arXiv subjects

Johanni Brea

Publications and source records attributed to Johanni Brea.

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↗

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

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↗

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↗

arXiv2022

Neural NID Rules

Abstract object properties and their relations are deeply rooted in human common sense, allowing people to predict the dynamics of the world even in situations that are novel but governed by familiar laws of physics. Standard machine learning models in model-based reinforcement learning are inadequate to generalize in this way. Inspired by the classic framework of noisy indeterministic deictic (NID) rules, we introduce here Neural NID, a method that learns abstract object properties and relations between objects with a suitably regularized graph neural network. We validate the greater generalization capability of Neural NID on simple benchmarks specifically designed to assess the transition dynamics learned by the model.

cs.LG↗

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↗

arXiv2021

Geometry of the Loss Landscape in Overparameterized Neural Networks: Symmetries and Invariances

We study how permutation symmetries in overparameterized multi-layer neural networks generate `symmetry-induced' critical points. Assuming a network with $ L $ layers of minimal widths $ r_1^*, \ldots, r_{L-1}^* $ reaches a zero-loss minimum at $ r_1^*! \cdots r_{L-1}^*! $ isolated points that are permutations of one another, we show that adding one extra neuron to each layer is sufficient to connect all these previously discrete minima into a single manifold. For a two-layer overparameterized network of width $ r^*+ h =: m $ we explicitly describe the manifold of global minima: it consists of $ T(r^*, m) $ affine subspaces of dimension at least $ h $ that are connected to one another. For a network of width $m$, we identify the number $G(r,m)$ of affine subspaces containing only symmetry-induced critical points that are related to the critical points of a smaller network of width $r<r^*$. Via a combinatorial analysis, we derive closed-form formulas for $ T $ and $ G $ and show that the number of symmetry-induced critical subspaces dominates the number of affine subspaces forming the global minima manifold in the mildly overparameterized regime (small $ h $) and vice versa in the vastly overparameterized regime ($h \gg r^*$). Our results provide new insights into the minimization of the non-convex loss function of overparameterized neural networks.

cs.LG↗

arXiv2020

Learning in Volatile Environments with the Bayes Factor Surprise

Surprise-based learning allows agents to rapidly adapt to non-stationary stochastic environments characterized by sudden changes. We show that exact Bayesian inference in a hierarchical model gives rise to a surprise-modulated trade-off between forgetting old observations and integrating them with the new ones. The modulation depends on a probability ratio, which we call "Bayes Factor Surprise", that tests the prior belief against the current belief. We demonstrate that in several existing approximate algorithms the Bayes Factor Surprise modulates the rate of adaptation to new observations. We derive three novel surprised-based algorithms, one in the family of particle filters, one in the family of variational learning, and the other in the family of message passing, that have constant scaling in observation sequence length and particularly simple update dynamics for any distribution in the exponential family. Empirical results show that these surprise-based algorithms estimate parameters better than alternative approximate approaches and reach levels of performance comparable to computationally more expensive algorithms. The Bayes Factor Surprise is related to but different from Shannon Surprise. In two hypothetical experiments, we make testable predictions for physiological indicators that dissociate the Bayes Factor Surprise from Shannon Surprise. The theoretical insight of casting various approaches as surprise-based learning, as well as the proposed online algorithms, may be applied to the analysis of animal and human behavior, and to reinforcement learning in non-stationary environments.

stat.ML↗

arXiv2019

Weight-space symmetry in deep networks gives rise to permutation saddles, connected by equal-loss valleys across the loss landscape

The permutation symmetry of neurons in each layer of a deep neural network gives rise not only to multiple equivalent global minima of the loss function, but also to first-order saddle points located on the path between the global minima. In a network of $d-1$ hidden layers with $n_k$ neurons in layers $k = 1, \ldots, d$, we construct smooth paths between equivalent global minima that lead through a `permutation point' where the input and output weight vectors of two neurons in the same hidden layer $k$ collide and interchange. We show that such permutation points are critical points with at least $n_{k+1}$ vanishing eigenvalues of the Hessian matrix of second derivatives indicating a local plateau of the loss function. We find that a permutation point for the exchange of neurons $i$ and $j$ transits into a flat valley (or generally, an extended plateau of $n_{k+1}$ flat dimensions) that enables all $n_k!$ permutations of neurons in a given layer $k$ at the same loss value. Moreover, we introduce high-order permutation points by exploiting the recursive structure in neural network functions, and find that the number of $K^{\text{th}}$-order permutation points is at least by a factor $\sum_{k=1}^{d-1}\frac{1}{2!^K}{n_k-K \choose K}$ larger than the (already huge) number of equivalent global minima. In two tasks, we illustrate numerically that some of the permutation points correspond to first-order saddles (`permutation saddles'): first, in a toy network with a single hidden layer on a function approximation task and, second, in a multilayer network on the MNIST task. Our geometric approach yields a lower bound on the number of critical points generated by weight-space symmetries and provides a simple intuitive link between previous mathematical results and numerical observations.

cs.LG↗

arXiv2019

GaussianProcesses.jl: A Nonparametric Bayes package for the Julia Language

Gaussian processes are a class of flexible nonparametric Bayesian tools that are widely used across the sciences, and in industry, to model complex data sources. Key to applying Gaussian process models is the availability of well-developed open source software, which is available in many programming languages. In this paper, we present a tutorial of the GaussianProcesses.jl package that has been developed for the Julia programming language. GaussianProcesses.jl utilises the inherent computational benefits of the Julia language, including multiple dispatch and just-in-time compilation, to produce a fast, flexible and user-friendly Gaussian processes package. The package provides many mean and kernel functions with supporting inference tools to fit exact Gaussian process models, as well as a range of alternative likelihood functions to handle non-Gaussian data (e.g. binary classification models) and sparse approximations for scalable Gaussian processes. The package makes efficient use of existing Julia packages to provide users with a range of optimization and plotting tools.

stat.CO↗

arXiv2019

Biologically plausible deep learning -- but how far can we go with shallow networks?

Training deep neural networks with the error backpropagation algorithm is considered implausible from a biological perspective. Numerous recent publications suggest elaborate models for biologically plausible variants of deep learning, typically defining success as reaching around 98% test accuracy on the MNIST data set. Here, we investigate how far we can go on digit (MNIST) and object (CIFAR10) classification with biologically plausible, local learning rules in a network with one hidden layer and a single readout layer. The hidden layer weights are either fixed (random or random Gabor filters) or trained with unsupervised methods (PCA, ICA or Sparse Coding) that can be implemented by local learning rules. The readout layer is trained with a supervised, local learning rule. We first implement these models with rate neurons. This comparison reveals, first, that unsupervised learning does not lead to better performance than fixed random projections or Gabor filters for large hidden layers. Second, networks with localized receptive fields perform significantly better than networks with all-to-all connectivity and can reach backpropagation performance on MNIST. We then implement two of the networks - fixed, localized, random & random Gabor filters in the hidden layer - with spiking leaky integrate-and-fire neurons and spike timing dependent plasticity to train the readout layer. These spiking models achieve > 98.2% test accuracy on MNIST, which is close to the performance of rate networks with one hidden layer trained with backpropagation. The performance of our shallow network models is comparable to most current biologically plausible models of deep learning. Furthermore, our results with a shallow spiking network provide an important reference and suggest the use of datasets other than MNIST for testing the performance of future models of biologically plausible deep learning.

cs.NE↗

arXiv2019

Learning to Generate Music with BachProp

As deep learning advances, algorithms of music composition increase in performance. However, most of the successful models are designed for specific musical structures. Here, we present BachProp, an algorithmic composer that can generate music scores in many styles given sufficient training data. To adapt BachProp to a broad range of musical styles, we propose a novel representation of music and train a deep network to predict the note transition probabilities of a given music corpus. In this paper, new music scores generated by BachProp are compared with the original corpora as well as with different network architectures and other related models. We show that BachProp captures important features of the original datasets better than other models and invite the reader to a qualitative comparison on a large collection of generated songs.

cs.SD↗

arXiv2018

On the choice of metric in gradient-based theories of brain function

The idea that the brain functions so as to minimize certain costs pervades theoretical neuroscience. Since a cost function by itself does not predict how the brain finds its minima, additional assumptions about the optimization method need to be made to predict the dynamics of physiological quantities. In this context, steepest descent (also called gradient descent) is often suggested as an algorithmic principle of optimization potentially implemented by the brain. In practice, researchers often consider the vector of partial derivatives as the gradient. However, the definition of the gradient and the notion of a steepest direction depend on the choice of a metric. Since the choice of the metric involves a large number of degrees of freedom, the predictive power of models that are based on gradient descent must be called into question, unless there are strong constraints on the choice of the metric. Here we provide a didactic review of the mathematics of gradient descent, illustrate common pitfalls of using gradient descent as a principle of brain function with examples from the literature and propose ways forward to constrain the metric.

q-bio.NC↗