A standard assumption in mean-field game (MFG) theory is that the coupling between the Hamilton-Jacobi equation and the transport equation is monotonically non-decreasing in the density of the population. In many cases, this assumption implies the existence and uniqueness of solutions. Here, we drop that assumption and construct explicit solutions for one-dimensional MFGs. These solutions exhibit phenomena not present in monotonically increasing MFGs: lowregularity, non-uniqueness, and the formation of regions with no agents.
Abstract-Here, we consider one-dimensional firstorder stationary mean-field games with congestion. These games arise when crowds face difficulty moving in highdensity regions. We look at both monotone decreasing and increasing interactions and construct explicit solutions using the current formulation. We observe new phenomena such as discontinuities, unhappiness traps and the nonexistence of solutions.
Stochastic gradient descent (SGD) for strongly convex functions converges at the rate O(1/k). However, achieving good results in practice requires tuning the parameters (for example the learning rate) of the algorithm.In this paper we propose a generalization of the Polyak step size, used for subgradient methods, to Stochastic gradient descent. We prove a non-asymptotic convergence at the rate O(1/k) with a rate constant which can be better than the corresponding rate constant for optimally scheduled SGD. We demonstrate that the method is effective in practice, and on convex optimization problems and on training deep neural networks, and compare to the theoretical rate.
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