Optimization - RDD-based API

• Mathematical description
  • Gradient descent
  • Stochastic gradient descent (SGD)
  • Update schemes for distributed SGD
  • Limited-memory BFGS (L-BFGS)
  • Choosing an Optimization Method
• Implementation in MLlib
  • Gradient descent and stochastic gradient descent
  • L-BFGS
• Developer’s notes

\[ \newcommand{\R}{\mathbb{R}} \newcommand{\E}{\mathbb{E}} \newcommand{\x}{\mathbf{x}} \newcommand{\y}{\mathbf{y}} \newcommand{\wv}{\mathbf{w}} \newcommand{\av}{\mathbf{\alpha}} \newcommand{\bv}{\mathbf{b}} \newcommand{\N}{\mathbb{N}} \newcommand{\id}{\mathbf{I}} \newcommand{\ind}{\mathbf{1}} \newcommand{\0}{\mathbf{0}} \newcommand{\unit}{\mathbf{e}} \newcommand{\one}{\mathbf{1}} \newcommand{\zero}{\mathbf{0}} \]

Mathematical description

Gradient descent

The simplest method to solve optimization problems of the form $\min_{\wv \in\R^d} \; f(\wv)$ is gradient descent. Such first-order optimization methods (including gradient descent and stochastic variants thereof) are well-suited for large-scale and distributed computation.

Gradient descent methods aim to find a local minimum of a function by iteratively taking steps in the direction of steepest descent, which is the negative of the derivative (called the gradient) of the function at the current point, i.e., at the current parameter value. If the objective function $f$ is not differentiable at all arguments, but still convex, then a sub-gradient is the natural generalization of the gradient, and assumes the role of the step direction. In any case, computing a gradient or sub-gradient of $f$ is expensive — it requires a full pass through the complete dataset, in order to compute the contributions from all loss terms.

Stochastic gradient descent (SGD)

Optimization problems whose objective function $f$ is written as a sum are particularly suitable to be solved using stochastic gradient descent (SGD). In our case, for the optimization formulations commonly used in supervised machine learning, \begin{equation} f(\wv) := \lambda\, R(\wv) + \frac1n \sum_{i=1}^n L(\wv;\x_i,y_i) \label{eq:regPrimal} \ . \end{equation} this is especially natural, because the loss is written as an average of the individual losses coming from each datapoint.

A stochastic subgradient is a randomized choice of a vector, such that in expectation, we obtain a true subgradient of the original objective function. Picking one datapoint $i\in[1..n]$ uniformly at random, we obtain a stochastic subgradient of $\eqref{eq:regPrimal}$, with respect to $\wv$ as follows: \[ f'_{\wv,i} := L'_{\wv,i} + \lambda\, R'_\wv \ , \] where $L'_{\wv,i} \in \R^d$ is a subgradient of the part of the loss function determined by the $i$-th datapoint, that is $L'_{\wv,i} \in \frac{\partial}{\partial \wv} L(\wv;\x_i,y_i)$. Furthermore, $R'_\wv$ is a subgradient of the regularizer $R(\wv)$, i.e. $R'_\wv \in \frac{\partial}{\partial \wv} R(\wv)$. The term $R'_\wv$ does not depend on which random datapoint is picked. Clearly, in expectation over the random choice of $i\in[1..n]$, we have that $f'_{\wv,i}$ is a subgradient of the original objective $f$, meaning that $\E\left[f'_{\wv,i}\right] \in \frac{\partial}{\partial \wv} f(\wv)$.

Running SGD now simply becomes walking in the direction of the negative stochastic subgradient $f'_{\wv,i}$, that is \begin{equation}\label{eq:SGDupdate} \wv^{(t+1)} := \wv^{(t)} - \gamma \; f'_{\wv,i} \ . \end{equation} Step-size. The parameter $\gamma$ is the step-size, which in the default implementation is chosen decreasing with the square root of the iteration counter, i.e. $\gamma := \frac{s}{\sqrt{t}}$ in the $t$-th iteration, with the input parameter $s=$ stepSize. Note that selecting the best step-size for SGD methods can often be delicate in practice and is a topic of active research.

Gradients. A table of (sub)gradients of the machine learning methods implemented in spark.mllib, is available in the classification and regression section.

Proximal Updates. As an alternative to just use the subgradient $R'(\wv)$ of the regularizer in the step direction, an improved update for some cases can be obtained by using the proximal operator instead. For the L1-regularizer, the proximal operator is given by soft thresholding, as implemented in L1Updater.

Update schemes for distributed SGD

The SGD implementation in GradientDescent uses a simple (distributed) sampling of the data examples. We recall that the loss part of the optimization problem $\eqref{eq:regPrimal}$ is $\frac1n \sum_{i=1}^n L(\wv;\x_i,y_i)$, and therefore $\frac1n \sum_{i=1}^n L'_{\wv,i}$ would be the true (sub)gradient. Since this would require access to the full data set, the parameter miniBatchFraction specifies which fraction of the full data to use instead. The average of the gradients over this subset, i.e. \[ \frac1{|S|} \sum_{i\in S} L'_{\wv,i} \ , \] is a stochastic gradient. Here $S$ is the sampled subset of size $|S|=$ miniBatchFraction $\cdot n$.

In each iteration, the sampling over the distributed dataset (RDD), as well as the computation of the sum of the partial results from each worker machine is performed by the standard spark routines.

If the fraction of points miniBatchFraction is set to 1 (default), then the resulting step in each iteration is exact (sub)gradient descent. In this case, there is no randomness and no variance in the used step directions. On the other extreme, if miniBatchFraction is chosen very small, such that only a single point is sampled, i.e. $|S|=$ miniBatchFraction $\cdot n = 1$, then the algorithm is equivalent to standard SGD. In that case, the step direction depends from the uniformly random sampling of the point.

Limited-memory BFGS (L-BFGS)

L-BFGS is an optimization algorithm in the family of quasi-Newton methods to solve the optimization problems of the form $\min_{\wv \in\R^d} \; f(\wv)$. The L-BFGS method approximates the objective function locally as a quadratic without evaluating the second partial derivatives of the objective function to construct the Hessian matrix. The Hessian matrix is approximated by previous gradient evaluations, so there is no vertical scalability issue (the number of training features) when computing the Hessian matrix explicitly in Newton’s method. As a result, L-BFGS often achieves more rapid convergence compared with other first-order optimization.

Choosing an Optimization Method

Linear methods use optimization internally, and some linear methods in spark.mllib support both SGD and L-BFGS. Different optimization methods can have different convergence guarantees depending on the properties of the objective function, and we cannot cover the literature here. In general, when L-BFGS is available, we recommend using it instead of SGD since L-BFGS tends to converge faster (in fewer iterations).

Implementation in MLlib

Gradient descent and stochastic gradient descent

Gradient descent methods including stochastic subgradient descent (SGD) as included as a low-level primitive in MLlib, upon which various ML algorithms are developed, see the linear methods section for example.

The SGD class GradientDescent sets the following parameters:

• Gradient is a class that computes the stochastic gradient of the function being optimized, i.e., with respect to a single training example, at the current parameter value. MLlib includes gradient classes for common loss functions, e.g., hinge, logistic, least-squares. The gradient class takes as input a training example, its label, and the current parameter value.
• Updater is a class that performs the actual gradient descent step, i.e. updating the weights in each iteration, for a given gradient of the loss part. The updater is also responsible to perform the update from the regularization part. MLlib includes updaters for cases without regularization, as well as L1 and L2 regularizers.
• stepSize is a scalar value denoting the initial step size for gradient descent. All updaters in MLlib use a step size at the t-th step equal to stepSize $/ \sqrt{t}$.
• numIterations is the number of iterations to run.
• regParam is the regularization parameter when using L1 or L2 regularization.
• miniBatchFraction is the fraction of the total data that is sampled in each iteration, to compute the gradient direction.
  • Sampling still requires a pass over the entire RDD, so decreasing miniBatchFraction may not speed up optimization much. Users will see the greatest speedup when the gradient is expensive to compute, for only the chosen samples are used for computing the gradient.

L-BFGS

L-BFGS is currently only a low-level optimization primitive in MLlib. If you want to use L-BFGS in various ML algorithms such as Linear Regression, and Logistic Regression, you have to pass the gradient of objective function, and updater into optimizer yourself instead of using the training APIs like LogisticRegressionWithSGD. See the example below. It will be addressed in the next release.

The L1 regularization by using L1Updater will not work since the soft-thresholding logic in L1Updater is designed for gradient descent. See the developer’s note.

The L-BFGS method LBFGS.runLBFGS has the following parameters:

• Gradient is a class that computes the gradient of the objective function being optimized, i.e., with respect to a single training example, at the current parameter value. MLlib includes gradient classes for common loss functions, e.g., hinge, logistic, least-squares. The gradient class takes as input a training example, its label, and the current parameter value.
• Updater is a class that computes the gradient and loss of objective function of the regularization part for L-BFGS. MLlib includes updaters for cases without regularization, as well as L2 regularizer.
• numCorrections is the number of corrections used in the L-BFGS update. 10 is recommended.
• maxNumIterations is the maximal number of iterations that L-BFGS can be run.
• regParam is the regularization parameter when using regularization.
• convergenceTol controls how much relative change is still allowed when L-BFGS is considered to converge. This must be nonnegative. Lower values are less tolerant and therefore generally cause more iterations to be run. This value looks at both average improvement and the norm of gradient inside Breeze LBFGS.

The return is a tuple containing two elements. The first element is a column matrix containing weights for every feature, and the second element is an array containing the loss computed for every iteration.

Here is an example to train binary logistic regression with L2 regularization using L-BFGS optimizer.

Developer’s notes

Since the Hessian is constructed approximately from previous gradient evaluations, the objective function can not be changed during the optimization process. As a result, Stochastic L-BFGS will not work naively by just using miniBatch; therefore, we don’t provide this until we have better understanding.

Updater is a class originally designed for gradient decent which computes the actual gradient descent step. However, we’re able to take the gradient and loss of objective function of regularization for L-BFGS by ignoring the part of logic only for gradient decent such as adaptive step size stuff. We will refactorize this into regularizer to replace updater to separate the logic between regularization and step update later.