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Felix Dangel

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14 papers

2 author rows

NeurIPS Conference 2025 Conference Paper

Collapsing Taylor Mode Automatic Differentiation

Felix Dangel
Tim Siebert
Marius Zeinhofer
Andrea Walther

Computing partial differential equation (PDE) operators via nested backpropagation is expensive, yet popular, and severely restricts their utility for scientific machine learning. Recent advances, like the forward Laplacian and randomizing Taylor mode automatic differentiation (AD), propose forward schemes to address this. We introduce an optimization technique for Taylor mode that 'collapses' derivatives by rewriting the computational graph, and demonstrate how to apply it to general linear PDE operators, and randomized Taylor mode. The modifications simply require propagating a sum up the computational graph, which could---or should---be done by a machine learning compiler, without exposing complexity to users. We implement our collapsing procedure and evaluate it on popular PDE operators, confirming it accelerates Taylor mode and outperforms nested backpropagation.

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ICML Conference 2025 Conference Paper

Fishers for Free? Approximating the Fisher Information Matrix by Recycling the Squared Gradient Accumulator

Yu Xin Li
Felix Dangel
Derek Tam
Colin Raffel

The diagonal of a model’s Fisher Information Matrix (the "Fisher") has frequently been used as a way to measure parameter sensitivity. Typically, the Fisher is estimated by computing the squared gradient of the model’s outputs with respect to its parameters, averaged over a few hundred or thousand examples — a process which incurs nontrivial computational costs. At the same time, adaptive gradient methods like the ubiquitous Adam optimizer compute a moving average of the squared gradient over the course of training. This paper therefore explores whether an approximation of the Fisher can be obtained "for free" by recycling the squared gradient accumulator that has already been computed over the course of training. Through a comprehensive set of experiments covering five applications of the Fisher, we demonstrate that the "Squisher" ( Squ ared gradient accumulator as an approximation of the F isher ) consistently performs similarly to the Fisher while outperforming baseline methods. Additionally, we clarify the exact differences between the Squisher and the Fisher and provide empirical quantification of their respective impact.

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ICML Conference 2025 Conference Paper

Hide & Seek: Transformer Symmetries Obscure Sharpness & Riemannian Geometry Finds It

Marvin F. da Silva
Felix Dangel
Sageev Oore

The concept of sharpness has been successfully applied to traditional architectures like MLPs and CNNs to predict their generalization. For transformers, however, recent work reported weak correlation between flatness and generalization. We argue that existing sharpness measures fail for transformers, because they have much richer symmetries in their attention mechanism that induce directions in parameter space along which the network or its loss remain identical. We posit that sharpness must account fully for these symmetries, and thus we redefine it on a quotient manifold that results from quotienting out the transformer symmetries, thereby removing their ambiguities. Leveraging tools from Riemannian geometry, we propose a fully general notion of sharpness, in terms of a geodesic ball on the symmetry-corrected quotient manifold. In practice, we need to resort to approximating the geodesics. Doing so up to first order yields existing adaptive sharpness measures, and we demonstrate that including higher-order terms is crucial to recover correlation with generalization. We present results on diagonal networks with synthetic data, and show that our geodesic sharpness reveals strong correlation for real-world transformers on both text and image classification tasks.

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NeurIPS Conference 2025 Conference Paper

Improving Energy Natural Gradient Descent through Woodbury, Momentum, and Randomization

Andrés Guzmán-Cordero
Felix Dangel
Gil Goldshlager
Marius Zeinhofer

Natural gradient methods significantly accelerate the training of Physics-Informed Neural Networks (PINNs), but are often prohibitively costly. We introduce a suite of techniques to improve the accuracy and efficiency of energy natural gradient descent (ENGD) for PINNs. First, we leverage the Woodbury formula to dramatically reduce the computational complexity of ENGD. Second, we adapt the Subsampled Projected-Increment Natural Gradient Descent algorithm from the variational Monte Carlo literature to accelerate the convergence. Third, we explore the use of randomized algorithms to further reduce the computational cost in the case of large batch sizes. We find that randomization accelerates progress in the early stages of training for low-dimensional problems, and we identify key barriers to attaining acceleration in other scenarios. Our numerical experiments demonstrate that our methods outperform previous approaches, achieving the same $L^2$ error as the original ENGD up to $75\times$ faster.

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ICLR Conference 2025 Conference Paper

What Does It Mean to Be a Transformer? Insights from a Theoretical Hessian Analysis

Weronika Ormaniec
Felix Dangel
Sidak Pal Singh

The Transformer architecture has inarguably revolutionized deep learning, overtaking classical architectures like multi-layer perceptions (MLPs) and convolutional neural networks (CNNs). At its core, the attention block differs in form and functionality from most other architectural components in deep learning—to the extent that, in comparison to MLPs/CNNs, Transformers are more often accompanied by adaptive optimizers, layer normalization, learning rate warmup, etc. The root causes behind these outward manifestations and the precise mechanisms that govern them remain poorly understood. In this work, we bridge this gap by providing a fundamental understanding of what distinguishes the Transformer from the other architectures—grounded in a theoretical comparison of the (loss) Hessian. Concretely, for a single self-attention layer, (a) we first entirely derive the Transformer’s Hessian and express it in matrix derivatives; (b) we then characterize it in terms of data, weight, and attention moment dependencies; and (c) while doing so further highlight the important structural differences to the Hessian of classical networks. Our results suggest that various common architectural and optimization choices in Transformers can be traced back to their highly non-linear dependencies on the data and weight matrices, which vary heterogeneously across parameters. Ultimately, our findings provide a deeper understanding of the Transformer’s unique optimization landscape and the challenges it poses.

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ICML Conference 2024 Conference Paper

Can We Remove the Square-Root in Adaptive Gradient Methods? A Second-Order Perspective

Wu Lin
Felix Dangel
Runa Eschenhagen
Juhan Bae
Richard E. Turner
Alireza Makhzani

Adaptive gradient optimizers like Adam(W) are the default training algorithms for many deep learning architectures, such as transformers. Their diagonal preconditioner is based on the gradient outer product which is incorporated into the parameter update via a square root. While these methods are often motivated as approximate second-order methods, the square root represents a fundamental difference. In this work, we investigate how the behavior of adaptive methods changes when we remove the root, i. e. strengthen their second-order motivation. Surprisingly, we find that such square-root-free adaptive methods close the generalization gap to SGD on convolutional architectures, while maintaining their root-based counterpart’s performance on transformers. The second-order perspective also has practical benefits for the development of non-diagonal adaptive methods through the concept of preconditioner invariance. In contrast to root-based methods like Shampoo, the root-free counterparts do not require numerically unstable matrix root decompositions and inversions, thus work well in half precision. Our findings provide new insights into the development of adaptive methods and raise important questions regarding the currently overlooked role of adaptivity for their success.

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NeurIPS Conference 2024 Conference Paper

Convolutions and More as Einsum: A Tensor Network Perspective with Advances for Second-Order Methods

Felix Dangel

Despite their simple intuition, convolutions are more tedious to analyze than dense layers, which complicates the transfer of theoretical and algorithmic ideas to convolutions. We simplify convolutions by viewing them as tensor networks (TNs) that allow reasoning about the underlying tensor multiplications by drawing diagrams, manipulating them to perform function transformations like differentiation, and efficiently evaluating them with einsum. To demonstrate their simplicity and expressiveness, we derive diagrams of various autodiff operations and popular curvature approximations with full hyper-parameter support, batching, channel groups, and generalization to any convolution dimension. Further, we provide convolution-specific transformations based on the connectivity pattern which allow to simplify diagrams before evaluation. Finally, we probe performance. Our TN implementation accelerates a recently-proposed KFAC variant up to 4. 5 x while removing the standard implementation's memory overhead, and enables new hardware-efficient tensor dropout for approximate backpropagation.

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NeurIPS Conference 2024 Conference Paper

Kronecker-Factored Approximate Curvature for Physics-Informed Neural Networks

Felix Dangel
Johannes Müller
Marius Zeinhofer

Physics-Informed Neural Networks (PINNs) are infamous for being hard to train. Recently, second-order methods based on natural gradient and Gauss-Newton methods have shown promising performance, improving the accuracy achieved by first-order methods by several orders of magnitude. While promising, the proposed methods only scale to networks with a few thousand parameters due to the high computational cost to evaluate, store, and invert the curvature matrix. We propose Kronecker-factored approximate curvature (KFAC) for PINN losses that greatly reduces the computational cost and allows scaling to much larger networks. Our approach goes beyond the popular KFAC for traditional deep learning problems as it captures contributions from a PDE's differential operator that are crucial for optimization. To establish KFAC for such losses, we use Taylor-mode automatic differentiation to describe the differential operator's computation graph as a forward network with shared weights which allows us to apply a variant of KFAC for networks with weight-sharing. Empirically, we find that our KFAC-based optimizers are competitive with expensive second-order methods on small problems, scale more favorably to higher-dimensional neural networks and PDEs, and consistently outperform first-order methods.

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ICML Conference 2024 Conference Paper

Revisiting Scalable Hessian Diagonal Approximations for Applications in Reinforcement Learning

Mohamed Elsayed 0003
Homayoon Farrahi
Felix Dangel
A. Rupam Mahmood

Second-order information is valuable for many applications but challenging to compute. Several works focus on computing or approximating Hessian diagonals, but even this simplification introduces significant additional costs compared to computing a gradient. In the absence of efficient exact computation schemes for Hessian diagonals, we revisit an early approximation scheme proposed by Becker and LeCun (1989, BL89), which has a cost similar to gradients and appears to have been overlooked by the community. We introduce HesScale, an improvement over BL89, which adds negligible extra computation. On small networks, we find that this improvement is of higher quality than all alternatives, even those with theoretical guarantees, such as unbiasedness, while being much cheaper to compute. We use this insight in reinforcement learning problems where small networks are used and demonstrate HesScale in second-order optimization and scaling the step-size parameter. In our experiments, HesScale optimizes faster than existing methods and improves stability through step-size scaling. These findings are promising for scaling second-order methods in larger models in the future.

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ICML Conference 2024 Conference Paper

Structured Inverse-Free Natural Gradient Descent: Memory-Efficient & Numerically-Stable KFAC

Wu Lin
Felix Dangel
Runa Eschenhagen
Kirill Neklyudov
Agustinus Kristiadi
Richard E. Turner
Alireza Makhzani

Second-order methods such as KFAC can be useful for neural net training. However, they are often memory-inefficient since their preconditioning Kronecker factors are dense, and numerically unstable in low precision as they require matrix inversion or decomposition. These limitations render such methods unpopular for modern mixed-precision training. We address them by (i) formulating an inverse-free KFAC update and (ii) imposing structures in the Kronecker factors, resulting in structured inverse-free natural gradient descent (SINGD). On modern neural networks, we show that SINGD is memory-efficient and numerically robust, in contrast to KFAC, and often outperforms AdamW even in half precision. Our work closes a gap between first- and second-order methods in modern low-precision training.

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NeurIPS Conference 2023 Conference Paper

The Geometry of Neural Nets' Parameter Spaces Under Reparametrization

Agustinus Kristiadi
Felix Dangel
Philipp Hennig

Model reparametrization, which follows the change-of-variable rule of calculus, is a popular way to improve the training of neural nets. But it can also be problematic since it can induce inconsistencies in, e. g. , Hessian-based flatness measures, optimization trajectories, and modes of probability densities. This complicates downstream analyses: e. g. one cannot definitively relate flatness with generalization since arbitrary reparametrization changes their relationship. In this work, we study the invariance of neural nets under reparametrization from the perspective of Riemannian geometry. From this point of view, invariance is an inherent property of any neural net if one explicitly represents the metric and uses the correct associated transformation rules. This is important since although the metric is always present, it is often implicitly assumed as identity, and thus dropped from the notation, then lost under reparametrization. We discuss implications for measuring the flatness of minima, optimization, and for probability-density maximization. Finally, we explore some interesting directions where invariance is useful.

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TMLR Journal 2023 Journal Article

ViViT: Curvature Access Through The Generalized Gauss-Newton’s Low-Rank Structure

Felix Dangel
Lukas Tatzel
Philipp Hennig

Curvature in form of the Hessian or its generalized Gauss-Newton (GGN) approximation is valuable for algorithms that rely on a local model for the loss to train, compress, or explain deep networks. Existing methods based on implicit multiplication via automatic differentiation or Kronecker-factored block diagonal approximations do not consider noise in the mini-batch. We present ViViT, a curvature model that leverages the GGN’s low-rank structure without further approximations. It allows for efficient computation of eigenvalues, eigenvectors, as well as per-sample first- and second-order directional derivatives. The representation is computed in parallel with gradients in one backward pass and offers a fine-grained cost-accuracy trade-off, which allows it to scale. We demonstrate this by conducting performance benchmarks and substantiate ViViT’s usefulness by studying the impact of noise on the GGN’s structural properties during neural network training.

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NeurIPS Conference 2021 Conference Paper

Cockpit: A Practical Debugging Tool for the Training of Deep Neural Networks

Frank Schneider
Felix Dangel
Philipp Hennig

When engineers train deep learning models, they are very much "flying blind". Commonly used methods for real-time training diagnostics, such as monitoring the train/test loss, are limited. Assessing a network's training process solely through these performance indicators is akin to debugging software without access to internal states through a debugger. To address this, we present Cockpit, a collection of instruments that enable a closer look into the inner workings of a learning machine, and a more informative and meaningful status report for practitioners. It facilitates the identification of learning phases and failure modes, like ill-chosen hyperparameters. These instruments leverage novel higher-order information about the gradient distribution and curvature, which has only recently become efficiently accessible. We believe that such a debugging tool, which we open-source for PyTorch, is a valuable help in troubleshooting the training process. By revealing new insights, it also more generally contributes to explainability and interpretability of deep nets.

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ICLR Conference 2020 Conference Paper

BackPACK: Packing more into Backprop

Felix Dangel
Frederik Kunstner
Philipp Hennig

Automatic differentiation frameworks are optimized for exactly one thing: computing the average mini-batch gradient. Yet, other quantities such as the variance of the mini-batch gradients or many approximations to the Hessian can, in theory, be computed efficiently, and at the same time as the gradient. While these quantities are of great interest to researchers and practitioners, current deep learning software does not support their automatic calculation. Manually implementing them is burdensome, inefficient if done naively, and the resulting code is rarely shared. This hampers progress in deep learning, and unnecessarily narrows research to focus on gradient descent and its variants; it also complicates replication studies and comparisons between newly developed methods that require those quantities, to the point of impossibility. To address this problem, we introduce BackPACK, an efficient framework built on top of PyTorch, that extends the backpropagation algorithm to extract additional information from first-and second-order derivatives. Its capabilities are illustrated by benchmark reports for computing additional quantities on deep neural networks, and an example application by testing several recent curvature approximations for optimization.

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