Stochastic gradient descent is a general optimization algorithm, but is typically used to fit the parameters of a machine learning model.

In standard (or "batch") gradient descent, the true gradient is used to update the parameters of the model. The true gradient is usually the sum of the gradients caused by each individual training example. The parameter vectors are adjusted by the negative of the true gradient multiplied by a step size. Therefore, batch gradient descent requires one sweep through the training set before any parameters can be changed.

In stochastic (or "on-line") gradient descent, the true gradient is approximated by the gradient of the cost function only evaluated on a single training example. The parameters are then adjusted by an amount proportional to this approximate gradient. Therefore, the parameters of the model are updated after each training example. For large data sets, on-line gradient descent can be much faster than batch gradient descent.

There is a compromise between the two forms, which is often called "mini-batches", where the true gradient is approximated by a sum over a small number of training examples.

Stochastic gradient descent is a form of stochastic approximation. The theory of stochastic approximations gives conditions on when stochastic gradient descent converges. If the step size scales as 1/T (where T is the number of gradient steps taken so far), then stochastic gradient descent is guaranteed to converge.

References

• Introduction to Stochastic Search and Optimization by James C. Spall, ISBN 0471330523, 2003
• Pattern Classification by Richard O. Duda, Peter E. Hart, David G. Stork, ISBN 0471056693, 2000