KPFlow: An Operator Perspective on Dynamic Collapse Under Gradient Descent Training of Recurrent Networks

• URL: http://arxiv.org/abs/2507.06381v1
• Date: Tue, 08 Jul 2025 20:33:15 GMT
• Title: KPFlow: An Operator Perspective on Dynamic Collapse Under Gradient Descent Training of Recurrent Networks
• Authors: James Hazelden, Laura Driscoll, Eli Shlizerman, Eric Shea-Brown,
• Abstract summary: We show how a gradient flow can be decomposed into a product that involves two operators.<n>We show how their interplay gives rise to low-dimensional latent dynamics under GD.<n>For multi-task training, we show that the operators can be used to measure how objectives relevant to individual sub-tasks align.
• Score: 9.512147747894026
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Gradient Descent (GD) and its variants are the primary tool for enabling efficient training of recurrent dynamical systems such as Recurrent Neural Networks (RNNs), Neural ODEs and Gated Recurrent units (GRUs). The dynamics that are formed in these models exhibit features such as neural collapse and emergence of latent representations that may support the remarkable generalization properties of networks. In neuroscience, qualitative features of these representations are used to compare learning in biological and artificial systems. Despite recent progress, there remains a need for theoretical tools to rigorously understand the mechanisms shaping learned representations, especially in finite, non-linear models. Here, we show that the gradient flow, which describes how the model's dynamics evolve over GD, can be decomposed into a product that involves two operators: a Parameter Operator, K, and a Linearized Flow Propagator, P. K mirrors the Neural Tangent Kernel in feed-forward neural networks, while P appears in Lyapunov stability and optimal control theory. We demonstrate two applications of our decomposition. First, we show how their interplay gives rise to low-dimensional latent dynamics under GD, and, specifically, how the collapse is a result of the network structure, over and above the nature of the underlying task. Second, for multi-task training, we show that the operators can be used to measure how objectives relevant to individual sub-tasks align. We experimentally and theoretically validate these findings, providing an efficient Pytorch package, \emph{KPFlow}, implementing robust analysis tools for general recurrent architectures. Taken together, our work moves towards building a next stage of understanding of GD learning in non-linear recurrent models.

Related papers

• Deep Learning Through A Telescoping Lens: A Simple Model Provides Empirical Insights On Grokking, Gradient Boosting & Beyond [61.18736646013446]
  In pursuit of a deeper understanding of its surprising behaviors, we investigate the utility of a simple yet accurate model of a trained neural network. Across three case studies, we illustrate how it can be applied to derive new empirical insights on a diverse range of prominent phenomena.
  arXiv  Detail & Related papers  (2024-10-31T22:54:34Z)
• Harnessing Neural Unit Dynamics for Effective and Scalable Class-Incremental Learning [38.09011520275557]
  Class-incremental learning (CIL) aims to train a model to learn new classes from non-stationary data streams without forgetting old ones. We propose a new kind of connectionist model by tailoring neural unit dynamics that adapt the behavior of neural networks for CIL.
  arXiv  Detail & Related papers  (2024-06-04T15:47:03Z)
• A Survey on Statistical Theory of Deep Learning: Approximation, Training Dynamics, and Generative Models [13.283281356356161]
  We review the literature on statistical theories of neural networks from three perspectives. Results on excess risks for neural networks are reviewed. Papers that attempt to answer how the neural network finds the solution that can generalize well on unseen data'' are reviewed.
  arXiv  Detail & Related papers  (2024-01-14T02:30:19Z)
• How neural networks learn to classify chaotic time series [77.34726150561087]
  We study the inner workings of neural networks trained to classify regular-versus-chaotic time series. We find that the relation between input periodicity and activation periodicity is key for the performance of LKCNN models.
  arXiv  Detail & Related papers  (2023-06-04T08:53:27Z)
• Identifying Equivalent Training Dynamics [3.793387630509845]
  We develop a framework for identifying conjugate and non-conjugate training dynamics. By leveraging advances in Koopman operator theory, we demonstrate that comparing Koopman eigenvalues can correctly identify a known equivalence between online mirror descent and online gradient descent. We then utilize our approach to: (a) identify non-conjugate training dynamics between shallow and wide fully connected neural networks; (b) characterize the early phase of training dynamics in convolutional neural networks; (c) uncover non-conjugate training dynamics in Transformers that do and do not undergo grokking.
  arXiv  Detail & Related papers  (2023-02-17T22:15:20Z)
• ConCerNet: A Contrastive Learning Based Framework for Automated Conservation Law Discovery and Trustworthy Dynamical System Prediction [82.81767856234956]
  This paper proposes a new learning framework named ConCerNet to improve the trustworthiness of the DNN based dynamics modeling. We show that our method consistently outperforms the baseline neural networks in both coordinate error and conservation metrics.
  arXiv  Detail & Related papers  (2023-02-11T21:07:30Z)
• The Underlying Correlated Dynamics in Neural Training [6.385006149689549]
  Training of neural networks is a computationally intensive task. We propose a model based on the correlation of the parameters' dynamics, which dramatically reduces the dimensionality. This representation enhances the understanding of the underlying training dynamics and can pave the way for designing better acceleration techniques.
  arXiv  Detail & Related papers  (2022-12-18T08:34:11Z)
• Decomposed Linear Dynamical Systems (dLDS) for learning the latent components of neural dynamics [6.829711787905569]
  We propose a new decomposed dynamical system model that represents complex non-stationary and nonlinear dynamics of time series data. Our model is trained through a dictionary learning procedure, where we leverage recent results in tracking sparse vectors over time. In both continuous-time and discrete-time instructional examples we demonstrate that our model can well approximate the original system.
  arXiv  Detail & Related papers  (2022-06-07T02:25:38Z)
• Sparse Flows: Pruning Continuous-depth Models [107.98191032466544]
  We show that pruning improves generalization for neural ODEs in generative modeling. We also show that pruning finds minimal and efficient neural ODE representations with up to 98% less parameters compared to the original network, without loss of accuracy.
  arXiv  Detail & Related papers  (2021-06-24T01:40:17Z)
• Statistical Mechanics of Deep Linear Neural Networks: The Back-Propagating Renormalization Group [4.56877715768796]
  We study the statistical mechanics of learning in Deep Linear Neural Networks (DLNNs) in which the input-output function of an individual unit is linear. We solve exactly the network properties following supervised learning using an equilibrium Gibbs distribution in the weight space. Our numerical simulations reveal that despite the nonlinearity, the predictions of our theory are largely shared by ReLU networks with modest depth.
  arXiv  Detail & Related papers  (2020-12-07T20:08:31Z)
• An Ode to an ODE [78.97367880223254]
  We present a new paradigm for Neural ODE algorithms, called ODEtoODE, where time-dependent parameters of the main flow evolve according to a matrix flow on the group O(d) This nested system of two flows provides stability and effectiveness of training and provably solves the gradient vanishing-explosion problem.
  arXiv  Detail & Related papers  (2020-06-19T22:05:19Z)
• Kernel and Rich Regimes in Overparametrized Models [69.40899443842443]
  We show that gradient descent on overparametrized multilayer networks can induce rich implicit biases that are not RKHS norms. We also demonstrate this transition empirically for more complex matrix factorization models and multilayer non-linear networks.
  arXiv  Detail & Related papers  (2020-02-20T15:43:02Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.