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Person# Francesco Spadaro

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This thesis is centered on questions coming from Machine Learning (ML) and Statistical Field Theory (SFT).In Machine Learning, we consider the subfield of Supervised Learning (SL), and in particular regression tasks where one tries to find a regressor that fits given labeled data points as well as possible while at the same time optimizing other constrains. We consider two very famous Regression Models: Kernel Methods and Random Features (RF) models. We show that RF models can be used as an effective way to implement Kernel Methods and discuss in details the robustness of this approximation. Furthermore we propose a new estimator for the analysis of the choice of kernels that it is based only on the acquired dataset.In SFT we focus on so-called two-dimensional Conformal Field Theories (CFTs). We study these theories via their connection with lattice models, detailing how they emerge as continuous limits of discrete models as well as the meaning of all the central quantities of the theory. Furthermore, we show how to connect CFTs with the theory of Schramm-Loewner Evolutions (SLEs). Finally, we focus on the semi-local theories associated with the Ising and Tricritical Ising models and analyze the probabilistic and geometric meaning of the holomorphic fermions present therein.

We consider several aspects of conjugating symmetry methods, including the method of invariants, with an asymptotic approach. In particular we consider how to extend to the stochastic setting several ideas which are well established in the deterministic one, such as conditional, partial and asymptotic symmetries. A number of explicit examples are presented.

Johanni Michael Brea, Wulfram Gerstner, Clément Hongler, Berfin Simsek, Francesco Spadaro

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.

2021