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Deep Learning · Algorithms

Yoav Gelberg, Yam Eitan, Aviv Navon, Aviv Shamsian, Theo Putterman, Michael Bronstein, Haggai Maron

Gradients of neural networks encode valuable information for optimization, editing, and analysis of models. Therefore, practitioners often treat gradients as inputs to task-specific algorithms, e.g., using gradient statistics for pruning or optimization. Recent works explore *learning* algorithms that operate directly on gradients but use architectures that are not specifically designed for gradient processing, hindering their applicability. In this paper, we present a principled approach for designing architectures that process gradients. Our approach is guided by three principles: (1) equivariant design that preserves neuron permutation symmetries, (2) processing sets of gradients across multiple data points to capture curvature information, and (3) efficient gradient representation through rank-1 decomposition. Based on these principles, we introduce GradMetaNet, a novel architecture for learning on gradients, constructed from simple equivariant blocks. We prove universality results for GradMetaNet, and show that previous approaches cannot approximate natural gradient-based functions that GradMetaNet can. We then demonstrate GradMetaNet's effectiveness on a diverse set of gradient-based tasks for *MLPs* and *transformers*, such as learned optimization, INR editing, and loss landscape curvature estimation.

Deep Learning · Graph Neural Networks

Pritish Chakraborty, Indradyumna Roy, Soumen Chakrabarti, Abir De

Retrieving graphs from a large corpus, that contain a subgraph isomorphic to a given query graph, is a core operation in many real-world applications. While recent multi-vector graph representations and scores based on set alignment and containment can provide accurate subgraph isomorphism tests, their use in retrieval remains limited by their need to score corpus graphs exhaustively. We introduce CoRGII (COntextual Representation of Graphs for Inverted Indexing), a graph indexing framework in which, starting with a contextual dense graph representation, a differentiable discretization module computes sparse binary codes over a learned latent vocabulary. This text document-like representation allows us to leverage classic, highly optimized inverted indexes, while supporting soft (vector) set containment scores. Improving on this paradigm further, we replace the classical impact score of a `word' on a graph (such as defined by TFIDF or BM25) with a data-driven, trainable impact score. Crucially, CoRGII is trained end-to-end using only binary relevance labels, without fine-grained supervision of query-to-document set alignments. Extensive experiments show that CoRGII provides better trade-offs between efficiency and accuracy, compared to several baselines.

Alexander Theus, Alessandro Cabodi, Sotiris Anagnostidis, Antonio Orvieto, Sidak Pal Singh, Valentina Boeva

Understanding the geometry of neural network loss landscapes is a central question in deep learning, with implications for generalization and optimization. A striking phenomenon is $\textit{linear mode connectivity}$ (LMC), where independently trained models can be connected by low- or zero-barrier paths, despite appearing to lie in separate loss basins. However, this is often obscured by symmetries in parameter space—such as neuron permutations—which make functionally equivalent models appear dissimilar. Prior work has predominantly focused on neuron reordering through permutations, but such approaches are limited in scope and fail to capture the richer symmetries exhibited by modern architectures such as Transformers. In this work, we introduce a unified framework that captures four symmetry classes—permutations, semi-permutations, orthogonal transformations, and general invertible maps—broadening the set of valid reparameterizations and subsuming many previous approaches as special cases. Crucially, this generalization enables, for the first time, the discovery of low- and zero-barrier linear interpolation paths between independently trained Vision Transformers and GPT-2 models. Furthermore, our framework extends beyond pairwise alignment, to multi-model and width-heterogeneous settings, enabling alignment across architectures of different sizes. These results reveal deeper structure in the loss landscape and underscore the importance of symmetry-aware analysis for understanding model space geometry.

Computer Vision · Everything Else

Yi Xiao, Qiangqiang Yuan, Kui Jiang, Wenke Huang, Qiang Zhang, Tingting Zheng, Chia-Wen Lin, Liangpei Zhang

Spiking neural networks (SNNs) are emerging as a promising alternative to traditional artificial neural networks (ANNs), offering biological plausibility and energy efficiency. Despite these merits, SNNs are frequently hampered by limited capacity and insufficient representation power, yet remain underexplored in remote sensing image (RSI) super-resolution (SR) tasks. In this paper, we first observe that spiking signals exhibit drastic intensity variations across diverse textures, highlighting an active learning state of the neurons. This observation motivates us to apply SNNs for efficient SR of RSIs. Inspired by the success of attention mechanisms in representing salient information, we devise the spiking attention block (SAB), a concise yet effective component that optimizes membrane potentials through inferred attention weights, which, in turn, regulates spiking activity for superior feature representation. Our key contributions include: 1) we bridge the independent modulation between temporal and channel dimensions, facilitating joint feature correlation learning, and 2) we access the global self-similar patterns in large-scale remote sensing imagery to infer spatial attention weights, incorporating effective priors for realistic and faithful reconstruction. Building upon SAB, we proposed SpikeSR, which achieves state-of-the-art performance across various remote sensing benchmarks such as AID, DOTA, and DIOR, while maintaining high computational efficiency. Code of SpikeSR will be available at https://github.com/XY-boy/SpikeSR.

Optimization · Stochastic & Convex Optimization

Patrick Cheridito, Jean-Loup Dupret, Donatien Hainaut

In this paper, we introduce a model-based deep-learning approach to solve finite-horizon continuous-time stochastic control problems with jumps. We iteratively train two neural networks: one to represent the optimal policy and the other to approximate the value function. Leveraging a continuous-time version of the dynamic programming principle, we derive two different training objectives based on the Hamilton--Jacobi--Bellman equation, ensuring that the networks capture the underlying stochastic dynamics. Empirical evaluations on different problems illustrate the accuracy and scalability of our approach, demonstrating its effectiveness in solving complex, high-dimensional stochastic control tasks.

Optimization · Neural Network Optimization

Zeju Qiu, Simon Buchholz, Tim Xiao, Maximilian Dax, Bernhard Schölkopf, Weiyang Liu

While large language models (LLMs) are driving the rapid advancement of artificial intelligence, effectively and reliably training these large models remains one of the field's most significant challenges. To address this challenge, we propose POET, a novel reParameterized training algorithm that uses Orthogonal Equivalence Transformation to optimize neurons. Specifically, POET reparameterizes each neuron with two learnable orthogonal matrices and a fixed random weight matrix. Because of its provable preservation of spectral properties of weight matrices, POET can stably optimize the objective function with improved generalization. We further develop efficient approximations that make POET flexible and scalable for training large-scale neural networks. Extensive experiments validate the effectiveness and scalability of POET in training LLMs.

Deep Learning · Graph Neural Networks

Shuaicheng Zhang, Haohui Wang, Junhong Lin, Xiaojie Guo, Yada Zhu, Si Zhang, Dongqi Fu, Dawei Zhou

Graph heterophily, where connected nodes have different labels, has attracted significant interest recently. Most existing works adopt a simplified approach - using low-pass filters for homophilic graphs and high-pass filters for heterophilic graphs. However, we discover that the relationship between graph heterophily and spectral filters is more complex - the optimal filter response varies across frequency components and does not follow a strict monotonic correlation with heterophily degree. This finding challenges conventional fixed filter designs and suggests the need for adaptive filtering to preserve expressiveness in graph embeddings. Formally, natural questions arise: Given a heterophilic graph $\mathcal{G}$ , how and to what extent will the varying heterophily degree of $\mathcal{G}$ affect the performance of GNNs? How can we design adaptive filters to fit those varying heterophilic connections? Our theoretical analysis reveals that the average frequency response of GNNs and graph heterophily degree do not follow a strict monotonic correlation, necessitating adaptive graph filters to guarantee good generalization performance. Hence, we propose HeroFilter, a simple yet powerful GNN, which extracts information across the heterophily spectrum and combines salient representations through adaptive mixing. HeroFilter's superior performance achieves up to 9.2% accuracy improvement over leading baselines across homophilic and heterophilic graphs.

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.

Trinity Chung, Yuchen Shen, Nathan Kong, Aran Nayebi

Tactile sensing remains far less understood in neuroscience and less effective in artificial systems compared to more mature modalities such as vision and language. We bridge these gaps by introducing a novel Encoder-Attender-Decoder (EAD) framework to systematically explore the space of task-optimized temporal neural networks trained on realistic tactile input sequences from a customized rodent whisker-array simulator. We identify convolutional recurrent neural networks (ConvRNNs) as superior encoders to purely feedforward and state-space architectures for tactile categorization. Crucially, these ConvRNN-encoder-based EAD models achieve neural representations closely matching rodent somatosensory cortex, saturating the explainable neural variability and revealing a clear linear relationship between supervised categorization performance and neural alignment. Furthermore, contrastive self-supervised ConvRNN-encoder-based EADs, trained with tactile-specific augmentations, match supervised neural fits, serving as an ethologically-relevant, label-free proxy. For neuroscience, our findings highlight nonlinear recurrent processing as important for general-purpose tactile representations in somatosensory cortex, providing the first quantitative characterization of the underlying inductive biases in this system. For embodied AI, our results emphasize the importance of recurrent EAD architectures to handle realistic tactile inputs, along with tailored self-supervised learning methods for achieving robust tactile perception with the same type of sensors animals use to sense in unstructured environments.

Deep Learning · Theory

Yinfeng Chen, Yuling Jiao, Rui Qiu, Zhou Yu

Linear sufficient dimension reduction, as exemplified by sliced inverse regression, has seen substantial development in the past thirty years. However, with the advent of more complex scenarios, nonlinear dimension reduction has gained considerable interest recently. This paper introduces a novel method for nonlinear sufficient dimension reduction, utilizing the generalized martingale difference divergence measure in conjunction with deep neural networks. The optimal solution of the proposed objective function is shown to be unbiased at the general level of $\sigma$-fields. And two optimization schemes, based on the fascinating deep neural networks, exhibit higher efficiency and flexibility compared to the classical eigendecomposition of linear operators.Moreover, we systematically investigate the slow rate and fast rate for the estimation error based on advanced $U$-process theory. Remarkably, the fast rate almost coincides with the minimax rate of nonparametric regression. The validity of our deep nonlinear sufficient dimension reduction methods is demonstrated through simulations and real data analysis.

Yuandong Tian

We prove rich algebraic structures of the solution space for 2-layer neural networks with quadratic activation and $L_2$ loss, trained on reasoning tasks in Abelian group (e.g., modular addition). Such a rich structure enables \emph{analytical} construction of global optimal solutions from partial solutions that only satisfy part of the loss, despite its high nonlinearity. We coin the framework as \ours{} (\emph{\underline{Co}mposing \underline{G}lobal \underline{S}olutions}). Specifically, we show that the weight space over different numbers of hidden nodes of the 2-layer network is equipped with a semi-ring algebraic structure, and the loss function to be optimized consists of \emph{sum potentials}, which are ring homomorphisms, allowing partial solutions to be composed into global ones by ring addition and multiplication. Our experiments show that around $95\%$ of the solutions obtained by gradient descent match exactly our theoretical constructions. Although the global solutions constructed only required a small number of hidden nodes, our analysis on gradient dynamics shows that overparameterization asymptotically decouples training dynamics and is beneficial. We further show that training dynamics favors simpler solutions under weight decay, and thus high-order global solutions such as perfect memorization are unfavorable. The code is open sourced\footnote{\url{https://github.com/facebookresearch/luckmatters/tree/yuandong3/ssl/real-dataset}}.

Deep Learning · Theory

Gerard Ben Arous, Murat Erdogdu, Nuri Mert Vural, Denny Wu

We study the optimization and sample complexity of gradient-based training of a two-layer neural network with quadratic activation function in the high-dimensional regime, where the data is generated as $y \propto \sum_{j=1}^{r}\lambda_j \sigma\left(\langle \boldsymbol{\theta_j}, \boldsymbol{x}\rangle\right), \boldsymbol{x} \sim \mathcal{N}(0,\boldsymbol{I}_d)$, where $\sigma$ is the 2nd Hermite polynomial, and $\lbrace \boldsymbol{\theta}_j \rbrace _{j=1}^{r} \subset \mathbb{R}^d$ are orthonormal signal directions. We consider the extensive-width regime $r \asymp d^\beta$ for $\beta \in (0, 1)$, and assume a power-law decay on the (non-negative) second-layer coefficients $\lambda_j\asymp j^{-\alpha}$ for $\alpha \geq 0$. We provide a sharp analysis of the SGD dynamics in the feature learning regime, for both the population limit and the finite-sample (online) discretization, and derive scaling laws for the prediction risk that highlight the power-law dependencies on the optimization time, the sample size, and the model width. Our analysis combines a precise characterization of the associated matrix Riccati differential equation with novel matrix monotonicity arguments to establish convergence guarantees for the infinite-dimensional effective dynamics.

Optimization · Neural Network Optimization

XINYU DING, Bangtian Liu, Siyu Liao, Zhongfeng Wang

Fast Fourier Transforms (FFT) are widely used to reduce memory and computational costs in deep learning. However, existing implementations, including standard FFT and real FFT (rFFT), cannot achieve true in-place computation. In particular, rFFT maps an input of size $n$ to a complex output of size $\frac{n}{2}+1$, causing dimensional mismatch and requiring additional memory allocation. We propose the first real-domain, fully in-place FFT framework (rdFFT) that preserves input-output dimensional consistency ($n \rightarrow n$). By leveraging butterfly operation symmetry and conjugate properties in the frequency domain, we design an implicit complex encoding scheme that eliminates intermediate cache usage entirely. Theoretically, our method reduces memory usage by 50\% compared to rFFTs. Moreover, it enables zero-cache parameter updates by utilizing the derivative property of the Fourier transform to compute matrix inverses efficiently without intermediate storage. Experiments on multiple natural language understanding tasks demonstrate the method’s effectiveness in maintaining model performance while significantly lowering memory overhead, offering a promising direction for frequency-domain lightweight adaptation.

Optimization · Neural Network Optimization

Tongtong Su, Yun Liao, Fengbo Zheng

Self-knowledge distillation (SKD) enables single-model training by distilling knowledge from the model's own output, eliminating the need for a separate teacher network required in conventional distillation methods. However, current SKD methods focus mainly on replicating common features in the student model, neglecting the extraction of key features that significantly enhance student learning. Inspired by this, we devise a self-knowledge distillation framework entitled Self-Distillation training via Proximal Gradient Optimization or SDPGO, which utilizes gradient information to identify and assign greater weight to features that significantly impact classification performance, enabling the network to learn the most relevant features during training. Specifically, the proposed framework refines the gradient information into a dynamically changing weighting factor to evaluate the distillation knowledge via the dynamic weight adjustment scheme. Meanwhile, we devise the sequential iterative learning module to dynamically optimize knowledge transfer by leveraging historical predictions and real-time gradients, stabilizing training through mini-batch-based KL divergence refinement while adaptively prioritizing task-critical features for efficient self-distillation. Comprehensive experiments on image classification, object detection, and semantic segmentation demonstrate that our method consistently surpasses recent state-of-the-art knowledge distillation techniques. Code is available at: https://github.com/nanxiaotong/SDGPO.

Optimization · Neural Network Optimization

Thomas Pethick, Wanyun Xie, Mete Erdogan, Kimon Antonakopoulos, Antonio Silveti-Falls, Volkan Cevher

This work introduces a hybrid non-Euclidean optimization method which generalizes gradient norm clipping by combining steepest descent and conditional gradient approaches. The method achieves the best of both worlds by establishing a descent property under a generalized notion of ($L_0$,$L_1$)-smoothness. Weight decay is incorporated in a principled manner by identifying a connection to the Frank-Wolfe short step. In the stochastic case, we show an order optimal $O(n^{-1/4})$ convergence rate by leveraging a momentum based gradient estimator. We discuss how to instantiate the algorithms for deep learning, which we dub Clipped Scion, and demonstrate their properties on image classification and language modeling. The code is available at https://github.com/LIONS-EPFL/ClippedScion.

Theory · Learning Theory

Roey Magen, Shuning Shang, Zhiwei Xu, Spencer Frei, Wei Hu, Gal Vardi

The phenomenon of benign overfitting, where a trained neural network perfectly fits noisy training data but still achieves near-optimal test performance, has been extensively studied in recent years for linear models and fully-connected/convolutional networks. In this work, we study benign overfitting in a single-head softmax attention model, which is the fundamental building block of Transformers. We prove that under appropriate conditions, the model exhibits benign overfitting in a classification setting already after two steps of gradient descent. Moreover, we show conditions where a minimum-norm/maximum-margin interpolator exhibits benign overfitting. We study how the overfitting behavior depends on the signal-to-noise ratio (SNR) of the data distribution, namely, the ratio between norms of signal and noise tokens, and prove that a sufficiently large SNR is both necessary and sufficient for benign overfitting.

General Machine Learning · Transfer, Multitask and Meta-learning

Yohan Jung, Hyungi Lee, Wenlong Chen, Thomas Möllenhoff, Yingzhen Li, Juho Lee, Mohammad Emtiyaz Khan

Despite recent progress, continual learning still does not match the performance of batch training. To avoid catastrophic forgetting, we need to build compact memory of essential past knowledge, but no clear solution has yet emerged, even for shallow neural networks with just one or two layers. In this paper, we present a new method to build compact memory for logistic regression. Our method is based on a result by Khan and Swaroop [2021] who show the existence of optimal memory for such models. We formulate the search for the optimal memory as Hessian-matching and propose a probabilistic PCA method to estimate them. Our approach can drastically improve accuracy compared to Experience Replay. For instance, on Split-ImageNet, we get 60% accuracy compared to 30% obtained by replay with memory-size equivalent to 0.3% of the data size. Increasing the memory size to 2% further boosts the accuracy to 74%, closing the gap to the batch accuracy of 77.6% on this task. Our work opens a new direction for building compact memory that can also be useful in the future for continual deep learning.

Theory · Reinforcement Learning and Planning

Tong Yang, Bo Dai, Lin Xiao, Yuejie Chi

Online reinforcement learning (RL) with complex function approximations such as transformers and deep neural networks plays a significant role in the modern practice of artificial intelligence. Despite its popularity and importance, balancing the fundamental trade-off between exploration and exploitation remains a long-standing challenge; in particular, we are still in lack of efficient and practical schemes that are backed by theoretical performance guarantees. Motivated by recent developments in exploration via optimistic regularization, this paper provides an interpretation of the principle of optimism through the lens of primal-dual optimization. From this fresh perspective, we set forth a new value-incentivized actor-critic (VAC) method, which optimizes a single easy-to-optimize objective integrating exploration and exploitation --- it promotes state-action and policy estimates that are both consistent with collected data transitions and result in higher value functions. Theoretically, the proposed VAC method has near-optimal regret guarantees under linear Markov decision processes (MDPs) in both finite-horizon and infinite-horizon settings, which can be extended to the general function approximation setting under appropriate assumptions.

Deep Learning · Graph Neural Networks

Sonia Mazelet, Rémi Flamary, Bertrand Thirion

Optimal transport between graphs, based on Gromov-Wasserstein and other extensions, is a powerful tool for comparing and aligning graph structures. However, solving the associated non-convex optimization problems is computationally expensive, which limits the scalability of these methods to large graphs. In this work, we present Unbalanced Learning of Optimal Transport (ULOT), a deep learning method that predicts optimal transport plans between two graphs. Our method is trained by minimizing the fused unbalanced Gromov-Wasserstein (FUGW) loss. We propose a novel neural architecture with cross-attention that is conditioned on the FUGW tradeoff hyperparameters. We evaluate ULOT on synthetic stochastic block model (SBM) graphs and on real cortical surface data obtained from fMRI. ULOT predicts transport plans with competitive loss up to two orders of magnitude faster than classical solvers. Furthermore, the predicted plan can be used as a warm start for classical solvers to accelerate their convergence. Finally, the predicted transport plan is fully differentiable with respect to the graph inputs and FUGW hyperparameters, enabling the optimization of functionals of the ULOT plan.

Applications · Physics

Jan Hagnberger, Daniel Musekamp, Mathias Niepert

Solving time-dependent Partial Differential Equations (PDEs) using a densely discretized spatial domain is a fundamental problem in various scientific and engineering disciplines, including modeling climate phenomena and fluid dynamics. However, performing these computations directly in the physical space often incurs significant computational costs. To address this issue, several neural surrogate models have been developed that operate in a compressed latent space to solve the PDE. While these approaches reduce computational complexity, they often use Transformer-based attention mechanisms to handle irregularly sampled domains, resulting in increased memory consumption. In contrast, convolutional neural networks allow memory-efficient encoding and decoding but are limited to regular discretizations. Motivated by these considerations, we propose CALM-PDE, a model class that efficiently solves arbitrarily discretized PDEs in a compressed latent space. We introduce a novel continuous convolution-based encoder-decoder architecture that uses an epsilon-neighborhood-constrained kernel and learns to apply the convolution operator to adaptive and optimized query points. We demonstrate the effectiveness of CALM-PDE on a diverse set of PDEs with both regularly and irregularly sampled spatial domains. CALM-PDE is competitive with or outperforms existing baseline methods while offering significant improvements in memory and inference time efficiency compared to Transformer-based methods.