Chiara Cammarota

Tony Bonnaire, Giulio Biroli, Chiara Cammarota

Gradient descent is commonly used to find minima in rough landscapes, particularly in recent machine learning applications. However, a theoretical understanding of why good solutions are found remains elusive, especially in strongly non-convex and high-dimensional settings. Here, we focus on the phase retrieval problem as a typical example, which has received a lot of attention recently in theoretical machine learning. We analyze the Hessian during gradient descent, identify a dynamical transition in its spectral properties, and relate it to the ability of escaping rough regions in the loss landscape. When the signal-to-noise ratio (SNR) is large enough, an informative negative direction exists in the Hessian at the beginning of the descent, i.e in the initial condition. While descending, a BBP transition in the spectrum takes place in finite time: the direction is lost, and the dynamics is trapped in a rugged region filled with marginally stable bad minima. Surprisingly, for finite system sizes, this window of negative curvature allows the system to recover the signal well before the theoretical SNR found for infinite sizes, emphasizing the central role of initialization and early-time dynamics for efficiently navigating rough landscapes.

Stefano Sarao Mannelli, Giulio Biroli, Chiara Cammarota et al.

Despite the widespread use of gradient-based algorithms for optimizing high-dimensional non-convex functions, understanding their ability of finding good minima instead of being trapped in spurious ones remains to a large extent an open problem. Here we focus on gradient flow dynamics for phase retrieval from random measurements. When the ratio of the number of measurements over the input dimension is small the dynamics remains trapped in spurious minima with large basins of attraction. We find analytically that above a critical ratio those critical points become unstable developing a negative direction toward the signal. By numerical experiments we show that in this regime the gradient flow algorithm is not trapped; it drifts away from the spurious critical points along the unstable direction and succeeds in finding the global minimum. Using tools from statistical physics we characterize this phenomenon, which is related to a BBP-type transition in the Hessian of the spurious minima.

Stefano Sarao Mannelli, Giulio Biroli, Chiara Cammarota et al.

Gradient-based algorithms are effective for many machine learning tasks, but despite ample recent effort and some progress, it often remains unclear why they work in practice in optimising high-dimensional non-convex functions and why they find good minima instead of being trapped in spurious ones. Here we present a quantitative theory explaining this behaviour in a spiked matrix-tensor model. Our framework is based on the Kac-Rice analysis of stationary points and a closed-form analysis of gradient-flow originating from statistical physics. We show that there is a well defined region of parameters where the gradient-flow algorithm finds a good global minimum despite the presence of exponentially many spurious local minima. We show that this is achieved by surfing on saddles that have strong negative direction towards the global minima, a phenomenon that is connected to a BBP-type threshold in the Hessian describing the critical points of the landscapes.

Giulio Biroli, Chiara Cammarota, Federico Ricci-Tersenghi

In many high-dimensional estimation problems the main task consists in minimizing a cost function, which is often strongly non-convex when scanned in the space of parameters to be estimated. A standard solution to flatten the corresponding rough landscape consists in summing the losses associated to different data points and obtain a smoother empirical risk. Here we propose a complementary method that works for a single data point. The main idea is that a large amount of the roughness is uncorrelated in different parts of the landscape. One can then substantially reduce the noise by evaluating an empirical average of the gradient obtained as a sum over many random independent positions in the space of parameters to be optimized. We present an algorithm, called Averaged Gradient Descent, based on this idea and we apply it to tensor PCA, which is a very hard estimation problem. We show that Averaged Gradient Descent over-performs physical algorithms such as gradient descent and approximate message passing and matches the best algorithmic thresholds known so far, obtained by tensor unfolding and methods based on sum-of-squares.

M. Baity-Jesi, L. Sagun, M. Geiger et al.

We analyze numerically the training dynamics of deep neural networks (DNN) by using methods developed in statistical physics of glassy systems. The two main issues we address are (1) the complexity of the loss landscape and of the dynamics within it, and (2) to what extent DNNs share similarities with glassy systems. Our findings, obtained for different architectures and datasets, suggest that during the training process the dynamics slows down because of an increasingly large number of flat directions. At large times, when the loss is approaching zero, the system diffuses at the bottom of the landscape. Despite some similarities with the dynamics of mean-field glassy systems, in particular, the absence of barrier crossing, we find distinctive dynamical behaviors in the two cases, showing that the statistical properties of the corresponding loss and energy landscapes are different. In contrast, when the network is under-parametrized we observe a typical glassy behavior, thus suggesting the existence of different phases depending on whether the network is under-parametrized or over-parametrized.