Alexandra Proca

openreview.netLearning dynamics in linear recurrent neural networksRecurrent neural networks (RNNs) are powerful models used widely in both machine learning and neuroscience to learn tasks with temporal dependencies and to model neural dynamics. However, despite...

How do task dynamics impact learning in networks with internal dynamics? Excited to share our ICML Oral paper on learning dynamics in linear RNNs! with @clementinedomine.bsky.social @mpshanahan.bsky.social and Pedro Mediano openreview.net/forum?id=KGO...

@ezekielwilliams.bsky.social

Submit your paper or extended abstract to UniReps 2026 in Paris! Deadline Oct. 4 AOE 🔵🔴

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Alexandra Proca reposted

Ezekiel Williams

10 June

1/7 Excited to share my last PhD article, just accepted to ICML 2026! In it, we (me, Alexandre Payeur, Guillaume Lajoie) used dynamical systems theory to study "local" learning in linear recurrent neural networks. See link for the paper, and thread for a brief summary. arxiv.org/abs/2606.00243

arxiv.orgDynamics and Representation Structure of Local Approximations to Gradient-Based Learning in Linear Recurrent Neural NetworksBiological and neuromorphic recurrent neural networks (RNNs) are subject to spatial and temporal locality constraints on the information that can plausibly be used during learning. A common strategy t...

Excited to share our ICLR Oral paper, co-lead with Pratyaksh Sharma, and with @lucas-prieto.bsky.social Pedro Mediano! We study how feature geometry is shaped by memory demands in RNNs, introducing the concept of temporal superposition. openreview.net/forum?id=7cM...

We induce spatial superposition by using 5D input (A-E) and temporal superposition by increasing k. As memory demand (k) increases, the RNN drops features in favor of representing others for a longer duration. This representational tradeoff leads to a strategy that is all-or-none.

In summary, we expand the study of superposition to recurrent architectures, enabling us to study how temporal information acts as a capacity constraint and affects feature geometry. Our work highlights how superposition affects dynamics and is shaped by time.

By varying sparsity, we observe a phase transition between dense and sparse-regime geometry, characterized by the angle that task-relevant features span k theta.

Adding a ReLU to the recurrence, nonlinear RNNs fully exploit the interference-free space by packing all intermediate feature directions in this space and implement “sharp” forgetting to remove old task-irrelevant feature directions.

Next, we add a ReLU to the readout. Feature directions that have negative projections onto the readout are cancelled out by the ReLU. In the sparse regime, this half-space becomes interference-free, incentivizing the RNN to pack all intermediate features into this half-space.

With a linear recurrence, the RNN is limited in expressivity, implementing a spiral sink solution. However, in the sparse regime, the largest feature directions are grouped into the interference-free space.

To better understand how learning incentivizes certain geometry, we derive a closed-form expression of the loss for a linear RNN, comprised of four interpretable terms.

We show that linear RNNs only implement a spiral sink solution, which is similar regardless of sparsity level, resulting in “smooth” forgetting by decaying old features into the origin. In the dense regime, RNNs with nonlinearities also find approximately this solution.

We first consider scalar inputs-outputs. Here, each feature is linearly represented by a feature direction vector in the hidden state. The model becomes bottlenecked once t is larger than the hidden state. Only feature directions that project onto w_y will affect the output.

Feature directions can interfere in two ways. Composition interference occurs when the activation of multiple features is linearly combined. Projection interference occurs when the activation of a feature direction is readout at the wrong time because it aligns with the readout.

ANNs trained on more sparse features than neurons compress data by representing features non-orthogonally in so-called superposition. Temporal information can also act as a capacity constraint. How does superposition behave and shape geometry in RNNs under memory demands?

We study RNNs trained on a delayed serial recall task, where k controls memory demand (the duration each input must be held in memory), varying sparsity, dimensionality, and nonlinearity.

Finally, although many results we present are based on SVD, we also derive a form based on an eigendecomposition, allowing for rotational dynamics and to which our framework naturally extends to. We use this to study learning in terms of polar coordinates in the complex plane.

In summary, this work provides a novel flexible framework for studying learning in linear RNNs, allowing us to generate new insights into their learning process and the solutions they find, and progress our understanding of cognition in dynamic task settings.

Motivated by this, we study task dynamics without zero-loss solutions and find that there exists a tradeoff between recurrent and feedforward computations that is characterized by a phase transition and leads to low-rank connectivity.

To study how recurrence might impact feature learning, we derive the NTK for finite-width LRNNs and evaluate its movement during training. We find that recurrence appears to facilitate kernel movement across many settings, suggesting a bias towards rich learning.

Additionally, these results predict behavior in networks performing integration tasks, where we relax our theoretical assumptions.

By analyzing the energy function, we identify an effective regularization term that incentivizes small weights, especially when task dynamics are not perfectly learnable.

We find that learning speed is dependent on both the scale of SVs and their temporal ordering, such that SVs occurring later in the trajectory have a greater impact on learning speed.

Next, we show that task dynamics determine a RNN’s ability to extrapolate to other sequence lengths and its hidden layer stability, even if there exists a perfect zero-loss solution.

We derive a form where the task dynamics are fully specified by the data correlation singular values (or eigenvalues) across time (t=1:T), and learning is characterized by a set of gradient flow equations and energy function that are decoupled across different dimensions.

Using this form, we derive solutions to the learning dynamics of the input-output modes and local approximations of the recurrent modes separately, and identify differences in the learning dynamics of recurrent networks compared to feedforward ones.

RNNs are popular in both ML and neuroscience to learn tasks with temporal dependencies and model neural dynamics. However, despite substantial work on RNNs, it's unknown how their underlying functional structures emerge from training on temporally-structured tasks.

@clementinedomine.bsky.social

We study a RNN that receives an input at each timestep and produces a final output at the last timestep (and generalize to the autoregressive case later). For each input at time t and the output, we can construct correlation matrices and compute their SVD (or eigendecomposition).

Alexandra Proca reposted

Clementine Domine 🍊 @CCN

4 Apr 2025

🚀 An other Exciting news! Our paper "From Lazy to Rich: Exact Learning Dynamics in Deep Linear Networks" has been accepted at ICLR 2025! arxiv.org/abs/2409.14623 A thread on how relative weight initialization shapes learning dynamics in deep networks. 🧵 (1/9)