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The twin peaks of learning neural networks
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.
High-dimensional manifold of solutions in neural networks: insights from statistical physics
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.
The star-shaped space of solutions of the spherical negative perceptron
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.
Typical and atypical solutions in non-convex neural networks with discrete and continuous weights
We analyze the geometry of the landscape of solutions in both models and find important similarities and differences.
Learning through atypical "phase transitions" in overparameterized neural networks
Current deep neural networks are highly overparameterized (up to billions of connection weights) and nonlinear.
Unveiling the structure of wide flat minima in neural networks
The success of deep learning has revealed the application potential of neural networks across the sciences and opened up fundamental theoretical problems.
Wide flat minima and optimal generalization in classifying high-dimensional Gaussian mixtures
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.
Properties of the geometry of solutions and capacity of multi-layer neural networks with Rectified Linear Units activations
Rectified Linear Units (ReLU) have become the main model for the neural units in current deep learning systems.
Random Combinatorial Optimization Problems: Mean Field and Finite-Dimensional Results
The second part of the thesis deals with mean-field combinatorial optimization problems.
Disordered Systems and Neural Networks
