its atomic unit, the pixel, follows a continuous 3D trajectory that unfolds over time, acting as the atomic primitive of dynamics. Recognizing this, we propose to represent any video as a Trajectory Field: a dense mapping that assigns each pixel in each frame to a parametric 3D trajectory. To this end, we introduce Trace Anything, a neural network that predicts the trajectory field in a feed-forward manner. Specifically, for each video frame, the model outputs a series of control point maps, defining parametric trajectories for each pixel. Together, our representation and model directly construct a 4D video representation in a single forward pass, without additional estimators or global alignment. We develop a synthetic data platform to construct a training dataset and a benchmark for trajectory field estimation. Experiments show that Trace Anything surpasses existing methods or performs competitively on the new benchmark and established point tracking benchmarks, with significant efficiency gains. Moreover, it facilitates downstream applications such as goal-conditioned manipulation, simple motion extrapolation, and spatio-temporal fusion. We will release the code, the model weights, and the data platform.

Applications · Neuroscience, Cognitive Science

Yifan Huang, Wei Fang, Zecheng Hao, Zhengyu Ma, Yonghong Tian

Deep spiking neural networks (SNNs) hold immense promise for low-power event-driven computing, but their direct training via backpropagation through time (BPTT) incurs prohibitive memory cost, which limits their scalability. Existing memory-saving approaches, such as online learning, BPTT-to-BP, and reversible networks, compromise accuracy, training speed, or applicability. In this work, we propose a novel and broadly applicable pipeline for memory-efficient SNN training that preserves BPTT's accuracy. Our pipeline integrates layer-wise gradient checkpointing with lossless spike compression to eliminate internal state storage and reduce the memory cost of per-layer input spikes. We also introduce a multi-stage checkpoint adjustment strategy that adaptively refines checkpoint placement based on profiling results to further optimize memory usage and improve training speed. Wrapped in an optimization pass, the pipeline automatically restructures the computation flow before training with minimal user effort. Extensive experiments on diverse architectures and tasks demonstrate up to $8\times$ memory efficiency gains with $\le 20\%$ speed reduction and no accuracy loss. Our method provides a practical solution for efficient and scalable SNN training. Code is available at https://github.com/AllenYolk/snn-gradient-checkpointing.

Optimization · Sampling and Optimization

Soumya Ram, Akhila Ram

Sampling from trained predictors is fundamental for interpretability and as a compute-light alternative to diffusion models, but local samplers struggle on the rugged, high-frequency functions such models learn. We observe that standard neural‑network training implicitly produces a coarse‑to‑fine sequence of models. Early checkpoints suppress high‑degree/ high‑frequency components (Boolean monomials; spherical harmonics under NTK), while later checkpoints restore detail. We exploit this by running a simple annealed sampler across the training trajectory, using early checkpoints for high‑mobility proposals and later ones for refinement. In the Boolean domain, this can turn the exponential bottleneck arising from rugged landscapes or needle gadgets into a near-linear one. In the continuous domain, under the NTK regime, this corresponds to smoothing under the NTK kernel. Requiring no additional compute, our method shows strong empirical gains across a variety of synthetic and real-world tasks, including constrained sampling tasks that diffusion models are unable to handle.