SGDOptim

A Julia package for Stochastic Gradient Descent (SGD) and its variants.

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With the advent of Big Data, Stochastic Gradient Descent (SGD) has become increasingly popular in recent years, especially in machine learning and related areas. This package implements the SGD algorithm and its variants under a generic setting to facilitate the use of SGD in practice.

Here is an example that demonstrates the use of this package in solving a ridge regression problem.

Optimization Algorithms

This package depends on EmpiricalRisks.jl, which provides the basic components, including predictors, loss functions, and regularizers.

On top of that, we provide a variety of algorithms, including SGD and its variants, and you may choose one that is suitable for your need:

For streaming settings:

• Stochastic Gradient Descent
• Accelerated Stochastic Gradient Descent
• Stochastic Proximal Gradient Descent

For distributed settings:

• Parallel Alternate Direction Methods of Multipliers (ADMM)
• ADMM with Variable Splitting

Learning rate:

The setting of the learning rate has significant impact on the algorithm's behavior. This package allows the learning rate setting to be provided as a function on t as a keyword argument.

The default setting is t -> 1.0 / (1.0 + t).

Key Functions

• sgd(rmodel, theta, stream; ...)

  Performs stochastic gradient descent to solve a (regularized) risk minimization problem.

  params	descriptions
  rmodel	the risk model, which can be constructed using riskmodel method.
  theta	The initial guess of the model parameter.
  stream	The input data stream. (See the Streams section below for details)

  This function also accepts keyword arguments:

  params	descriptions
  reg	the regularizer (default = ZeroReg(), means no regularization). See the documentation on regularizers for details.
  lrate	the learning rate rule, which should be a function of t (default as mentioned above).
  callback	the callback function, which will be invoked during iterations. default is simple_trace. See the Callbacks section below for detail.
  cbinterval	the interval of invoking the callback, i.e. the function invokes the callback every cbinterval iterations. (default is 0, meaning that it never invokes the callback).

Streams

Unlike conventional methods, SGD and its variants look at a single sample or a small batch of samples at each iteration. In other words, data are viewed as a stream of samples or minibatches.

This package provides a variety of ways to construct data streams. Each data stream is essentially an iterator that implements the start, done, and next methods (see [here]( <http://julia.readthedocs.org/en/latest/stdlib/collections/#iteration) for details of Julia's iteration patterns). Each item from a data stream can be either a sample (as a pair of input and output) or a mini-batch (as a pair of multi-input array and multi-output array).

Note: All SGD algorithms in this package support both sample streams and mini-batch streams. At each iteration, the algorithm works on a single item from the stream, which can be either a sample or a mini-batch.

The package provides several methods to construct streams of samples or minibatches.

• sample_seq(X, Y[, ord])

  Wrap an input array X and an output array Y into a stream of individual samples.

  Each item of the stream is a pair, comprised of an item from X and a corresponding item from Y. If X is a vector, then each item of X is a scalar, if X is a matrix, then each item of X is a column vector. The same applies to Y.

  The ord argument is an instance of AbstractVector that specifies the order in which the samples are scanned. If ord is omitted, it is, by default, set to the natural order, namely, 1:n, where n is the number of samples in the data set.

• minibatch_seq(X, Y, bsize[, ord])

  Wrap an input array X and an output array Y into a stream of mini-batches of size bsize or smaller.

  For example, if X and Y have 28 samples, by setting bsize to 10, we partition the data set into three minibatches, respectively corresponding to the indices 1:10, 11:20, and 21:28.

  The ord argument specifies the order in which the mini-batches are used. For example, if ord is set to [3, 2, 1], it first takes the 3rd batch, then 2nd, and finally 1st. If ord is omitted, it is, by default, set to the natural order, namely, 1:m, where m is the number of mini-batches.

Callbacks

The algorithms provided in this package interoperate with the rest of the world through callbacks. In particular, it allows a third party (e.g. a higher-level script, a user, a GUI, etc) to monitor the progress of the optimization and take proper actions.

Generally, a callback is an arbitrary function (or closure) that can be called in the following way:

callback(theta, t, n, v)

params	descriptions
theta	The current solution.
t	The number of elapsed iterations.
n	The number of samples that have been used.
v	The objective value of the last item, which can be an objective evaluated on a single sample or the total objective value evaluated on the last batch of samples.

The package already provides some callbacks for simple use:

• simple_trace

  Simply print the optimization trace, including the number of iterations, and the average loss of the last iteration.

  This is the default choice for most algorithms.

• gtcompare_trace(theta_g)

  In addition to printing the optimization trace, it also computes and shows the deviation from a given oracle theta_g.

  Note: gtcompare_trace is a high-level function, and gtcompare_trace(theta_g) produces a callback function.