saddle point

Optimization

A stationary point of a function — one where the gradientgradient, n.: A list of slopes showing which way, and how steeply, a quantity rises as each of its inputs changes. Training nudges an AI's settings in the opposite direction, downhill. is zero — that is neither a local minimum nor a local maximum, because the function curves upward along some directions and downward along others; for smooth functions the HessianHessian, n.: A table of 'slopes of slopes' that describes how a surface curves: whether a point is the bottom of a bowl, the top of a hill or a mountain pass. there has eigenvalues of both signs. In high-dimensional non-convex problems such as neural-network training, saddle points vastly outnumber local minimalocal minimum, n.: A spot lower than everything immediately around it, like the bottom of a small valley, even though a deeper valley may exist somewhere else., and an influential 2014 analysis argued that they and the plateaus around them, rather than bad minima, are the main cause of slow training. Plain gradient descentgradient descent, n.: A method for training models by repeatedly nudging every setting a small step in whichever direction most reduces the error, like a hiker who cannot see the valley but keeps stepping wherever the ground tilts down. can linger near saddles because gradients there are tiny, but noise from stochastic gradient descentstochastic gradient descent, n.: A faster version of gradient descent that estimates which way is downhill from a small random sample of the training data at each step, instead of checking the whole dataset. and momentummomentum, n.: A training trick that lets each update keep part of the previous update's direction, like a ball rolling downhill that builds up speed and coasts over small bumps. tends to push iterates off along descending directions.

The loss plateau was a saddle point; a little gradient noise was enough to escape it.

Game theory

A solution of a min–max problem: a point that is a minimum with respect to one player's variables and a maximum with respect to the other's, so neither can improve alone. Training a generative adversarial networkgenerative adversarial network, n.: Two neural networks trained as rivals: one forges fake images, the other tries to spot the fakes. As both improve, the forger's output becomes realistic enough to fool people. seeks such a point, and John von Neumann's 1928 minimax theorem guarantees one for finite two-player zero-sum gamesgame theory, n.: The mathematics of strategic decisions, where each player's best move depends on what the others do. It is used to study competition and cooperation among people, firms, nations and AI agents. in mixed strategies.