arXiv subjects

William T. Redman

Publications and source records attributed to William T. Redman.

10 recordsLinked to original sources

arXiv2026

Shortcut Solutions Learned by Transformers Impair Continual Compositional Reasoning

Identifying and exploiting common features across domains is at the heart of the human ability to make analogies, and is believed to be crucial for the ability to continually learn. To do this successfully, general and flexible computational strategies must be developed. While the extent to which Transformer neural network models can perform compositional reasoning has been the subject of intensive recent investigation, little work has been done to systematically understand how well these models can leverage their representations to learn new, related experiences. To address this gap, we expand the previously developed Learning Equality and Group Operations (LEGO) framework to a continual learning (CL) setting ("continual LEGO"). Using this continual LEGO experimental paradigm, we study the capability of feedforward and recurrent Transformer models to perform CL. We find that BERT, a canonical feedforward Transformer model, learns shortcut solutions that limits its ability to generalize and prevents strong forward transfer to new experiences. In contrast, we find evidence supporting the hypothesis that ALBERT, a recurrent version of BERT, learns a For loop-esque solution, which leads to better CL performance. When applying BERT and ALBERT models to a CL setting that requires composition across experiences, we find that both model families fail. Our investigation suggests that ALBERT models can have their performance drop rescued by use of training strategies that combine data across experiences, but this is not true for BERT models, where a detrimental shortcut solution becomes entrenched with initial training. Our results demonstrate that the recurrent ALBERT model may have an inductive bias better suited for CL and motivate future investigation of the interplay between Transformer architecture and computational solutions that emerge in modern models and tasks.

cs.LG↗

arXiv2026

Interpreting Reinforcement Learning Model Behavior via Koopman with Control

Reinforcement learning (RL) models have shown the capability of learning complex behaviors, but quantitatively assessing those behaviors - which is critical for safety assurance and the discovery of novel strategies - is challenging. By viewing RL models as control systems, we hypothesize that data-driven approximations of their associated Koopman operators may provide dynamical information about their behavior, thus enabling greater interpretability. To test this, we apply the Koopman with control framework to RL models trained on several standard benchmark environments and demonstrate that properties of the fit linear control models, such as stability and controllability, evolve during training in a task dependent manner. Comparing these metrics across different training epochs or across differently optimized RL models enables an understanding of how they differ. In addition, we find cases where - even when the reward achieved by the RL model is static - the stability and controllability is nonetheless evolving, predicting increased reward with further training. This suggests that these metrics may be able to serve as hidden progress measures, a core idea in mechanistic interpretability. Taken together, our results illustrate that the Koopman with control framework provides a comprehensive way in which to analyze and interpret the behavior of RL models, particularly across training.

math.OC↗

arXiv2025

Distance by de-correlation: Computing distance with heterogeneous grid cells

Encoding the distance between locations in space is essential for accurate navigation. Grid cells, a functional class of neurons in medial entorhinal cortex, are believed to support this computation. Inspired by recent work finding populations of grid cells to have small, but robust heterogeneity in their grid properties, we hypothesize that distance coding can be achieved by a simple de-correlation of population activity. We develop a mathematical theory for describing this de-correlation in one-dimension, showing that its predictions are consistent with simulations of noisy grid cells. Our simulations highlight a non-intuitive prediction of such a distance by de-correlation framework. Namely, some further distances are better encoded than some nearer distances. We find preliminary evidence of this ``sweet spot'' in previously published rodent behavioral experiments and demonstrate that a decoder which estimates distance from the de-correlation of populations of simulated noisy grid cells leads to a similar pattern of errors. Finally, we extend our theory to two-dimensions and, by simulating noisy grid cells in two-dimensions, find that there exists a trade-off between the range of distances that can be encoded by de-correlation of population activity and the distinguishability between different distances, which is controlled by the amount of variability in grid properties. We show that the previously measured average amount of grid property variability strikes a balance, enabling the encoding of distances up to several meters. Our work provides new insight on how grid cells can underlie the encoding of distance and why grid cells may have small amounts of heterogeneity in their grid properties.

q-bio.NC↗

arXiv2025

On How Iterative Magnitude Pruning Discovers Local Receptive Fields in Fully Connected Neural Networks

Since its use in the Lottery Ticket Hypothesis, iterative magnitude pruning (IMP) has become a popular method for extracting sparse subnetworks that can be trained to high performance. Despite its success, the mechanism that drives the success of IMP remains unclear. One possibility is that IMP is capable of extracting subnetworks with good inductive biases that facilitate performance. Supporting this idea, recent work showed that applying IMP to fully connected neural networks (FCNs) leads to the emergence of local receptive fields (RFs), a feature of mammalian visual cortex and convolutional neural networks that facilitates image processing. However, it remains unclear why IMP would uncover localized features in the first place. Inspired by results showing that training on synthetic images with highly non-Gaussian statistics (e.g., sharp edges) is sufficient to drive the emergence of local RFs in FCNs, we hypothesize that IMP iteratively increases the non-Gaussian statistics of FCN representations, creating a feedback loop that enhances localization. Here, we demonstrate first that non-Gaussian input statistics are indeed necessary for IMP to discover localized RFs. We then develop a new method for measuring the effect of individual weights on the statistics of the FCN representations ("cavity method"), which allows us to show that IMP systematically increases the non-Gaussianity of pre-activations, leading to the formation of localized RFs. Our work, which is the first to study the effect of IMP on the statistics of the representations of neural networks, sheds parsimonious light on one way in which IMP can drive the formation of strong inductive biases.

cs.LG↗

arXiv2025

Koopman Learning with Episodic Memory

Koopman operator theory has found significant success in learning models of complex, real-world dynamical systems, enabling prediction and control. The greater interpretability and lower computational costs of these models, compared to traditional machine learning methodologies, make Koopman learning an especially appealing approach. Despite this, little work has been performed on endowing Koopman learning with the ability to leverage its own failures. To address this, we equip Koopman methods -- developed for predicting non-autonomous time-series -- with an episodic memory mechanism, enabling global recall of (or attention to) periods in time where similar dynamics previously occurred. We find that a basic implementation of Koopman learning with episodic memory leads to significant improvements in prediction on synthetic and real-world data. Our framework has considerable potential for expansion, allowing for future advances, and opens exciting new directions for Koopman learning.

math.DS↗

arXiv2024

Identifying Equivalent Training Dynamics

Study of the nonlinear evolution deep neural network (DNN) parameters undergo during training has uncovered regimes of distinct dynamical behavior. While a detailed understanding of these phenomena has the potential to advance improvements in training efficiency and robustness, the lack of methods for identifying when DNN models have equivalent dynamics limits the insight that can be gained from prior work. Topological conjugacy, a notion from dynamical systems theory, provides a precise definition of dynamical equivalence, offering a possible route to address this need. However, topological conjugacies have historically been challenging to compute. By leveraging advances in Koopman operator theory, we develop a framework for identifying conjugate and non-conjugate training dynamics. To validate our approach, 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. Our results, across a range of DNN architectures, illustrate the flexibility of our framework and highlight its potential for shedding new light on training dynamics.

cs.LG↗

arXiv2022

Algorithmic (Semi-)Conjugacy via Koopman Operator Theory

Iterative algorithms are of utmost importance in decision and control. With an ever growing number of algorithms being developed, distributed, and proprietarized, there is a similarly growing need for methods that can provide classification and comparison. By viewing iterative algorithms as discrete-time dynamical systems, we leverage Koopman operator theory to identify (semi-)conjugacies between algorithms using their spectral properties. This provides a general framework with which to classify and compare algorithms.

cs.DS↗

arXiv2022

Universality of Winning Tickets: A Renormalization Group Perspective

Foundational work on the Lottery Ticket Hypothesis has suggested an exciting corollary: winning tickets found in the context of one task can be transferred to similar tasks, possibly even across different architectures. This has generated broad interest, but methods to study this universality are lacking. We make use of renormalization group theory, a powerful tool from theoretical physics, to address this need. We find that iterative magnitude pruning, the principal algorithm used for discovering winning tickets, is a renormalization group scheme, and can be viewed as inducing a flow in parameter space. We demonstrate that ResNet-50 models with transferable winning tickets have flows with common properties, as would be expected from the theory. Similar observations are made for BERT models, with evidence that their flows are near fixed points. Additionally, we leverage our framework to study winning tickets transferred across ResNet architectures, observing that smaller models have flows with more uniform properties than larger models, complicating transfer between them.

cs.LG↗

arXiv2022

An Operator Theoretic View on Pruning Deep Neural Networks

The discovery of sparse subnetworks that are able to perform as well as full models has found broad applied and theoretical interest. While many pruning methods have been developed to this end, the naïve approach of removing parameters based on their magnitude has been found to be as robust as more complex, state-of-the-art algorithms. The lack of theory behind magnitude pruning's success, especially pre-convergence, and its relation to other pruning methods, such as gradient based pruning, are outstanding open questions in the field that are in need of being addressed. We make use of recent advances in dynamical systems theory, namely Koopman operator theory, to define a new class of theoretically motivated pruning algorithms. We show that these algorithms can be equivalent to magnitude and gradient based pruning, unifying these seemingly disparate methods, and find that they can be used to shed light on magnitude pruning's performance during the early part of training.

cs.LG↗

arXiv2021

On Koopman Mode Decomposition and Tensor Component Analysis

Koopman mode decomposition and tensor component analysis (also known as CANDECOMP/PARAFAC or canonical polyadic decomposition) are two popular approaches of decomposing high dimensional data sets into low dimensional modes that capture the most relevant features and/or dynamics. Despite their similar goal, the two methods are largely used by different scientific communities and formulated in distinct mathematical languages. We examine the two together and show that, under a certain (reasonable) condition on the data, the theoretical decomposition given by tensor component analysis is the \textit{same} as that given by Koopman mode decomposition. This provides a "bridge" with which the two communities should be able to more effectively communicate. When this condition is not met, Koopman mode decomposition still provides a tensor decomposition with an \textit{a priori} computable error, providing an alternative to the non-convex optimization that tensor component analysis requires. Our work provides new possibilities for algorithmic approaches to Koopman mode decomposition and tensor component analysis, provides a new perspective on the success of tensor component analysis, and builds upon a growing body of work showing that dynamical systems, and Koopman operator theory in particular, can be useful for problems that have historically made use of optimization theory.

math.NA↗