Recent generalizations of the Hopfield model of associative memories are able to store a number $P$ of random patterns that grows exponentially with the number $N$ of neurons, $P=\exp(\alpha N)$. Besides the huge storage capacity, another interesting feature of these networks is their connection to the attention mechanism which is part of the Transformer architectures widely applied in deep learning. In this work, we consider a generic family of pattern ensembles, and thanks to the statistical mechanics analysis of an auxiliary Random Energy Model, we are able to provide exact asymptotic thresholds for the retrieval of a typical pattern, $\alpha_1$, and lower bounds for the maximum of the load $\alpha$ for which all patterns can be retrieved, $\alpha_c$. Additionally, we characterize the size of the basins of attractions. We discuss in detail the cases of Gaussian and spherical patterns, and show that they display rich and qualitatively different phase diagrams.

The Exponential Capacity of Dense Associative Memories

It was recently shown that almost all solutions in the symmetric binary perceptron are isolated, even at low constraint densities, suggesting that finding typical solutions is hard. In contrast, some algorithms have been shown empirically to succeed in finding solutions at low density. This phenomenon has been justified numerically by the existence of subdominant and dense connected regions of solutions, which are accessible by simple learning algorithms. In this paper, we establish formally such a phenomenon for both the symmetric and asymmetric binary perceptrons. We show that at low constraint density (equivalently for overparametrized perceptrons), there exists indeed a subdominant connected cluster of solutions with almost maximal diameter, and that an efficient multiscale majority algorithm can find solutions in such a cluster with high probability, settling in particular an open problem posed by Perkins-Xu '21. In addition, even close to the critical threshold, we show that there exist clusters of linear diameter for the symmetric perceptron, as well as for the asymmetric perceptron under additional assumptions.

Binary perceptron: efficient algorithms can find solutions in a rare well-connected cluster.

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 escape the bias-variance 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 nonconvex neural network models. 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 generalisation performance. We find that there exist a gap between the SAT/UNSAT interpolation transition where solutions begin to exist and the point where algorithms start to find solutions, i.e. where accessible solutions appear. This second phase transition coincides with the discontinuous appearance of atypical solutions that are locally extremely entropic, i.e., flat regions of the weight space that are particularly solution-dense and have good generalization properties. Although exponentially rare compared to typical solutions (which are narrower and extremely difficult to sample), entropic solutions are accessible to the algorithms used in learning. We can characterize the generalization error of different solutions and optimize the Bayesian prediction, for data generated from a structurally different network. Numerical tests on observables suggested by the theory confirm that the scenario extends to realistic deep networks.

Learning through atypical "phase transitions" in overparameterized neural networks.

We apply digitized quantum annealing (QA) and quantum approximate optimization algorithm (QAOA) to a paradigmatic task of supervised learning in artificial neural networks: the optimization of synaptic weights for the binary perceptron. At variance with the usual QAOA applications to MaxCut, or to quantum spin-chains ground-state preparation, here the classical cost function is characterized by highly nonlocal multispin interactions. Yet, we provide evidence for the existence of optimal smooth solutions for the QAOA parameters, which are transferable among typical instances of the same problem, and we prove numerically an enhanced performance of QAOA over traditional QA. We also investigate on the role of the classical cost-function landscape geometry in this problem. By artificially breaking this geometrical structure, we show that the detrimental effect of a gap-closing transition, encountered in QA, is also negatively affecting the performance of our QAOA implementation. 

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Quantum approximate optimization algorithm applied to the binary perceptron

Binary perceptron is a fundamental model of supervised learning for the non-convex optimization, which is a root of the popular deep learning. Binary perceptron is able to achieve a classification of random high-dimensional data by computing the marginal probabilities of binary synapses. The relationship between the algorithmic instability and the equilibrium analysis of the model remains elusive. Here, we establish the relationship by showing that the instability condition around the algorithmic fixed point is identical to the instability for breaking the replica symmetric saddle point solution of the free energy function. Therefore, our analysis provides insights towards bridging the gap between non-convex learning dynamics and statistical mechanics properties of more complex neural networks.

Equivalence between algorithmic instability and transition to replica symmetry breaking in perceptron learning systems

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 which needs to be understood in depth. Such algorithms are able to fit the data almost perfectly, 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 from the coalescence of minima that correspond to high-margin classifications. Despite being exponentially rare compared to zero-margin solutions, 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.

Unveiling the structure of wide flat minima in neural networks.

The Hidden-Manifold Hopfield Model and a learning phase transition

The Hopfield model is a paradigmatic model of neural networks that has been analyzed for many decades in the statistical physics, neuroscience, and machine learning communities. Inspired by the manifold hypothesis in machine learning, we propose and investigate a generalization of the standard setting that we name Random-Features Hopfield Model. Here $P$ binary patterns of length $N$ are generated by applying to Gaussian vectors sampled in a latent space of dimension $D$ a random projection followed by a non-linearity. Using the replica method from statistical physics, we derive the phase diagram of the model in the limit $P,N,D\to\infty$ with fixed ratios $\alpha=P/N$ and $\alpha_D=D/N$. Besides the usual retrieval phase, where the patterns can be dynamically recovered from some initial corruption, we uncover a new phase where the features characterizing the projection can be recovered instead. We call this phenomena the learning phase transition, as the features are not explicitly given to the model but rather are inferred from the patterns in an unsupervised fashion.

Storage and Learning phase transitions in the Random-Features Hopfield Model

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 atypically robust 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.

Clarissa Lauditi

Papers

Unveiling the structure of wide flat minima in neural networks.

The Hidden-Manifold Hopfield Model and a learning phase transition

Learning through atypical "phase transitions" in overparameterized neural networks.

Storage and Learning phase transitions in the Random-Features Hopfield Model

The star-shaped space of solutions of the spherical negative perceptron