Enrico M. Malatesta

Marco Benedetti, Andrej Bogdanov, Enrico M. Malatesta et al.

We initiate the study of the algorithmic complexity of finding collisions in single-layer binary neural networks. Given a random matrix $\mathbf{A} \in \mathbb{R}^{m\times n}$, an input $\mathbf{x} \in \{-1,1\}^n$ is mapped to a binary output vector $φ(\mathbf{A}\mathbf{x})\in \{-1,1\}^m$, where $φ$ is an activation function with constant behavior on $[κ, \infty)$ for some threshold $κ\geq 0$. We identify the threshold scale $κ=Θ(1/\sqrtα)$, where $α=m/n$, as separating two complementary phenomena. When $κ\ll 1/\sqrtα$, we give a simple online algorithm that efficiently produces extensive collisions. When $κ\gg 1/\sqrtα$, for a natural \emph{randomized} non-periodic activation and suitable oscillation complexity, we prove that the extensive-collision space exhibits an overlap gap property (OGP), yielding an exponential lower bound against online algorithms. Ours is the first work to use the overlap gap property as a rigorous criterion for collision resistance. The key difference between collision finding and average-case search is that collision finding has a new ``worst-case'' aspect: the collision finder has full control over the choice of colliding pairs. Our lower bound is proved in the online model; extending such guarantees to broader classes of algorithms, including spectral, algebraic, lattice-based, or quantum methods, remains an open direction.

Carlo Baldassi, Enrico M. Malatesta, Gabriele Perugini et al.

We study the binary and continuous negative-margin perceptrons as simple non-convex neural network models learning random rules and associations. We analyze the geometry of the landscape of solutions in both models and find important similarities and differences. Both models exhibit subdominant minimizers which are extremely flat and wide. These minimizers coexist with a background of dominant solutions which are composed by an exponential number of algorithmically inaccessible small clusters for the binary case (the frozen 1-RSB phase) or a hierarchical structure of clusters of different sizes for the spherical case (the full RSB phase). In both cases, when a certain threshold in constraint density is crossed, the local entropy of the wide flat minima becomes non-monotonic, indicating a break-up of the space of robust solutions into disconnected components. This has a strong impact on the behavior of algorithms in binary models, which cannot access the remaining isolated clusters. For the spherical case the behaviour is different, since even beyond the disappearance of the wide flat minima the remaining solutions are shown to always be surrounded by a large number of other solutions at any distance, up to capacity. Indeed, we exhibit numerical evidence that algorithms seem to find solutions up to the SAT/UNSAT transition, that we compute here using an 1RSB approximation. For both models, the generalization performance as a learning device is shown to be greatly improved by the existence of wide flat minimizers even when trained in the highly underconstrained regime of very negative margins.

Silvio Kalaj, Clarissa Lauditi, Gabriele Perugini et al.

It has been recently shown that a learning transition happens when a Hopfield Network stores examples generated as superpositions of random features, where new attractors corresponding to such features appear in the model. In this work we reveal that the network also develops attractors corresponding to previously unseen examples generated with the same set of features. We explain this surprising behaviour in terms of spurious states of the learned features: we argue that, increasing the number of stored examples beyond the learning transition, the model also learns to mix the features to represent both stored and previously unseen examples. We support this claim with the computation of the phase diagram of the model.

Enrico M. Malatesta

In these pedagogic notes I review the statistical mechanics approach to neural networks, focusing on the paradigmatic example of the perceptron architecture with binary an continuous weights, in the classification setting. I will review the Gardner's approach based on replica method and the derivation of the SAT/UNSAT transition in the storage setting. Then, I discuss some recent works that unveiled how the zero training error configurations are geometrically arranged, and how this arrangement changes as the size of the training set increases. I also illustrate how different regions of solution space can be explored analytically and how the landscape in the vicinity of a solution can be characterized. I give evidence how, in binary weight models, algorithmic hardness is a consequence of the disappearance of a clustered region of solutions that extends to very large distances. Finally, I demonstrate how the study of linear mode connectivity between solutions can give insights into the average shape of the solution manifold.

Jean Barbier, Federica Gerace, Alessandro Ingrosso et al.

Understanding the generalization abilities of neural networks for simple input-output distributions is crucial to account for their learning performance on real datasets. The classical teacher-student setting, where a network is trained from data obtained thanks to a label-generating teacher model, serves as a perfect theoretical test bed. In this context, a complete theoretical account of the performance of fully connected one-hidden layer networks in the presence of generic activation functions is lacking. In this work, we develop such a general theory for narrow networks, i.e. networks with a large number of hidden units, yet much smaller than the input dimension. Using methods from statistical physics, we provide closed-form expressions for the typical performance of both finite temperature (Bayesian) and empirical risk minimization estimators, in terms of a small number of weight statistics. In doing so, we highlight the presence of a transition where hidden neurons specialize when the number of samples is sufficiently large and proportional to the number of parameters of the network. Our theory accurately predicts the generalization error of neural networks trained on regression or classification tasks with either noisy full-batch gradient descent (Langevin dynamics) or full-batch gradient descent.

Elizaveta Demyanenko, Christoph Feinauer, Enrico M. Malatesta et al.

Recent works demonstrated the existence of a double-descent phenomenon for the generalization error of neural networks, where highly overparameterized models escape overfitting and achieve good test performance, at odds with the standard bias-variance trade-off described by statistical learning theory. In the present work, we explore a link between this phenomenon and the increase of complexity and sensitivity of the function represented by neural networks. In particular, we study the Boolean mean dimension (BMD), a metric developed in the context of Boolean function analysis. Focusing on a simple teacher-student setting for the random feature model, we derive a theoretical analysis based on the replica method that yields an interpretable expression for the BMD, in the high dimensional regime where the number of data points, the number of features, and the input size grow to infinity. We find that, as the degree of overparameterization of the network is increased, the BMD reaches an evident peak at the interpolation threshold, in correspondence with the generalization error peak, and then slowly approaches a low asymptotic value. The same phenomenology is then traced in numerical experiments with different model classes and training setups. Moreover, we find empirically that adversarially initialized models tend to show higher BMD values, and that models that are more robust to adversarial attacks exhibit a lower BMD.

Brandon Livio Annesi, Clarissa Lauditi, Carlo Lucibello et al.

Empirical studies on the landscape of neural networks have shown that low-energy configurations are often found in complex connected structures, where zero-energy paths between pairs of distant solutions can be constructed. Here we consider the spherical negative perceptron, a prototypical non-convex neural network model framed as a continuous constraint satisfaction problem. We introduce a general analytical method for computing energy barriers in the simplex with vertex configurations sampled from the equilibrium. We find that in the over-parameterized regime the solution manifold displays simple connectivity properties. There exists a large geodesically convex component that is attractive for a wide range of optimization dynamics. Inside this region we identify a subset of atypical high-margin solutions that are geodesically connected with most other solutions, giving rise to a star-shaped geometry. We analytically characterize the organization of the connected space of solutions and show numerical evidence of a transition, at larger constraint densities, where the aforementioned simple geodesic connectivity breaks down.

Carlo Baldassi, Clarissa Lauditi, Enrico M. Malatesta et al.

Current deep neural networks are highly overparameterized (up to billions of connection weights) and nonlinear. Yet they can fit data almost perfectly through variants of gradient descent algorithms and achieve unexpected levels of prediction accuracy without overfitting. These are formidable results that defy predictions of statistical learning and pose conceptual challenges for non-convex optimization. In this paper, we use methods from statistical physics of disordered systems to analytically study the computational fallout of overparameterization in non-convex binary neural network models, trained on data generated from a structurally simpler but "hidden" network. As the number of connection weights increases, we follow the changes of the geometrical structure of different minima of the error loss function and relate them to learning and generalization performance. A first transition happens at the so-called interpolation point, when solutions begin to exist (perfect fitting becomes possible). This transition reflects the properties of typical solutions, which however are in sharp minima and hard to sample. After a gap, a second transition occurs, with the discontinuous appearance of a different kind of "atypical" structures: wide regions of the weight space that are particularly solution-dense and have good generalization properties. The two kinds of solutions coexist, with the typical ones being exponentially more numerous, but empirically we find that efficient algorithms sample the atypical, rare ones. This suggests that the atypical phase transition is the relevant one for learning. The results of numerical tests with realistic networks on observables suggested by the theory are consistent with this scenario.

Carlo Baldassi, Clarissa Lauditi, Enrico M. Malatesta et al.

The success of deep learning has revealed the application potential of neural networks across the sciences and opened up fundamental theoretical problems. In particular, the fact that learning algorithms based on simple variants of gradient methods are able to find near-optimal minima of highly nonconvex loss functions is an unexpected feature of neural networks. Moreover, such algorithms are able to fit the data even in the presence of noise, and yet they have excellent predictive capabilities. Several empirical results have shown a reproducible correlation between the so-called flatness of the minima achieved by the algorithms and the generalization performance. At the same time, statistical physics results have shown that in nonconvex networks a multitude of narrow minima may coexist with a much smaller number of wide flat minima, which generalize well. Here we show that wide flat minima arise as complex extensive structures, from the coalescence of minima around "high-margin" (i.e., locally robust) configurations. Despite being exponentially rare compared to zero-margin ones, high-margin minima tend to concentrate in particular regions. These minima are in turn surrounded by other solutions of smaller and smaller margin, leading to dense regions of solutions over long distances. Our analysis also provides an alternative analytical method for estimating when flat minima appear and when algorithms begin to find solutions, as the number of model parameters varies.

Carlo Baldassi, Enrico M. Malatesta, Matteo Negri et al.

We analyze the connection between minimizers with good generalizing properties and high local entropy regions of a threshold-linear classifier in Gaussian mixtures with the mean squared error loss function. We show that there exist configurations that achieve the Bayes-optimal generalization error, even in the case of unbalanced clusters. We explore analytically the error-counting loss landscape in the vicinity of a Bayes-optimal solution, and show that the closer we get to such configurations, the higher the local entropy, implying that the Bayes-optimal solution lays inside a wide flat region. We also consider the algorithmically relevant case of targeting wide flat minima of the (differentiable) mean squared error loss. Our analytical and numerical results show not only that in the balanced case the dependence on the norm of the weights is mild, but also, in the unbalanced case, that the performances can be improved.

Carlo Baldassi, Enrico M. Malatesta, Riccardo Zecchina

Rectified Linear Units (ReLU) have become the main model for the neural units in current deep learning systems. This choice has been originally suggested as a way to compensate for the so called vanishing gradient problem which can undercut stochastic gradient descent (SGD) learning in networks composed of multiple layers. Here we provide analytical results on the effects of ReLUs on the capacity and on the geometrical landscape of the solution space in two-layer neural networks with either binary or real-valued weights. We study the problem of storing an extensive number of random patterns and find that, quite unexpectedly, the capacity of the network remains finite as the number of neurons in the hidden layer increases, at odds with the case of threshold units in which the capacity diverges. Possibly more important, a large deviation approach allows us to find that the geometrical landscape of the solution space has a peculiar structure: while the majority of solutions are close in distance but still isolated, there exist rare regions of solutions which are much more dense than the similar ones in the case of threshold units. These solutions are robust to perturbations of the weights and can tolerate large perturbations of the inputs. The analytical results are corroborated by numerical findings.