Contribution of task-irrelevant stimuli to drift of neural representations

TL;DR

The paper addresses representational drift in online lifelong learning, showing that task-irrelevant stimuli can drive long-term changes in task-relevant representations even under stable performance. It develops a theoretical diffusion framework that decomposes dynamics into fast normal fluctuations and slow tangential diffusion on the solution manifold, and analytically derives drift rates for four canonical models (Oja's rule, Similarity Matching, autoencoder, and a supervised two-layer network). Across synthetic Gaussian data, nonlinear networks, and MNIST, the results show drift scales with the variance and dimension of the task-irrelevant subspace and differs qualitatively from Gaussian synaptic-noise predictions, yielding testable geometry- and dimension-dependent signatures. The work thus links stimulus structure, task, and learning rule to representational drift and suggests using drift as a signal to infer underlying computations in neural systems, with implications for interpreting lifelong learning and designing robust AI systems.

Abstract

Biological and artificial learners are inherently exposed to a stream of data and experience throughout their lifetimes and must constantly adapt to, learn from, or selectively ignore the ongoing input. Recent findings reveal that, even when the performance remains stable, the underlying neural representations can change gradually over time, a phenomenon known as representational drift. Studying the different sources of data and noise that may contribute to drift is essential for understanding lifelong learning in neural systems. However, a systematic study of drift across architectures and learning rules, and the connection to task, are missing. Here, in an online learning setup, we characterize drift as a function of data distribution, and specifically show that the learning noise induced by task-irrelevant stimuli, which the agent learns to ignore in a given context, can create long-term drift in the representation of task-relevant stimuli. Using theory and simulations, we demonstrate this phenomenon both in Hebbian-based learning -- Oja's rule and Similarity Matching -- and in stochastic gradient descent applied to autoencoders and a supervised two-layer network. We consistently observe that the drift rate increases with the variance and the dimension of the data in the task-irrelevant subspace. We further show that this yields different qualitative predictions for the geometry and dimension-dependency of drift than those arising from Gaussian synaptic noise. Overall, our study links the structure of stimuli, task, and learning rule to representational drift and could pave the way for using drift as a signal for uncovering underlying computation in the brain.

Figures (7)

• Figure 1: Demonstration of the effect of task-irrelevant stimuli on representational drift. A one-layer network is continuously presented with input samples and updated using Oja's learning rule long after convergence ($n=50$, $m=30$, $\eta=0.025$). a) Grassmann distance between the $m$-principal subspace of the data and row space of $\bm{W}$. b) Norms of representations for a task-relevant ($\bm{x}_{||} \in \mathcal{X}_{||}$, blue), and a task-irrelevant stimuli ($\bm{x}_{\perp} \in \mathcal{X}_{||}$, orange). Note that, $\bm{y}_i$'s for $\bm{x}_{\perp}$ fluctuate around zero which leads to a non-zero steady-state norm. c) Decay of average cosine similarity for task-relevant stimuli representations, under two values of $\lambda_{\perp}$. Each curve is averaged for $m=30$ stimuli; shaded regions indicate the standard error of the mean). d) Same as c) but for $n=50$ and $n=80$.
• Figure 2: Neural network models studied in this work. a) Multi-dimensional Oja network, b) Similarity Matching network, c) autoencoder with a bottleneck, and d) a two-layer network. The first three networks are trained in an unsupervised way, while the two-layer network is trained with supervised learning.
• Figure 3: Drift rate for different architectures as a function of: a) learning rate $\eta$, b) task-irrelevant variance $\lambda_{\perp}$, and c) the dimension of the task-irrelevant space, $n-m^*$. Note that $m^*=m$ for Oja and Similarity Matching (SM), and $m^*=p$ for the autoencoder (AE). For panels a) and b), $n=5, m=3$; for panel c), $\eta=0.001$ and $\sqrt{\lambda_{\perp}}=0.01$.
• Figure 4: Example of drift in a non-linear network. a) Schematic of a two-layer network trained to represent the position on the ring ($\theta$). b) Reorganization of receptive fields after continued training, as shown by two snapshots $t_1=1.25\!\times\!10^6$ and $t_2=5\!\times\! 10^6$. The right panel shows the stability of kernel $K(\theta_1,\theta_2) = \bm{h}(\theta_1)^T \bm{h}(\theta_2)$ across time. c) Decay of average cosine similarity and its dependence on $\lambda_{\perp}$ (averages were performed over representations of 5 uniformly chosen angles and 80 runs, shaded area: standard error of the mean). Unless otherwise specified, for all simulations: $n=8$, $p=10$, $m=2$, $\eta=0.2$, $\gamma=0.05$ and $\lambda_{\perp} = 0.25$.
• Figure 5: Drift in MNIST data. a) Two snapshots of hidden layer representations for 10 sample digits in a linear autoencoder ($n=m=784$, $p=2$, $\eta=0.01$, $t_1=1.6\!\times\!10^4$ and $t_2 = 10^5$). To show the fluctuations alongside the drift, a time window of $t_w=5000$ preceding each snapshot is considered and representations at 100 uniformly chosen times are overlaid. b) Normalized rate of drift as a function of output dimension ($m$) for Oja and Similarity Matching (SM), and bottleneck dimension ($p$) for autoencoder (AE). For the results in this panel, projections of MNIST data onto its top $n=20$ principal components are used as input. The plots show the drift rate for the representation of the top principal vector (shaded area: standard deviation over multiple runs).