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Deep Learning · Everything Else

Qian-Yuan Tang, Yufei Gu, Yunfeng Cai, Mingming Sun, Ping Li, zhou Xun, Zeke Xie

It is well-known that the Hessian of deep loss landscape matters to optimization and generalization of deep learning. Previous studies reported a rough Hessian structure in deep learning, which consists of two components, a small number of large eigenvalues and a large number of nearly-zero eigenvalues. To the best of our knowledge, we are the first to report that a simple but overlooked power-law Hessian structure exists in well-trained deep neural networks, including Convolutional Neural Networks (CNNs) and Large Language Models (LLMs). Moreover, we provide a maximum-entropy theoretical interpretation for the power-law Hessian structure and theoretically demonstrate the existence of robust and low-dimensional subspace of deep neural networks. Our extensive experiments using the proposed power-law spectral method demonstrate that the power-law Hessian spectra critically relate to multiple important behaviors of deep learning, including optimization, generalization, and overparameterization. Notably, we discover that the power-law Hessian structure of a given LLM can effectively predict generalization during training, while conventional sharpness-based generalization measures that often works well on CNNs become nearly useless for as a generalization predictor of LLMs.

Optimization · Large Scale, Parallel and Distributed

Hongyao Chen, Tianyang Xu, Xiaojun Wu, Josef Kittler

Batch Normalisation (BN) is widely used in conventional deep neural network training to harmonise the input-output distributions for each batch of data.However, federated learning, a distributed learning paradigm, faces the challenge of dealing with non-independent and identically distributed data among the client nodes. Due to the lack of a coherent methodology for updating BN statistical parameters, standard BN degrades the federated learning performance.To this end, it is urgent to explore an alternative normalisation solution for federated learning. In this work, we resolve the dilemma of the BN layer in federated learning by developing a customised normalisation approach, Hybrid Batch Normalisation (HBN). HBN separates the update of statistical parameters (*i.e.*, means and variances used for evaluation) from that of learnable parameters (*i.e.*, parameters that require gradient updates), obtaining unbiased estimates of global statistical parameters in distributed scenarios. In contrast with the existing solutions, we emphasise the supportive power of global statistics for federated learning. The HBN layer introduces a learnable hybrid distribution factor, allowing each computing node to adaptively mix the statistical parameters of the current batch with the global statistics. Our HBN can serve as a powerful plugin to advance federated learning performance.It reflects promising merits across a wide range of federated learning settings, especially for small batch sizes and heterogeneous data. Code is available at https://github.com/Hongyao-Chen/HybridBN.

Theory

Fabiola Ricci, Lorenzo Bardone, Sebastian Goldt

Deep neural networks learn structured features from complex, non-Gaussian inputs, but the mechanisms behind this process remain poorly understood. Our work is motivated by the observation that the first-layer filters learnt by deep convolutional neural networks from natural images resemble those learnt by independent component analysis (ICA), a simple unsupervised method that seeks the most non-Gaussian projections of its inputs. This similarity suggests that ICA provides a simple, yet principled model for studying feature learning. Here, we leverage this connection to investigate the interplay between data structure and optimisation in feature learning for the most popular ICA algorithm, FastICA, and stochastic gradient descent (SGD), which is used to train deep networks. We rigorously establish that FastICA requires at least $n\gtrsim d^4$ samples to recover a single non-Gaussian direction from $d$-dimensional inputs on a simple synthetic data model. We show that vanilla online SGD outperforms FastICA, and prove that the optimal sample complexity $n\gtrsim d^2$ can be reached by smoothing the loss, albeit in a data-dependent way. We finally demonstrate the existence of a search phase for FastICA on ImageNet, and discuss how the strong non-Gaussianity of said images compensates for the poor sample complexity of FastICA.

Applications · Chemistry, Physics, and Earth Sciences

Runzhong Wang, Rui-Xi Wang, Mrunali Manjrekar, Connor Coley

Molecular machine learning has gained popularity with the advancements of geometric deep learning. In parallel, retrieval-augmented generation has become a principled approach commonly used with language models. However, the optimal integration of retrieval augmentation into molecular machine learning remains unclear. Graph neural networks stand to benefit from clever matching to understand the structural alignment of retrieved molecules to a query molecule. Neural graph matching offers a compelling solution by explicitly modeling node and edge affinities between two structural graphs while employing a noise-robust, end-to-end neural network to learn affinity metrics. We apply this approach to mass spectrum simulation and introduce MARASON, a novel model that incorporates neural graph matching to enhance a fragmentation-based neural network. Experimental results highlight the effectiveness of our design, with MARASON achieving 27% top-1 accuracy, a substantial improvement over the non-retrieval state-of-the-art accuracy of 19%. Moreover, MARASON outperforms both naive retrieval-augmented generation methods and traditional graph matching approaches. Code is publicly available at https://github.com/coleygroup/ms-pred.

Deep Learning · Graph Neural Networks

Antonis Vasileiou, Ben Finkelshtein, Floris Geerts, Ron Levie, Christopher Morris

The expressive power of message-passing graph neural networks (MPNNs) is reasonably well understood, primarily through combinatorial techniques from graph isomorphism testing. However, MPNNs' generalization abilities---making meaningful predictions beyond the training set---remain less explored. Current generalization analyses often overlook graph structure, limit the focus to specific aggregation functions, and assume the impractical, hard-to-optimize $0$-$1$ loss function. Here, we extend recent advances in graph similarity theory to assess the influence of graph structure, aggregation, and loss functions on MPNNs' generalization abilities. Our empirical study supports our theoretical insights, improving our understanding of MPNNs' generalization properties.

Deep Learning · Graph Neural Networks

Cheng Xin, Fan Xu, Xin Ding, Jie Gao, Jiaxin Ding

Graph Neural Networks (GNNs) have shown remarkable success across various scientific fields,yet their adoption in critical decision-making is often hindered by a lack of interpretability. Recently,intrinsic interpretable GNNs have been studied to provide insights into model predictions by identifying rationale substructures in graphs. However, existing methods face challenges when the underlying rationale subgraphs are complex and varied. In this work, we propose TopInG: Topologically Interpretable Graph Learning, a novel topological framework that leverages persistent homology to identify persistent rationale subgraphs. TopInG employs a rationale filtration learning approach to model an autoregressive generating process of rationale subgraphs, and introduces a self-adjusted topological constraint, termed topological discrepancy, to enforce a persistent topological distinction between rationale subgraphs and irrelevant counterparts. We provide theoretical guarantees that our loss function is uniquely optimized by the ground truth under specific conditions. Extensive experiments demonstrate TopInG's effectiveness in tackling key challenges, such as handling variform rationale subgraphs, balancing predictive performance with interpretability, and mitigating spurious correlations. Results show that our approach improves upon state-of-the-artmethods on both predictive accuracy and interpretation quality.

Theory · Learning Theory

Ari Karchmer, Eran Malach

We study the relationship between gradient-based optimization of parametric models (e.g., neural networks) and optimization of linear combinations of random features. Our main result shows that if a parametric model can be learned using mini-batch stochastic gradient descent (bSGD) without making assumptions about the data distribution, then with high probability, the target function can also be approximated using a polynomial-sized combination of random features. The size of this combination depends on the number of gradient steps and numerical precision used in the bSGD process. This finding reveals fundamental limitations of distribution-free learning in neural networks trained by gradient descent, highlighting why making assumptions about data distributions is often crucial in practice.Along the way, we also introduce a new theoretical framework called average probabilistic dimension complexity (adc), which extends the probabilistic dimension complexity developed by Kamath et al. (2020). We prove that adc has a polynomial relationship with statistical query dimension, and use this relationship to demonstrate an infinite separation between adc and standard dimension complexity.

Deep Learning · Theory

Niket Patel, Guido Montufar

We define the *local complexity* of a neural network with continuous piecewise linear activations as a measure of the density of linear regions over an input data distribution. We show theoretically that ReLU networks that learn low-dimensional feature representations have a lower local complexity. This allows us to connect recent empirical observations on feature learning at the level of the weight matrices with concrete properties of the learned functions. In particular, we show that the local complexity serves as an upper bound on the total variation of the function over the input data distribution and thus that feature learning can be related to adversarial robustness. Lastly, we consider how optimization drives ReLU networks towards solutions with lower local complexity. Overall, this work contributes a theoretical framework towards relating geometric properties of ReLU networks to different aspects of learning such as feature learning and representation cost.

Deep Learning · Graph Neural Networks

Xunkai Li, Zhengyu Wu, Kaichi Yu, Hongchao Qin, Guang Zeng, Rong-Hua Li, Guoren Wang

Although directed graphs (digraphs) offer strong modeling capabilities for complex topological systems, existing DiGraph Neural Networks (DiGNNs) struggle to fully capture the concealed rich structural information. This data-level limitation results in model-level sub-optimal predictive performance and underscores the necessity of further exploring the potential correlations between the directed edges (topology) and node profiles (features and labels) from a data-centric perspective, thereby empowering model-centric neural networks with stronger encoding capabilities. In this paper, we propose **E**ntropy-driven **D**igraph knowl**E**dge distillatio**N** (EDEN), which can serve as a data-centric digraph learning paradigm or a model-agnostic hot-and-plug data-centric Knowledge Distillation (KD) module. EDEN implements data-centric machine learning by constructing a coarse-grained Hierarchical Knowledge Tree (HKT) using proposed hierarchical encoding theory, and refining HKT through mutual information analysis of node profiles to guide knowledge distillation during training. As a general framework, EDEN naturally extends to undirected graphs and consistently delivers strong performance. Extensive experiments on 14 (di)graph datasets—spanning both homophily and heterophily settings—and across four downstream tasks show that EDEN achieves SOTA results and significantly enhances existing (Di)GNNs.

Applications

Patara Trirat, Wonyong Jeong, Sung Ju Hwang

Automated machine learning (AutoML) accelerates AI development by automating tasks in the development pipeline, such as optimal model search and hyperparameter tuning. Existing AutoML systems often require technical expertise to set up complex tools, which is in general time-consuming and requires a large amount of human effort. Therefore, recent works have started exploiting large language models (LLM) to lessen such burden and increase the usability of AutoML frameworks via a natural language interface, allowing non-expert users to build their data-driven solutions. These methods, however, are usually designed only for a particular process in the AI development pipeline and do not efficiently use the inherent capacity of the LLMs. This paper proposes *AutoML-Agent*, a novel multi-agent framework tailored for full-pipeline AutoML, i.e., from data retrieval to model deployment. *AutoML-Agent* takes user's task descriptions, facilitates collaboration between specialized LLM agents, and delivers deployment-ready models. Unlike existing work, instead of devising a single plan, we introduce a retrieval-augmented planning strategy to enhance exploration to search for more optimal plans. We also decompose each plan into sub-tasks (e.g., data preprocessing and neural network design) each of which is solved by a specialized agent we build via prompting executing in parallel, making the search process more efficient. Moreover, we propose a multi-stage verification to verify executed results and guide the code generation LLM in implementing successful solutions. Extensive experiments on seven downstream tasks using fourteen datasets show that *AutoML-Agent* achieves a higher success rate in automating the full AutoML process, yielding systems with good performance throughout the diverse domains.

Reinforcement Learning

David Pfau, Ian Davies, Diana Borsa, João Madeira Araujo, Brendan Tracey, Hado van Hasselt

We introduce Wasserstein Policy Optimization (WPO), an actor-critic algorithm for reinforcement learning in continuous action spaces. WPO can be derived as an approximation to Wasserstein gradient flow over the space of all policies projected into a finite-dimensional parameter space (e.g., the weights of a neural network), leading to a simple and completely general closed-form update. The resulting algorithm combines many properties of deterministic and classic policy gradient methods. Like deterministic policy gradients, it exploits knowledge of the *gradient* of the action-value function with respect to the action. Like classic policy gradients, it can be applied to stochastic policies with arbitrary distributions over actions -- without using the reparameterization trick. We show results on the DeepMind Control Suite and a magnetic confinement fusion task which compare favorably with state-of-the-art continuous control methods.

Position · Understanding

David A. Danhofer, Davide DAscenzo, Rafael Dubach, Tomaso A Poggio

Overparametrized Deep Neural Networks (DNNs) have demonstrated remarkable success in a wide variety of domains too high-dimensional for classical shallow networks subject to the curse of dimensionality. However, open questions about fundamental principles, that govern the learning dynamics of DNNs, remain. In this position paper we argue that it is the ability of DNNs to exploit the compositionally sparse structure of the target function driving their success. As such, DNNs can leverage the property that most practically relevant functions can be composed from a small set of constituent functions, each of which relies only on a low-dimensional subset of all inputs. We show that this property is shared by all efficiently Turing-computable functions and is therefore highly likely present in all current learning problems. While some promising theoretical insights on questions concerned with approximation and generalization exist in the setting of compositionally sparse functions, several important questions on the learnability and optimization of DNNs remain. Completing the picture of the role of compositional sparsity in deep learning is essential to a comprehensive theory of artificial—and even general—intelligence.

Optimization · Large Scale, Parallel and Distributed

Anh Duc Nguyen, Ilia Markov, Zhengqing Wu, Ali Ramezani-Kebrya, Kimon Antonakopoulos, Dan Alistarh, Volkan Cevher

Modern deep neural networks exhibit heterogeneity across numerous layers of various types such as residuals, multi-head attention, etc., due to varying structures (dimensions, activation functions, etc.), distinct representation characteristics, which impact predictions. We develop a general layer-wise quantization framework with tight variance and code-length bounds, adapting to the heterogeneities over the course of training. We then apply a new layer-wise quantization technique within distributed variational inequalities (VIs), proposing a novel Quantized Optimistic Dual Averaging (QODA) algorithm with adaptive learning rates, which achieves competitive convergence rates for monotone VIs. We empirically show that QODA achieves up to a $150$% speedup over the baselines in end-to-end training time for training Wasserstein GAN on $12+$ GPUs.

General Machine Learning · Unsupervised and Semi-supervised Learning

Jonathan Geuter, Gregor Kornhardt, Ingimar Tomasson, Vaios Laschos

Optimal Transport (OT) problems are a cornerstone of many applications, but solving them is computationally expensive. To address this problem, we propose UNOT (Universal Neural Optimal Transport), a novel framework capable of accurately predicting (entropic) OT distances and plans between discrete measures of variable resolution for a given cost function. UNOT builds on Fourier Neural Operators, a universal class of neural networks that map between function spaces and that are discretization-invariant, which enables our network to process measures of varying sizes. The network is trained adversarially using a second, generating network and a self-supervised bootstrapping loss. We theoretically justify the use of FNOs, prove that our generator is universal, and that minimizing the bootstrapping loss provably minimizes the ground truth loss. Through extensive experiments, we show that our network not only accurately predicts optimal transport distances and plans across a wide range of datasets, but also captures the geometry of the Wasserstein space correctly. Furthermore, we show that our network can be used as a state-of-the-art initialization for the Sinkhorn algorithm, significantly outperforming existing approaches.

Deep Learning · Graph Neural Networks

YuQing Xie, Ameya Daigavane, Mit Kotak, Tess Smidt

$E(3)$-equivariant neural networks have demonstrated success across a wide range of 3D modelling tasks. A fundamental operation in these networks is the tensor product, which interacts two geometric features in an equivariant manner to create new features. Due to the high computational complexity of the tensor product, significant effort has been invested to optimize the runtime of this operation. For example, Luo et al. (2024) recently proposed the Gaunt tensor product (GTP) which promises a significant speedup. In this work, we provide a careful, systematic analysis of a number of tensor product operations. In particular, we emphasize that different tensor products are not performing the same operation. The reported speedups typically come at the cost of expressivity. We introduce measures of expressivity and interactability to characterize these differences. In addition, we realized the original implementation of GTP can be greatly simplified by directly using a spherical grid at no cost in asymptotic runtime. This spherical grid approach is faster on our benchmarks and in actual training of the MACE interatomic potential by 30\%. Finally, we provide the first systematic microbenchmarks of the various tensor product operations. We find that the theoretical runtime guarantees can differ wildly from empirical performance, demonstrating the need for careful application-specific benchmarking. Code is available at https://github.com/atomicarchitects/PriceofFreedom

Deep Learning · Robustness

Jonas Möller, Lukas Pirch, Felix Weissberg, Sebastian Baunsgaard, Thorsten Eisenhofer, Konrad Rieck

Linear algebra is a cornerstone of neural network inference. The efficiency of popular frameworks, such as TensorFlow and PyTorch, critically depends on backend libraries providing highly optimized matrix multiplications and convolutions. A diverse range of these backends exists across platforms, including Intel MKL, Nvidia CUDA, and Apple Accelerate. Although these backends provide equivalent functionality, subtle variations in their implementations can lead to seemingly negligible differences during inference. In this paper, we investigate these minor discrepancies and demonstrate how they can be selectively amplified by adversaries. Specifically, we introduce *Chimera examples*, inputs to models that elicit conflicting predictions depending on the employed backend library. These inputs can even be constructed with integer values, creating a vulnerability exploitable from real-world input domains. We analyze the prevalence and extent of the underlying attack surface and propose corresponding defenses to mitigate this threat.

Deep Learning · Theory

Pedro Cisneros-Velarde, Bhavesh Shrimali, Arindam Banerjee

Neural Operators that directly learn mappings between function spaces, such as Deep Operator Networks (DONs) and Fourier Neural Operators (FNOs), have received considerable attention. Despite the universal approximation guarantees for DONs and FNOs, there is currently no optimization convergence guarantee for learning such networks using gradient descent (GD). In this paper, we address this open problem by presenting a unified framework for optimization based on GD and applying it to establish convergence guarantees for both DONs and FNOs. In particular, we show that the losses associated with both of these neural operators satisfy two conditions—restricted strong convexity (RSC) and smoothness—that guarantee a decrease on their loss values due to GD. Remarkably, these two conditions are satisfied for each neural operator due to different reasons associated with the architectural differences of the respective models. One takeaway that emerges from the theory is that wider networks benefit optimization convergence guarantees for both DONs and FNOs. We present empirical results on canonical operator learning problems to support our theoretical results and find that larger widths benefit training.

General Machine Learning · Online Learning, Active Learning and Bandits

Yifeng Wang, Xueying Zhan, Siyu Huang

As deep learning continues to evolve, the need for data efficiency becomes increasingly important. Considering labeling large datasets is both time-consuming and expensive, active learning (AL) provides a promising solution to this challenge by iteratively selecting the most informative subsets of examples to train deep neural networks, thereby reducing the labeling cost. However, the effectiveness of different AL algorithms can vary significantly across data scenarios, and determining which AL algorithm best fits a given task remains a challenging problem. This work presents the first differentiable AL strategy search method, named AutoAL, which is designed on top of existing AL sampling strategies. AutoAL consists of two neural nets, named SearchNet and FitNet, which are optimized concurrently under a differentiable bi-level optimization framework. For any given task, SearchNet and FitNet are iteratively co-optimized using the labeled data, learning how well a set of candidate AL algorithms perform on that task. With the optimal AL strategies identified, SearchNet selects a small subset from the unlabeled pool for querying their annotations, enabling efficient training of the task model. Experimental results demonstrate that AutoAL consistently achieves superior accuracy compared to all candidate AL algorithms and other selective AL approaches, showcasing its potential for adapting and integrating multiple existing AL methods across diverse tasks and domains.

Optimization

Thien Nguyen, Huy Nguyen

We introduce two complementary techniques for efficient optimization that reduce memory requirements while accelerating training oflarge-scale neural networks. The first technique, Subset-Norm step size, generalizes AdaGrad-Norm and AdaGrad(-Coordinate) through step-size sharing. Subset-Norm (SN) reduces AdaGrad's memory footprint from $O(d)$ to $O(\sqrt{d})$, where $d$ is the model size. For non-convex smooth objectives under coordinate-wise sub-gaussian noise, we show a noise-adapted high-probability convergence guarantee with improved dimensional dependence of SN over existing methods. Our second technique, Subspace-Momentum, reduces the momentum state's memory footprint by restricting momentum to a low-dimensional subspace while performing SGD in the orthogonal complement. We prove high-probability convergence rates for Subspace-Momentum under standard assumptions. Empirical evaluation on pre-training and fine-tuning LLMs demonstrates the effectiveness of our methods. For instance, combining Subset-Norm with Subspace-Momentum achieves Adam's validation perplexity for LLaMA 1B in approximately half the trainingtokens (6.8B vs 13.1B) while reducing Adam's optimizer-states memory footprint by more than 80\% with minimal additional hyperparameter tuning.

Deep Learning · Graph Neural Networks

Yu He, Ellen Vitercik

Neural Algorithmic Reasoning (NAR) trains neural networks to simulate classical algorithms, enabling structured and interpretable reasoning over complex data. While prior research has predominantly focused on learning exact algorithms for polynomial-time-solvable problems, extending NAR to harder problems remains an open challenge. In this work, we introduce a general NAR framework grounded in the primal-dual paradigm, a classical method for designing efficient approximation algorithms. By leveraging a bipartite representation between primal and dual variables, we establish an alignment between primal-dual algorithms and Graph Neural Networks. Furthermore, we incorporate optimal solutions from small instances to greatly enhance the model’s reasoning capabilities. Our empirical results demonstrate that our model not only simulates but also outperforms approximation algorithms for multiple tasks, exhibiting robust generalization to larger and out-of-distribution graphs. Moreover, we highlight the framework’s practical utility by integrating it with commercial solvers and applying it to real-world datasets.