Benoit Dherin

Benoit Dherin, Huiyi Hu, Jie Ren et al.

We introduce a new deep generative model useful for uncertainty quantification: the Morse neural network, which generalizes the unnormalized Gaussian densities to have modes of high-dimensional submanifolds instead of just discrete points. Fitting the Morse neural network via a KL-divergence loss yields 1) a (unnormalized) generative density, 2) an OOD detector, 3) a calibration temperature, 4) a generative sampler, along with in the supervised case 5) a distance aware-classifier. The Morse network can be used on top of a pre-trained network to bring distance-aware calibration w.r.t the training data. Because of its versatility, the Morse neural networks unifies many techniques: e.g., the Entropic Out-of-Distribution Detector of (Macêdo et al., 2021) in OOD detection, the one class Deep Support Vector Description method of (Ruff et al., 2018) in anomaly detection, or the Contrastive One Class classifier in continuous learning (Sun et al., 2021). The Morse neural network has connections to support vector machines, kernel methods, and Morse theory in topology.

Benoit Dherin, Michael Munn, Mihaela Rosca et al.

In many contexts, simpler models are preferable to more complex models and the control of this model complexity is the goal for many methods in machine learning such as regularization, hyperparameter tuning and architecture design. In deep learning, it has been difficult to understand the underlying mechanisms of complexity control, since many traditional measures are not naturally suitable for deep neural networks. Here we develop the notion of geometric complexity, which is a measure of the variability of the model function, computed using a discrete Dirichlet energy. Using a combination of theoretical arguments and empirical results, we show that many common training heuristics such as parameter norm regularization, spectral norm regularization, flatness regularization, implicit gradient regularization, noise regularization and the choice of parameter initialization all act to control geometric complexity, providing a unifying framework in which to characterize the behavior of deep learning models.

Mihaela Rosca, Yan Wu, Chongli Qin et al.

The recipe behind the success of deep learning has been the combination of neural networks and gradient-based optimization. Understanding the behavior of gradient descent however, and particularly its instability, has lagged behind its empirical success. To add to the theoretical tools available to study gradient descent we propose the principal flow (PF), a continuous time flow that approximates gradient descent dynamics. To our knowledge, the PF is the only continuous flow that captures the divergent and oscillatory behaviors of gradient descent, including escaping local minima and saddle points. Through its dependence on the eigendecomposition of the Hessian the PF sheds light on the recently observed edge of stability phenomena in deep learning. Using our new understanding of instability we propose a learning rate adaptation method which enables us to control the trade-off between training stability and test set evaluation performance.

Hanna Mazzawi, Xavi Gonzalvo, Michael Wunder et al.

In recent years, deep learning has made remarkable progress in a wide range of domains, with a particularly notable impact on natural language processing tasks. One of the challenges associated with training deep neural networks in the context of LLMs is the need for large amounts of computational resources and time. To mitigate this, network growing algorithms offer potential cost savings, but their underlying mechanisms are poorly understood. We present two notable contributions in this paper. First, we present Deep Fusion, an efficient approach to network training that leverages pre-trained initializations of smaller networks. Second, we propose a theoretical framework using backward error analysis to illustrate the dynamics of mid-training network growth. Our experiments show how Deep Fusion is a practical and effective approach that not only accelerates the training process but also reduces computational requirements, maintaining or surpassing traditional training methods' performance in various NLP tasks and T5 model sizes. Finally, we validate our theoretical framework, which guides the optimal use of Deep Fusion, showing that with carefully optimized training dynamics, it significantly reduces both training time and resource consumption.

Benoit Dherin

Using backward error analysis, we compute implicit training biases in multitask and continual learning settings for neural networks trained with stochastic gradient descent. In particular, we derive modified losses that are implicitly minimized during training. They have three terms: the original loss, accounting for convergence, an implicit flatness regularization term proportional to the learning rate, and a last term, the conflict term, which can theoretically be detrimental to both convergence and implicit regularization. In multitask, the conflict term is a well-known quantity, measuring the gradient alignment between the tasks, while in continual learning the conflict term is a new quantity in deep learning optimization, although a basic tool in differential geometry: The Lie bracket between the task gradients.

Benoit Dherin, Michael Munn, Hanna Mazzawi et al.

One of the most striking features of Large Language Models (LLM) is their ability to learn in context. Namely at inference time an LLM is able to learn new patterns without any additional weight update when these patterns are presented in the form of examples in the prompt, even if these patterns were not seen during training. The mechanisms through which this can happen are still largely unknown. In this work, we show that the stacking of a self-attention layer with an MLP, allows the transformer block to implicitly modify the weights of the MLP layer according to the context. We argue through theory and experimentation that this simple mechanism may be the reason why LLMs can learn in context and not only during training. Specifically, we show under mild simplifying assumptions how a transformer block implicitly transforms a context into a low-rank weight-update of the MLP layer.

Michael Munn, Benoit Dherin, Javier Gonzalvo

Many of the recent remarkable advances in computer vision and language models can be attributed to the success of transfer learning via the pre-training of large foundation models. However, a theoretical framework which explains this empirical success is incomplete and remains an active area of research. Flatness of the loss surface and neural collapse have recently emerged as useful pre-training metrics which shed light on the implicit biases underlying pre-training. In this paper, we explore the geometric complexity of a model's learned representations as a fundamental mechanism that relates these two concepts. We show through experiments and theory that mechanisms which affect the geometric complexity of the pre-trained network also influence the neural collapse. Furthermore, we show how this effect of the geometric complexity generalizes to the neural collapse of new classes as well, thus encouraging better performance on downstream tasks, particularly in the few-shot setting.

Benoit Dherin, Benny Avelin, Anders Karlsson et al.

Despite exceptional achievements, training neural networks remains computationally expensive and is often plagued by instabilities that can degrade convergence. While learning rate schedules can help mitigate these issues, finding optimal schedules is time-consuming and resource-intensive. This work explores theoretical issues concerning training stability in the constant-learning-rate (i.e., without schedule) and small-batch-size regime. Surprisingly, we show that the order of gradient updates affects stability and convergence in gradient-based optimizers. We illustrate this new line of thinking using backward-SGD, which processes batch gradient updates like SGD but in reverse order. Our theoretical analysis shows that in contractive regions (e.g., around minima) backward-SGD converges to a point while the standard forward-SGD generally only converges to a distribution. This leads to improved stability and convergence which we demonstrate experimentally. While full backward-SGD is computationally intensive in practice, it highlights opportunities to exploit reverse training dynamics (or more generally alternate iteration orders) to improve training. To our knowledge, this represents a new and unexplored avenue in deep learning optimization.

Benoit Dherin, Mihaela Rosca

We characterize regions of a loss surface as corridors when the continuous curves of steepest descent -- the solutions of the gradient flow -- become straight lines. We show that corridors provide insights into gradient-based optimization, since corridors are exactly the regions where gradient descent and the gradient flow follow the same trajectory, while the loss decreases linearly. As a result, inside corridors there are no implicit regularization effects or training instabilities that have been shown to occur due to the drift between gradient descent and the gradient flow. Using the loss linear decrease on corridors, we devise a learning rate adaptation scheme for gradient descent; we call this scheme Corridor Learning Rate (CLR). The CLR formulation coincides with a special case of Polyak step-size, discovered in the context of convex optimization. The Polyak step-size has been shown recently to have also good convergence properties for neural networks; we further confirm this here with results on CIFAR-10 and ImageNet.

Adrian Goldwaser, Michael Munn, Javier Gonzalvo et al.

Recent research has established that the impact of context in a vanilla transformer can be represented implicitly by forming a token-dependent, rank-1 patch to its MLP weights. This work extends that foundational theory to the diverse architectures of modern Large Language Models. We first demonstrate a precise, analytical solution for a Gemma-style transformer block, proving that the entire effect of a context can be perfectly mapped to rank-1 patches on its MLP weight matrices and a patch to the RMSNorm scale. We then generalize this result, providing a constructive proof and algorithm for multi-layer models. To unify these findings, we introduce a general framework centered on two core properties: input controllability and output controllability. We prove that a perfect implicit weight patch is possible for any MLP block where the inner function is input-controllable and the outer function is output-controllable. This provides a simpler and more powerful lens for understanding how transformer models transmute prompts into effective weights. This setup generalizes to a wide range of modern LLM architectures including gating, pre-/post-norm, mixture of experts and sequential/parallel transformer blocks.

Hanna Mazzawi, Benoit Dherin, Michael Munn et al.

A growing body of research has demonstrated that the behavior of large language models can be effectively controlled at inference time by directly modifying their internal states, either through vector additions to their activations or through updates to their weight matrices. These techniques, while powerful, are often guided by empirical heuristics, such as deriving steering vectors from the average activations of contrastive prompts. This work provides a theoretical foundation for these interventions, explaining how they emerge from the fundamental computations of the transformer architecture. Building on the recent finding that a prompt's influence can be mathematically mapped to implicit weight updates (Dherin et al., 2025), we generalize this theory to deep, multi-block transformers. We show how the information contained in any chunk of a user prompt is represented and composed internally through weight vectors and weight matrices. We then derive a principled method for condensing this information into token-independent thought vectors and thought matrices. These constructs provide a theoretical explanation for existing vector- and matrix-based model editing techniques and offer a direct, computationally-grounded method for transmuting textual input into reusable weight updates.

Benoit Dherin, Michael Munn

Deep residual architectures, such as ResNet and the Transformer, have enabled models of unprecedented depth, yet a formal understanding of why depth is so effective remains an open question. A popular intuition, following Veit et al. (2016), is that these residual networks behave like ensembles of many shallower models. Our key finding is an explicit analytical formula that verifies this ensemble perspective, proving that increasing network depth is mathematically equivalent to expanding the size of this implicit ensemble. Furthermore, our expansion reveals a hierarchical ensemble structure in which the combinatorial growth of computation paths leads to an explosion in the output signal, explaining the historical necessity of normalization layers in training deep models. This insight offers a first principles explanation for the historical dependence on normalization layers and sheds new light on a family of successful normalization-free techniques like SkipInit and Fixup. However, while these previous approaches infer scaling factors through optimizer analysis or a heuristic analogy to Batch Normalization, our work offers the first explanation derived directly from the network's inherent functional structure. Specifically, our Residual Expansion Theorem reveals that scaling each residual module provides a principled solution to taming the combinatorial explosion inherent to these architectures. We further show that this scaling acts as a capacity controls that also implicitly regularizes the model's complexity.

Benoit Dherin, Michael Munn, Hanna Mazzawi et al.

Modern deep learning algorithms use variations of gradient descent as their main learning methods. Gradient descent can be understood as the simplest Ordinary Differential Equation (ODE) solver; namely, the Euler method applied to the gradient flow differential equation. Since Euler, many ODE solvers have been devised that follow the gradient flow equation more precisely and more stably. Runge-Kutta (RK) methods provide a family of very powerful explicit and implicit high-order ODE solvers. However, these higher-order solvers have not found wide application in deep learning so far. In this work, we evaluate the performance of higher-order RK solvers when applied in deep learning, study their limitations, and propose ways to overcome these drawbacks. In particular, we explore how to improve their performance by naturally incorporating key ingredients of modern neural network optimizers such as preconditioning, adaptive learning rates, and momentum.

Benoit Dherin, Michael Munn, David G. T. Barrett

In over-parameterized deep neural networks there can be many possible parameter configurations that fit the training data exactly. However, the properties of these interpolating solutions are poorly understood. We argue that over-parameterized neural networks trained with stochastic gradient descent are subject to a Geometric Occam's Razor; that is, these networks are implicitly regularized by the geometric model complexity. For one-dimensional regression, the geometric model complexity is simply given by the arc length of the function. For higher-dimensional settings, the geometric model complexity depends on the Dirichlet energy of the function. We explore the relationship between this Geometric Occam's Razor, the Dirichlet energy and other known forms of implicit regularization. Finally, for ResNets trained on CIFAR-10, we observe that Dirichlet energy measurements are consistent with the action of this implicit Geometric Occam's Razor.

Mihaela Rosca, Yan Wu, Benoit Dherin et al.

Gradient-based methods for two-player games produce rich dynamics that can solve challenging problems, yet can be difficult to stabilize and understand. Part of this complexity originates from the discrete update steps given by simultaneous or alternating gradient descent, which causes each player to drift away from the continuous gradient flow -- a phenomenon we call discretization drift. Using backward error analysis, we derive modified continuous dynamical systems that closely follow the discrete dynamics. These modified dynamics provide an insight into the notorious challenges associated with zero-sum games, including Generative Adversarial Networks. In particular, we identify distinct components of the discretization drift that can alter performance and in some cases destabilize the game. Finally, quantifying discretization drift allows us to identify regularizers that explicitly cancel harmful forms of drift or strengthen beneficial forms of drift, and thus improve performance of GAN training.

Samuel L. Smith, Benoit Dherin, David G. T. Barrett et al.

For infinitesimal learning rates, stochastic gradient descent (SGD) follows the path of gradient flow on the full batch loss function. However moderately large learning rates can achieve higher test accuracies, and this generalization benefit is not explained by convergence bounds, since the learning rate which maximizes test accuracy is often larger than the learning rate which minimizes training loss. To interpret this phenomenon we prove that for SGD with random shuffling, the mean SGD iterate also stays close to the path of gradient flow if the learning rate is small and finite, but on a modified loss. This modified loss is composed of the original loss function and an implicit regularizer, which penalizes the norms of the minibatch gradients. Under mild assumptions, when the batch size is small the scale of the implicit regularization term is proportional to the ratio of the learning rate to the batch size. We verify empirically that explicitly including the implicit regularizer in the loss can enhance the test accuracy when the learning rate is small.

David G. T. Barrett, Benoit Dherin

Gradient descent can be surprisingly good at optimizing deep neural networks without overfitting and without explicit regularization. We find that the discrete steps of gradient descent implicitly regularize models by penalizing gradient descent trajectories that have large loss gradients. We call this Implicit Gradient Regularization (IGR) and we use backward error analysis to calculate the size of this regularization. We confirm empirically that implicit gradient regularization biases gradient descent toward flat minima, where test errors are small and solutions are robust to noisy parameter perturbations. Furthermore, we demonstrate that the implicit gradient regularization term can be used as an explicit regularizer, allowing us to control this gradient regularization directly. More broadly, our work indicates that backward error analysis is a useful theoretical approach to the perennial question of how learning rate, model size, and parameter regularization interact to determine the properties of overparameterized models optimized with gradient descent.