Surfing on minima of isostatic landscapes: avalanches and unjamming transition

Abstract

Recently, we showed that optimization problems, both in infinite as well as in finite dimensions, for continuous variables and soft excluded volume constraints, can display entire isostatic phases where local minima of the cost function are marginally stable configurations endowed with non-linear excitations [, ]. In this work we describe an athermal adiabatic algorithm to explore with continuity the corresponding rough high-dimensional landscape. We concentrate on a prototype problem of this kind, the spherical perceptron optimization problem with linear cost function (hinge loss). This algorithm allows to ‘surf’ between isostatic marginally stable configurations and to investigate some properties of such landscape. In particular we focus on the statistics of avalanches occurring when local minima are destabilized. We show that when perturbing such minima, the system undergoes plastic rearrangements whose size is power law distributed and we characterize the corresponding critical exponent. Finally we investigate the critical properties of the unjamming transition, showing that the linear interaction potential gives rise to logarithmic behavior in the scaling of energy and pressure as a function of the distance from the unjamming point. For some quantities, the logarithmic corrections can be gauged out. This is the case of the number of soft constraints that are violated as a function of the distance from jamming which follows a non-trivial power law behavior.

Gradient descent dynamics and the jamming transition in infinite dimensions

“…A special class of such loss minimization problems is obtained whenever the loss function associated to a single data point is continuous and perfectly vanishing when the data point is correctly classified, and positive otherwise, e.g. the so-called squared hinge loss [18][19][20][21][22]. The question then becomes whether a choice of parameters θ exists, such that all data points in the training set are perfectly classified, leading to a zero loss.…”