Stochastic gradient descent
In classical statistics, sum-minimization problems arise in least squares and in maximum-likelihood estimation (for independent observations). The general class of estimators that arise as minimizers of sums are called M-estimators. However, in statistics, it has been long recognized that requiring even local minimization is too restrictive for some problems of maximum-likelihood estimation, as shown for example by Thomas Ferguson's example. Therefore, contemporary statistical theorists often consider stationary points of the likelihood function (or zeros of its derivative, the score function, and other estimating equations).
When used to minimize the above function, a standard (or "batch") gradient descent method would perform the following iterations :
where is a step size (sometimes called the learning rate in machine learning).
In many cases, the summand functions have a simple form that enables inexpensive evaluations of the sum-function and the sum gradient. For example, in statistics, one-parameter exponential families allow economical function-evaluations and gradient-evaluations.
However, in other cases, evaluating the sum-gradient may require expensive evaluations of the gradients from all summand functions. When the training set is enormous and no simple formulas exist, evaluating the sums of gradients becomes very expensive, because evaluating the gradient requires evaluating all the summand functions' gradients. To economize on the computational cost at every iteration, stochastic gradient descent samples a subset of summand functions at every step. This is very effective in the case of large-scale machine learning problems.
In stochastic (or "on-line") gradient descent, the true gradient of is approximated by a gradient at a single example:
As the algorithm sweeps through the training set, it performs the above update for each training example. Several passes over the training set are made until the algorithm converges. Typical implementations may also randomly shuffle training examples at each pass and use an adaptive learning rate.
In pseudocode, stochastic gradient descent with shuffling of training set at each pass can be presented as follows:
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.
The convergence of stochastic gradient descent has been analyzed using the theories of convex minimization and of stochastic approximation. Briefly, when the learning rates decrease with an appropriate rate, and subject to relatively mild assumptions, stochastic gradient descent converges almost surely to a global minimum when the objective function is convex or pseudoconvex, and otherwise converges almost surely to a local minimum.  This is in fact a consequence of the Robbins-Siegmund theorem.
Let's suppose we want to fit a straight line to a training set of two-dimensional points using least squares. The objective function to be minimized is:
The last line in the above pseudocode for this specific problem will become:
Stochastic gradient descent is a popular algorithm for training a wide range of models in machine learning, including (linear) support vector machines, logistic regression (see, e.g., Vowpal Wabbit) and graphical models. It competes with the L-BFGS algorithm, which is also widely used. SGD has been used since at least 1960 for training linear regression models, originally under the name ADALINE.
Another popular stochastic gradient descent algorithm is the least mean squares (LMS) adaptive filter.
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