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 title: Optimization - RDD-based API
 displayTitle: Optimization - RDD-based API
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 * Table of contents
 {:toc}
 
 `\[
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 ## Mathematical description
 
 ### Gradient descent
 The simplest method to solve optimization problems of the form `$\min_{\wv \in\R^d} \; f(\wv)$`
 is [gradient descent](http://en.wikipedia.org/wiki/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](http://en.wikipedia.org/wiki/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 <a
 href="mllib-classification-regression.html">supervised machine learning</a>,
 `\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 <a href="mllib-classification-regression.html">classification and regression</a> 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](api/scala/org/apache/spark/mllib/optimization/L1Updater.html).
 
 
 ### Update schemes for distributed SGD
 The SGD implementation in
 [GradientDescent](api/scala/org/apache/spark/mllib/optimization/GradientDescent.html) 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](rdd-programming-guide.html#resilient-distributed-datasets-rdds)), 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](http://en.wikipedia.org/wiki/Limited-memory_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](mllib-linear-methods.html) 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 
 <a href="mllib-linear-methods.html">linear methods</a> 
 section for example.
 
 The SGD class
 [GradientDescent](api/scala/org/apache/spark/mllib/optimization/GradientDescent.html)
 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](api/scala/org/apache/spark/mllib/classification/LogisticRegressionWithSGD.html).
 See the example below. It will be addressed in the next release. 
 
 The L1 regularization by using 
 [L1Updater](api/scala/org/apache/spark/mllib/optimization/L1Updater.html) 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](api/scala/org/apache/spark/mllib/optimization/LBFGS.html)
 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](https://github.com/scalanlp/breeze/blob/master/math/src/main/scala/breeze/optimize/LBFGS.scala).
 
 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. 
 
 <div class="codetabs">
 
 <div data-lang="scala" markdown="1">
 Refer to the [`LBFGS` Scala docs](api/scala/org/apache/spark/mllib/optimization/LBFGS.html) and [`SquaredL2Updater` Scala docs](api/scala/org/apache/spark/mllib/optimization/SquaredL2Updater.html) for details on the API.
 
 {% include_example scala/org/apache/spark/examples/mllib/LBFGSExample.scala %}
 </div>
 
 <div data-lang="java" markdown="1">
 Refer to the [`LBFGS` Java docs](api/java/org/apache/spark/mllib/optimization/LBFGS.html) and [`SquaredL2Updater` Java docs](api/java/org/apache/spark/mllib/optimization/SquaredL2Updater.html) for details on the API.
 
 {% include_example java/org/apache/spark/examples/mllib/JavaLBFGSExample.java %}
 </div>
 </div>
 
 ## 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. 

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