arXiv subjects

Wulfram Gerstner

Publications and source records attributed to Wulfram Gerstner.

At least 19 recordsLinked to original sources

arXiv2026

Teacher Geometry Shapes Learnability in Teacher-Student Networks

Teacher-student systems, in which a teacher neural network generates training labels so that a student neural network can learn to implement the same function, are widely used as an abstract setting to study learning. However, the structure of the teachers is often overlooked by assuming randomly-generated, normally-distributed parameters. This hides substantial variation in how learnable different teachers are. We formalize learnability as the success rate of converging to the global minimum, as a function of overparameterization, learning algorithm, student initialization distribution, and teacher geometry. We both identify an easy distribution that maximizes node dissimilarity and a hard distribution that minimizes it, and show that these two distributions induce markedly different success rates across a large range of settings and for different activation functions. To explain the gap, we study the loss landscape of small neural networks that contain two distinct kinds of suboptimal local minima, out-of-bounds (OOB) minima at the edge of the data distribution and interior minima within. Assuming infinite data and a fast readout layer, we analytically reduce the loss landscape of small networks to two dimensions, showing that the region of attraction of interior minima changes as a function of teacher structure. In larger networks, maximally dissimilar teachers induce more interior minima, while minimally dissimilar teachers induce more OOB minima. Motivated by these analyses, we show that differentially increasing the learning rate of the readout layer and decreasing the learning rate of the inner biases increases success rates. These findings provide an important step in narrowing the gap between the study of teacher-student networks and more structured functions that arise in practice.

cs.LG↗

arXiv2026

Self-supervised local learning rules learn the hidden hierarchical structure of high-dimensional data

The brain learns abstract representations of high-dimensional sensory input, but the plasticity rules that enable such learning are unknown. We study biologically plausible algorithms on the Random Hierarchy Model (RHM), an artificial dataset designed to investigate how deep neural networks learn the intrinsic hierarchical structure of high-dimensional data. We focus on two types of local learning rules that avoid both a long convergence time and the use of a symmetric error network. The first type uses direct feedback signals to approximate error propagation from the output layer. The second type uses layerwise self-supervised contrastive or non-contrastive loss functions that do not explicitly approximate errors at the output layer. We show that all rules of the first type fail to solve the tasks of the RHM and trace this failure back to input-specific nonlinearities (`masking') that are implemented in full backpropagation and are essential for learning complex tasks. However, algorithms of the second type are able to learn the hierarchical hidden structure of the RHM tasks and are as data-efficient as supervised backpropagation training, while being compatible with known rules of synaptic plasticity in cortex.

cs.LG↗

arXiv2026

Can Local Learning Match Self-Supervised Backpropagation?

While end-to-end self-supervised learning with backpropagation (global BP-SSL) has become central for training modern AI systems, theories of local self-supervised learning (local-SSL) have struggled to build functional representations in deep neural networks. To establish a link between global and local rules, we first develop a theory for deep linear networks: we identify conditions for local-SSL algorithms (like Forward-forward or CLAPP) to implement exactly the same weight update as a global BP-SSL. Starting from the theoretical insights, we then develop novel variants of local-SSL algorithms to approximate global BP-SSL in deep non-linear convolutional neural networks. Variants that improve the similarity between gradient updates of local-SSL with those of global BP-SSL also show better performance on image datasets (CIFAR-10, STL-10, and Tiny ImageNet). The best local-SSL rule with the CLAPP loss function matches the performance of a comparable global BP-SSL with InfoNCE or CPC-like loss functions, and improves upon state-of-the-art for local SSL on these benchmarks.

cs.LG↗

arXiv2025

Data Augmentation Techniques to Reverse-Engineer Neural Network Weights from Input-Output Queries

Network weights can be reverse-engineered given enough informative samples of a network's input-output function. In a teacher-student setup, this translates into collecting a dataset of the teacher mapping -- querying the teacher -- and fitting a student to imitate such mapping. A sensible choice of queries is the dataset the teacher is trained on. But current methods fail when the teacher parameters are more numerous than the training data, because the student overfits to the queries instead of aligning its parameters to the teacher. In this work, we explore augmentation techniques to best sample the input-output mapping of a teacher network, with the goal of eliciting a rich set of representations from the teacher hidden layers. We discover that standard augmentations such as rotation, flipping, and adding noise, bring little to no improvement to the identification problem. We design new data augmentation techniques tailored to better sample the representational space of the network's hidden layers. With our augmentations we extend the state-of-the-art range of recoverable network sizes. To test their scalability, we show that we can recover networks of up to 100 times more parameters than training data-points.

cs.AI↗

arXiv2025

Flat Channels to Infinity in Neural Loss Landscapes

The loss landscapes of neural networks contain minima and saddle points that may be connected in flat regions or appear in isolation. We identify and characterize a special structure in the loss landscape: channels along which the loss decreases extremely slowly, while the output weights of at least two neurons, $a_i$ and $a_j$, diverge to $\pm$infinity, and their input weight vectors, $\mathbf{w_i}$ and $\mathbf{w_j}$, become equal to each other. At convergence, the two neurons implement a gated linear unit: $a_i\sigma(\mathbf{w_i} \cdot \mathbf{x}) + a_j\sigma(\mathbf{w_j} \cdot \mathbf{x}) \rightarrow \sigma(\mathbf{w} \cdot \mathbf{x}) + (\mathbf{v} \cdot \mathbf{x}) \sigma'(\mathbf{w} \cdot \mathbf{x})$. Geometrically, these channels to infinity are asymptotically parallel to symmetry-induced lines of critical points. Gradient flow solvers, and related optimization methods like SGD or ADAM, reach the channels with high probability in diverse regression settings, but without careful inspection they look like flat local minima with finite parameter values. Our characterization provides a comprehensive picture of these quasi-flat regions in terms of gradient dynamics, geometry, and functional interpretation. The emergence of gated linear units at the end of the channels highlights a surprising aspect of the computational capabilities of fully connected layers.

cs.LG↗

arXiv2024

Should Under-parameterized Student Networks Copy or Average Teacher Weights?

Any continuous function $f^*$ can be approximated arbitrarily well by a neural network with sufficiently many neurons $k$. We consider the case when $f^*$ itself is a neural network with one hidden layer and $k$ neurons. Approximating $f^*$ with a neural network with $n< k$ neurons can thus be seen as fitting an under-parameterized "student" network with $n$ neurons to a "teacher" network with $k$ neurons. As the student has fewer neurons than the teacher, it is unclear, whether each of the $n$ student neurons should copy one of the teacher neurons or rather average a group of teacher neurons. For shallow neural networks with erf activation function and for the standard Gaussian input distribution, we prove that "copy-average" configurations are critical points if the teacher's incoming vectors are orthonormal and its outgoing weights are unitary. Moreover, the optimum among such configurations is reached when $n-1$ student neurons each copy one teacher neuron and the $n$-th student neuron averages the remaining $k-n+1$ teacher neurons. For the student network with $n=1$ neuron, we provide additionally a closed-form solution of the non-trivial critical point(s) for commonly used activation functions through solving an equivalent constrained optimization problem. Empirically, we find for the erf activation function that gradient flow converges either to the optimal copy-average critical point or to another point where each student neuron approximately copies a different teacher neuron. Finally, we find similar results for the ReLU activation function, suggesting that the optimal solution of underparameterized networks has a universal structure.

cs.LG↗

arXiv2023

Trial matching: capturing variability with data-constrained spiking neural networks

Simultaneous behavioral and electrophysiological recordings call for new methods to reveal the interactions between neural activity and behavior. A milestone would be an interpretable model of the co-variability of spiking activity and behavior across trials. Here, we model a mouse cortical sensory-motor pathway in a tactile detection task reported by licking with a large recurrent spiking neural network (RSNN), fitted to the recordings via gradient-based optimization. We focus specifically on the difficulty to match the trial-to-trial variability in the data. Our solution relies on optimal transport to define a distance between the distributions of generated and recorded trials. The technique is applied to artificial data and neural recordings covering six cortical areas. We find that the resulting RSNN can generate realistic cortical activity and predict jaw movements across the main modes of trial-to-trial variability. Our analysis also identifies an unexpected mode of variability in the data corresponding to task-irrelevant movements of the mouse.

q-bio.NC↗

arXiv2023

High-performance deep spiking neural networks with 0.3 spikes per neuron

Communication by rare, binary spikes is a key factor for the energy efficiency of biological brains. However, it is harder to train biologically-inspired spiking neural networks (SNNs) than artificial neural networks (ANNs). This is puzzling given that theoretical results provide exact mapping algorithms from ANNs to SNNs with time-to-first-spike (TTFS) coding. In this paper we analyze in theory and simulation the learning dynamics of TTFS-networks and identify a specific instance of the vanishing-or-exploding gradient problem. While two choices of SNN mappings solve this problem at initialization, only the one with a constant slope of the neuron membrane potential at threshold guarantees the equivalence of the training trajectory between SNNs and ANNs with rectified linear units. We demonstrate that training deep SNN models achieves the exact same performance as that of ANNs, surpassing previous SNNs on image classification datasets such as MNIST/Fashion-MNIST, CIFAR10/CIFAR100 and PLACES365. Our SNN accomplishes high-performance classification with less than 0.3 spikes per neuron, lending itself for an energy-efficient implementation. We show that fine-tuning SNNs with our robust gradient descent algorithm enables their optimization for hardware implementations with low latency and resilience to noise and quantization.

cs.NE↗

arXiv2023

Context selectivity with dynamic availability enables lifelong continual learning

"You never forget how to ride a bike", -- but how is that possible? The brain is able to learn complex skills, stop the practice for years, learn other skills in between, and still retrieve the original knowledge when necessary. The mechanisms of this capability, referred to as lifelong learning (or continual learning, CL), are unknown. We suggest a bio-plausible meta-plasticity rule building on classical work in CL which we summarize in two principles: (i) neurons are context selective, and (ii) a local availability variable partially freezes the plasticity if the neuron was relevant for previous tasks. In a new neuro-centric formalization of these principles, we suggest that neuron selectivity and neuron-wide consolidation is a simple and viable meta-plasticity hypothesis to enable CL in the brain. In simulation, this simple model balances forgetting and consolidation leading to better transfer learning than contemporary CL algorithms on image recognition and natural language processing CL benchmarks.

cs.LG↗

arXiv2023

Expand-and-Cluster: Parameter Recovery of Neural Networks

Can we identify the weights of a neural network by probing its input-output mapping? At first glance, this problem seems to have many solutions because of permutation, overparameterisation and activation function symmetries. Yet, we show that the incoming weight vector of each neuron is identifiable up to sign or scaling, depending on the activation function. Our novel method 'Expand-and-Cluster' can identify layer sizes and weights of a target network for all commonly used activation functions. Expand-and-Cluster consists of two phases: (i) to relax the non-convex optimisation problem, we train multiple overparameterised student networks to best imitate the target function; (ii) to reverse engineer the target network's weights, we employ an ad-hoc clustering procedure that reveals the learnt weight vectors shared between students -- these correspond to the target weight vectors. We demonstrate successful weights and size recovery of trained shallow and deep networks with less than 10\% overhead in the layer size and describe an `ease-of-identifiability' axis by analysing 150 synthetic problems of variable difficulty.

cs.NE↗

arXiv2023

Emergent rate-based dynamics in duplicate-free populations of spiking neurons

Can Spiking Neural Networks (SNNs) approximate the dynamics of Recurrent Neural Networks (RNNs)? Arguments in classical mean-field theory based on laws of large numbers provide a positive answer when each neuron in the network has many "duplicates", i.e. other neurons with almost perfectly correlated inputs. Using a disordered network model that guarantees the absence of duplicates, we show that duplicate-free SNNs can converge to RNNs, thanks to the concentration of measure phenomenon. This result reveals a general mechanism underlying the emergence of rate-based dynamics in large SNNs.

q-bio.NC↗

arXiv2023

MLPGradientFlow: going with the flow of multilayer perceptrons (and finding minima fast and accurately)

MLPGradientFlow is a software package to solve numerically the gradient flow differential equation $\dot θ= -\nabla \mathcal L(θ; \mathcal D)$, where $θ$ are the parameters of a multi-layer perceptron, $\mathcal D$ is some data set, and $\nabla \mathcal L$ is the gradient of a loss function. We show numerically that adaptive first- or higher-order integration methods based on Runge-Kutta schemes have better accuracy and convergence speed than gradient descent with the Adam optimizer. However, we find Newton's method and approximations like BFGS preferable to find fixed points (local and global minima of $\mathcal L$) efficiently and accurately. For small networks and data sets, gradients are usually computed faster than in pytorch and Hessian are computed at least $5\times$ faster. Additionally, the package features an integrator for a teacher-student setup with bias-free, two-layer networks trained with standard Gaussian input in the limit of infinite data. The code is accessible at https://github.com/jbrea/MLPGradientFlow.jl.

cs.LG↗

arXiv2023

Mesoscopic modeling of hidden spiking neurons

Can we use spiking neural networks (SNN) as generative models of multi-neuronal recordings, while taking into account that most neurons are unobserved? Modeling the unobserved neurons with large pools of hidden spiking neurons leads to severely underconstrained problems that are hard to tackle with maximum likelihood estimation. In this work, we use coarse-graining and mean-field approximations to derive a bottom-up, neuronally-grounded latent variable model (neuLVM), where the activity of the unobserved neurons is reduced to a low-dimensional mesoscopic description. In contrast to previous latent variable models, neuLVM can be explicitly mapped to a recurrent, multi-population SNN, giving it a transparent biological interpretation. We show, on synthetic spike trains, that a few observed neurons are sufficient for neuLVM to perform efficient model inversion of large SNNs, in the sense that it can recover connectivity parameters, infer single-trial latent population activity, reproduce ongoing metastable dynamics, and generalize when subjected to perturbations mimicking photo-stimulation.

q-bio.NC↗

arXiv2022

An Exact Mapping From ReLU Networks to Spiking Neural Networks

Deep spiking neural networks (SNNs) offer the promise of low-power artificial intelligence. However, training deep SNNs from scratch or converting deep artificial neural networks to SNNs without loss of performance has been a challenge. Here we propose an exact mapping from a network with Rectified Linear Units (ReLUs) to an SNN that fires exactly one spike per neuron. For our constructive proof, we assume that an arbitrary multi-layer ReLU network with or without convolutional layers, batch normalization and max pooling layers was trained to high performance on some training set. Furthermore, we assume that we have access to a representative example of input data used during training and to the exact parameters (weights and biases) of the trained ReLU network. The mapping from deep ReLU networks to SNNs causes zero percent drop in accuracy on CIFAR10, CIFAR100 and the ImageNet-like data sets Places365 and PASS. More generally our work shows that an arbitrary deep ReLU network can be replaced by an energy-efficient single-spike neural network without any loss of performance.

cs.NE↗

arXiv2022

Correlation-invariant synaptic plasticity

Cortical populations of neurons develop sparse representations adapted to the statistics of the environment. While existing synaptic plasticity models reproduce some of the observed receptive-field properties, a major obstacle is the sensitivity of Hebbian learning to omnipresent spurious correlations in cortical networks which can overshadow relevant latent input features. Here we develop a theory for synaptic plasticity that is invariant to second-order correlations in the input. Going beyond classical Hebbian learning, we show how Hebbian long-term depression (LTD) cancels the sensitivity to second-order correlations, so that receptive fields become aligned with features hidden in higher-order statistics. Our simulations demonstrate how correlation-invariance enables biologically realistic models to develop sparse population codes, despite diverse levels of variability and heterogeneity. The theory advances our understanding of local unsupervised learning in cortical circuits and assigns a specific functional role to synaptic LTD mechanisms in pyramidal neurons.

q-bio.NC↗

arXiv2022

A taxonomy of surprise definitions

Surprising events trigger measurable brain activity and influence human behavior by affecting learning, memory, and decision-making. Currently there is, however, no consensus on the definition of surprise. Here we identify 18 mathematical definitions of surprise in a unifying framework. We first propose a technical classification of these definitions into three groups based on their dependence on an agent's belief, show how they relate to each other, and prove under what conditions they are indistinguishable. Going beyond this technical analysis, we propose a taxonomy of surprise definitions and classify them into four conceptual categories based on the quantity they measure: (i) 'prediction surprise' measures a mismatch between a prediction and an observation; (ii) 'change-point detection surprise' measures the probability of a change in the environment; (iii) 'confidence-corrected surprise' explicitly accounts for the effect of confidence; and (iv) 'information gain surprise' measures the belief-update upon a new observation. The taxonomy poses the foundation for principled studies of the functional roles and physiological signatures of surprise in the brain.

q-bio.NC↗

arXiv2022

Kernel Memory Networks: A Unifying Framework for Memory Modeling

We consider the problem of training a neural network to store a set of patterns with maximal noise robustness. A solution, in terms of optimal weights and state update rules, is derived by training each individual neuron to perform either kernel classification or interpolation with a minimum weight norm. By applying this method to feed-forward and recurrent networks, we derive optimal models, termed kernel memory networks, that include, as special cases, many of the hetero- and auto-associative memory models that have been proposed over the past years, such as modern Hopfield networks and Kanerva's sparse distributed memory. We modify Kanerva's model and demonstrate a simple way to design a kernel memory network that can store an exponential number of continuous-valued patterns with a finite basin of attraction. The framework of kernel memory networks offers a simple and intuitive way to understand the storage capacity of previous memory models, and allows for new biological interpretations in terms of dendritic non-linearities and synaptic cross-talk.

cs.NE↗

arXiv2021

Fitting summary statistics of neural data with a differentiable spiking network simulator

Fitting network models to neural activity is an important tool in neuroscience. A popular approach is to model a brain area with a probabilistic recurrent spiking network whose parameters maximize the likelihood of the recorded activity. Although this is widely used, we show that the resulting model does not produce realistic neural activity. To correct for this, we suggest to augment the log-likelihood with terms that measure the dissimilarity between simulated and recorded activity. This dissimilarity is defined via summary statistics commonly used in neuroscience and the optimization is efficient because it relies on back-propagation through the stochastically simulated spike trains. We analyze this method theoretically and show empirically that it generates more realistic activity statistics. We find that it improves upon other fitting algorithms for spiking network models like GLMs (Generalized Linear Models) which do not usually rely on back-propagation. This new fitting algorithm also enables the consideration of hidden neurons which is otherwise notoriously hard, and we show that it can be crucial when trying to infer the network connectivity from spike recordings.

stat.ML↗