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| <h1 class="title">Optimization - RDD-based API</h1> |
| |
| |
| <ul id="markdown-toc"> |
| <li><a href="#mathematical-description" id="markdown-toc-mathematical-description">Mathematical description</a> <ul> |
| <li><a href="#gradient-descent" id="markdown-toc-gradient-descent">Gradient descent</a></li> |
| <li><a href="#stochastic-gradient-descent-sgd" id="markdown-toc-stochastic-gradient-descent-sgd">Stochastic gradient descent (SGD)</a></li> |
| <li><a href="#update-schemes-for-distributed-sgd" id="markdown-toc-update-schemes-for-distributed-sgd">Update schemes for distributed SGD</a></li> |
| <li><a href="#limited-memory-bfgs-l-bfgs" id="markdown-toc-limited-memory-bfgs-l-bfgs">Limited-memory BFGS (L-BFGS)</a></li> |
| <li><a href="#choosing-an-optimization-method" id="markdown-toc-choosing-an-optimization-method">Choosing an Optimization Method</a></li> |
| </ul> |
| </li> |
| <li><a href="#implementation-in-mllib" id="markdown-toc-implementation-in-mllib">Implementation in MLlib</a> <ul> |
| <li><a href="#gradient-descent-and-stochastic-gradient-descent" id="markdown-toc-gradient-descent-and-stochastic-gradient-descent">Gradient descent and stochastic gradient descent</a></li> |
| <li><a href="#l-bfgs" id="markdown-toc-l-bfgs">L-BFGS</a></li> |
| </ul> |
| </li> |
| <li><a href="#developers-notes" id="markdown-toc-developers-notes">Developer’s notes</a></li> |
| </ul> |
| |
| <p><code class="language-plaintext highlighter-rouge">\[ |
| \newcommand{\R}{\mathbb{R}} |
| \newcommand{\E}{\mathbb{E}} |
| \newcommand{\x}{\mathbf{x}} |
| \newcommand{\y}{\mathbf{y}} |
| \newcommand{\wv}{\mathbf{w}} |
| \newcommand{\av}{\mathbf{\alpha}} |
| \newcommand{\bv}{\mathbf{b}} |
| \newcommand{\N}{\mathbb{N}} |
| \newcommand{\id}{\mathbf{I}} |
| \newcommand{\ind}{\mathbf{1}} |
| \newcommand{\0}{\mathbf{0}} |
| \newcommand{\unit}{\mathbf{e}} |
| \newcommand{\one}{\mathbf{1}} |
| \newcommand{\zero}{\mathbf{0}} |
| \]</code></p> |
| |
| <h2 id="mathematical-description">Mathematical description</h2> |
| |
| <h3 id="gradient-descent">Gradient descent</h3> |
| <p>The simplest method to solve optimization problems of the form <code class="language-plaintext highlighter-rouge">$\min_{\wv \in\R^d} \; f(\wv)$</code> |
| is <a href="http://en.wikipedia.org/wiki/Gradient_descent">gradient descent</a>. |
| Such first-order optimization methods (including gradient descent and stochastic variants |
| thereof) are well-suited for large-scale and distributed computation.</p> |
| |
| <p>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 |
| <a href="http://en.wikipedia.org/wiki/Gradient">gradient</a>) of the function at the current point, i.e., at |
| the current parameter value. |
| If the objective function <code class="language-plaintext highlighter-rouge">$f$</code> is not differentiable at all arguments, but still convex, then a |
| <em>sub-gradient</em> |
| 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 <code class="language-plaintext highlighter-rouge">$f$</code> is expensive — it requires a full |
| pass through the complete dataset, in order to compute the contributions from all loss terms.</p> |
| |
| <h3 id="stochastic-gradient-descent-sgd">Stochastic gradient descent (SGD)</h3> |
| <p>Optimization problems whose objective function <code class="language-plaintext highlighter-rouge">$f$</code> is written as a sum are particularly |
| suitable to be solved using <em>stochastic gradient descent (SGD)</em>. |
| In our case, for the optimization formulations commonly used in <a href="mllib-classification-regression.html">supervised machine learning</a>, |
| <code class="language-plaintext highlighter-rouge">\begin{equation} |
| f(\wv) := |
| \lambda\, R(\wv) + |
| \frac1n \sum_{i=1}^n L(\wv;\x_i,y_i) |
| \label{eq:regPrimal} |
| \ . |
| \end{equation}</code> |
| this is especially natural, because the loss is written as an average of the individual losses |
| coming from each datapoint.</p> |
| |
| <p>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 <code class="language-plaintext highlighter-rouge">$i\in[1..n]$</code> uniformly at random, we obtain a stochastic subgradient of |
| <code class="language-plaintext highlighter-rouge">$\eqref{eq:regPrimal}$</code>, with respect to <code class="language-plaintext highlighter-rouge">$\wv$</code> as follows: |
| <code class="language-plaintext highlighter-rouge">\[ |
| f'_{\wv,i} := L'_{\wv,i} + \lambda\, R'_\wv \ , |
| \]</code> |
| where <code class="language-plaintext highlighter-rouge">$L'_{\wv,i} \in \R^d$</code> is a subgradient of the part of the loss function determined by the |
| <code class="language-plaintext highlighter-rouge">$i$</code>-th datapoint, that is <code class="language-plaintext highlighter-rouge">$L'_{\wv,i} \in \frac{\partial}{\partial \wv} L(\wv;\x_i,y_i)$</code>. |
| Furthermore, <code class="language-plaintext highlighter-rouge">$R'_\wv$</code> is a subgradient of the regularizer <code class="language-plaintext highlighter-rouge">$R(\wv)$</code>, i.e. <code class="language-plaintext highlighter-rouge">$R'_\wv \in |
| \frac{\partial}{\partial \wv} R(\wv)$</code>. The term <code class="language-plaintext highlighter-rouge">$R'_\wv$</code> does not depend on which random |
| datapoint is picked. |
| Clearly, in expectation over the random choice of <code class="language-plaintext highlighter-rouge">$i\in[1..n]$</code>, we have that <code class="language-plaintext highlighter-rouge">$f'_{\wv,i}$</code> is |
| a subgradient of the original objective <code class="language-plaintext highlighter-rouge">$f$</code>, meaning that <code class="language-plaintext highlighter-rouge">$\E\left[f'_{\wv,i}\right] \in |
| \frac{\partial}{\partial \wv} f(\wv)$</code>.</p> |
| |
| <p>Running SGD now simply becomes walking in the direction of the negative stochastic subgradient |
| <code class="language-plaintext highlighter-rouge">$f'_{\wv,i}$</code>, that is |
| <code class="language-plaintext highlighter-rouge">\begin{equation}\label{eq:SGDupdate} |
| \wv^{(t+1)} := \wv^{(t)} - \gamma \; f'_{\wv,i} \ . |
| \end{equation}</code> |
| <strong>Step-size.</strong> |
| The parameter <code class="language-plaintext highlighter-rouge">$\gamma$</code> is the step-size, which in the default implementation is chosen |
| decreasing with the square root of the iteration counter, i.e. <code class="language-plaintext highlighter-rouge">$\gamma := \frac{s}{\sqrt{t}}$</code> |
| in the <code class="language-plaintext highlighter-rouge">$t$</code>-th iteration, with the input parameter <code class="language-plaintext highlighter-rouge">$s=$ stepSize</code>. Note that selecting the best |
| step-size for SGD methods can often be delicate in practice and is a topic of active research.</p> |
| |
| <p><strong>Gradients.</strong> |
| A table of (sub)gradients of the machine learning methods implemented in <code class="language-plaintext highlighter-rouge">spark.mllib</code>, is available in |
| the <a href="mllib-classification-regression.html">classification and regression</a> section.</p> |
| |
| <p><strong>Proximal Updates.</strong> |
| As an alternative to just use the subgradient <code class="language-plaintext highlighter-rouge">$R'(\wv)$</code> 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 |
| <a href="api/scala/org/apache/spark/mllib/optimization/L1Updater.html">L1Updater</a>.</p> |
| |
| <h3 id="update-schemes-for-distributed-sgd">Update schemes for distributed SGD</h3> |
| <p>The SGD implementation in |
| <a href="api/scala/org/apache/spark/mllib/optimization/GradientDescent.html">GradientDescent</a> uses |
| a simple (distributed) sampling of the data examples. |
| We recall that the loss part of the optimization problem <code class="language-plaintext highlighter-rouge">$\eqref{eq:regPrimal}$</code> is |
| <code class="language-plaintext highlighter-rouge">$\frac1n \sum_{i=1}^n L(\wv;\x_i,y_i)$</code>, and therefore <code class="language-plaintext highlighter-rouge">$\frac1n \sum_{i=1}^n L'_{\wv,i}$</code> would |
| be the true (sub)gradient. |
| Since this would require access to the full data set, the parameter <code class="language-plaintext highlighter-rouge">miniBatchFraction</code> specifies |
| which fraction of the full data to use instead. |
| The average of the gradients over this subset, i.e. |
| <code class="language-plaintext highlighter-rouge">\[ |
| \frac1{|S|} \sum_{i\in S} L'_{\wv,i} \ , |
| \]</code> |
| is a stochastic gradient. Here <code class="language-plaintext highlighter-rouge">$S$</code> is the sampled subset of size <code class="language-plaintext highlighter-rouge">$|S|=$ miniBatchFraction |
| $\cdot n$</code>.</p> |
| |
| <p>In each iteration, the sampling over the distributed dataset |
| (<a href="rdd-programming-guide.html#resilient-distributed-datasets-rdds">RDD</a>), as well as the |
| computation of the sum of the partial results from each worker machine is performed by the |
| standard spark routines.</p> |
| |
| <p>If the fraction of points <code class="language-plaintext highlighter-rouge">miniBatchFraction</code> 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 <code class="language-plaintext highlighter-rouge">miniBatchFraction</code> is chosen very small, such that only a single point |
| is sampled, i.e. <code class="language-plaintext highlighter-rouge">$|S|=$ miniBatchFraction $\cdot n = 1$</code>, then the algorithm is equivalent to |
| standard SGD. In that case, the step direction depends from the uniformly random sampling of the |
| point.</p> |
| |
| <h3 id="limited-memory-bfgs-l-bfgs">Limited-memory BFGS (L-BFGS)</h3> |
| <p><a href="http://en.wikipedia.org/wiki/Limited-memory_BFGS">L-BFGS</a> is an optimization |
| algorithm in the family of quasi-Newton methods to solve the optimization problems of the form |
| <code class="language-plaintext highlighter-rouge">$\min_{\wv \in\R^d} \; f(\wv)$</code>. 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.</p> |
| |
| <h3 id="choosing-an-optimization-method">Choosing an Optimization Method</h3> |
| |
| <p><a href="mllib-linear-methods.html">Linear methods</a> use optimization internally, and some linear methods in <code class="language-plaintext highlighter-rouge">spark.mllib</code> 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).</p> |
| |
| <h2 id="implementation-in-mllib">Implementation in MLlib</h2> |
| |
| <h3 id="gradient-descent-and-stochastic-gradient-descent">Gradient descent and stochastic gradient descent</h3> |
| <p>Gradient descent methods including stochastic subgradient descent (SGD) as |
| included as a low-level primitive in <code class="language-plaintext highlighter-rouge">MLlib</code>, upon which various ML algorithms |
| are developed, see the |
| <a href="mllib-linear-methods.html">linear methods</a> |
| section for example.</p> |
| |
| <p>The SGD class |
| <a href="api/scala/org/apache/spark/mllib/optimization/GradientDescent.html">GradientDescent</a> |
| sets the following parameters:</p> |
| |
| <ul> |
| <li><code class="language-plaintext highlighter-rouge">Gradient</code> 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.</li> |
| <li><code class="language-plaintext highlighter-rouge">Updater</code> 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.</li> |
| <li><code class="language-plaintext highlighter-rouge">stepSize</code> 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 |
| <code class="language-plaintext highlighter-rouge">stepSize $/ \sqrt{t}$</code>.</li> |
| <li><code class="language-plaintext highlighter-rouge">numIterations</code> is the number of iterations to run.</li> |
| <li><code class="language-plaintext highlighter-rouge">regParam</code> is the regularization parameter when using L1 or L2 regularization.</li> |
| <li><code class="language-plaintext highlighter-rouge">miniBatchFraction</code> is the fraction of the total data that is sampled in |
| each iteration, to compute the gradient direction. |
| <ul> |
| <li>Sampling still requires a pass over the entire RDD, so decreasing <code class="language-plaintext highlighter-rouge">miniBatchFraction</code> 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.</li> |
| </ul> |
| </li> |
| </ul> |
| |
| <h3 id="l-bfgs">L-BFGS</h3> |
| <p>L-BFGS is currently only a low-level optimization primitive in <code class="language-plaintext highlighter-rouge">MLlib</code>. 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 |
| <a href="api/scala/org/apache/spark/mllib/classification/LogisticRegressionWithSGD.html">LogisticRegressionWithSGD</a>. |
| See the example below. It will be addressed in the next release.</p> |
| |
| <p>The L1 regularization by using |
| <a href="api/scala/org/apache/spark/mllib/optimization/L1Updater.html">L1Updater</a> will not work since the |
| soft-thresholding logic in L1Updater is designed for gradient descent. See the developer’s note.</p> |
| |
| <p>The L-BFGS method |
| <a href="api/scala/org/apache/spark/mllib/optimization/LBFGS.html">LBFGS.runLBFGS</a> |
| has the following parameters:</p> |
| |
| <ul> |
| <li><code class="language-plaintext highlighter-rouge">Gradient</code> 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.</li> |
| <li><code class="language-plaintext highlighter-rouge">Updater</code> 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.</li> |
| <li><code class="language-plaintext highlighter-rouge">numCorrections</code> is the number of corrections used in the L-BFGS update. 10 is |
| recommended.</li> |
| <li><code class="language-plaintext highlighter-rouge">maxNumIterations</code> is the maximal number of iterations that L-BFGS can be run.</li> |
| <li><code class="language-plaintext highlighter-rouge">regParam</code> is the regularization parameter when using regularization.</li> |
| <li><code class="language-plaintext highlighter-rouge">convergenceTol</code> 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 <a href="https://github.com/scalanlp/breeze/blob/master/math/src/main/scala/breeze/optimize/LBFGS.scala">Breeze LBFGS</a>.</li> |
| </ul> |
| |
| <p>The <code class="language-plaintext highlighter-rouge">return</code> 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.</p> |
| |
| <p>Here is an example to train binary logistic regression with L2 regularization using |
| L-BFGS optimizer.</p> |
| |
| <div class="codetabs"> |
| |
| <div data-lang="scala"> |
| <p>Refer to the <a href="api/scala/org/apache/spark/mllib/optimization/LBFGS.html"><code class="language-plaintext highlighter-rouge">LBFGS</code> Scala docs</a> and <a href="api/scala/org/apache/spark/mllib/optimization/SquaredL2Updater.html"><code class="language-plaintext highlighter-rouge">SquaredL2Updater</code> Scala docs</a> for details on the API.</p> |
| |
| <div class="highlight"><pre class="codehilite"><code><span class="k">import</span> <span class="nn">org.apache.spark.mllib.classification.LogisticRegressionModel</span> |
| <span class="k">import</span> <span class="nn">org.apache.spark.mllib.evaluation.BinaryClassificationMetrics</span> |
| <span class="k">import</span> <span class="nn">org.apache.spark.mllib.linalg.Vectors</span> |
| <span class="k">import</span> <span class="nn">org.apache.spark.mllib.optimization.</span><span class="o">{</span><span class="nc">LBFGS</span><span class="o">,</span> <span class="nc">LogisticGradient</span><span class="o">,</span> <span class="nc">SquaredL2Updater</span><span class="o">}</span> |
| <span class="k">import</span> <span class="nn">org.apache.spark.mllib.util.MLUtils</span> |
| |
| <span class="k">val</span> <span class="nv">data</span> <span class="k">=</span> <span class="nv">MLUtils</span><span class="o">.</span><span class="py">loadLibSVMFile</span><span class="o">(</span><span class="n">sc</span><span class="o">,</span> <span class="s">"data/mllib/sample_libsvm_data.txt"</span><span class="o">)</span> |
| <span class="k">val</span> <span class="nv">numFeatures</span> <span class="k">=</span> <span class="nv">data</span><span class="o">.</span><span class="py">take</span><span class="o">(</span><span class="mi">1</span><span class="o">)(</span><span class="mi">0</span><span class="o">).</span><span class="py">features</span><span class="o">.</span><span class="py">size</span> |
| |
| <span class="c1">// Split data into training (60%) and test (40%).</span> |
| <span class="k">val</span> <span class="nv">splits</span> <span class="k">=</span> <span class="nv">data</span><span class="o">.</span><span class="py">randomSplit</span><span class="o">(</span><span class="nc">Array</span><span class="o">(</span><span class="mf">0.6</span><span class="o">,</span> <span class="mf">0.4</span><span class="o">),</span> <span class="n">seed</span> <span class="k">=</span> <span class="mi">11L</span><span class="o">)</span> |
| |
| <span class="c1">// Append 1 into the training data as intercept.</span> |
| <span class="k">val</span> <span class="nv">training</span> <span class="k">=</span> <span class="nf">splits</span><span class="o">(</span><span class="mi">0</span><span class="o">).</span><span class="py">map</span><span class="o">(</span><span class="n">x</span> <span class="k">=></span> <span class="o">(</span><span class="nv">x</span><span class="o">.</span><span class="py">label</span><span class="o">,</span> <span class="nv">MLUtils</span><span class="o">.</span><span class="py">appendBias</span><span class="o">(</span><span class="nv">x</span><span class="o">.</span><span class="py">features</span><span class="o">))).</span><span class="py">cache</span><span class="o">()</span> |
| |
| <span class="k">val</span> <span class="nv">test</span> <span class="k">=</span> <span class="nf">splits</span><span class="o">(</span><span class="mi">1</span><span class="o">)</span> |
| |
| <span class="c1">// Run training algorithm to build the model</span> |
| <span class="k">val</span> <span class="nv">numCorrections</span> <span class="k">=</span> <span class="mi">10</span> |
| <span class="k">val</span> <span class="nv">convergenceTol</span> <span class="k">=</span> <span class="mi">1</span><span class="n">e</span><span class="o">-</span><span class="mi">4</span> |
| <span class="k">val</span> <span class="nv">maxNumIterations</span> <span class="k">=</span> <span class="mi">20</span> |
| <span class="k">val</span> <span class="nv">regParam</span> <span class="k">=</span> <span class="mf">0.1</span> |
| <span class="k">val</span> <span class="nv">initialWeightsWithIntercept</span> <span class="k">=</span> <span class="nv">Vectors</span><span class="o">.</span><span class="py">dense</span><span class="o">(</span><span class="k">new</span> <span class="nc">Array</span><span class="o">[</span><span class="kt">Double</span><span class="o">](</span><span class="n">numFeatures</span> <span class="o">+</span> <span class="mi">1</span><span class="o">))</span> |
| |
| <span class="nf">val</span> <span class="o">(</span><span class="n">weightsWithIntercept</span><span class="o">,</span> <span class="n">loss</span><span class="o">)</span> <span class="k">=</span> <span class="nv">LBFGS</span><span class="o">.</span><span class="py">runLBFGS</span><span class="o">(</span> |
| <span class="n">training</span><span class="o">,</span> |
| <span class="k">new</span> <span class="nc">LogisticGradient</span><span class="o">(),</span> |
| <span class="k">new</span> <span class="nc">SquaredL2Updater</span><span class="o">(),</span> |
| <span class="n">numCorrections</span><span class="o">,</span> |
| <span class="n">convergenceTol</span><span class="o">,</span> |
| <span class="n">maxNumIterations</span><span class="o">,</span> |
| <span class="n">regParam</span><span class="o">,</span> |
| <span class="n">initialWeightsWithIntercept</span><span class="o">)</span> |
| |
| <span class="k">val</span> <span class="nv">model</span> <span class="k">=</span> <span class="k">new</span> <span class="nc">LogisticRegressionModel</span><span class="o">(</span> |
| <span class="nv">Vectors</span><span class="o">.</span><span class="py">dense</span><span class="o">(</span><span class="nv">weightsWithIntercept</span><span class="o">.</span><span class="py">toArray</span><span class="o">.</span><span class="py">slice</span><span class="o">(</span><span class="mi">0</span><span class="o">,</span> <span class="nv">weightsWithIntercept</span><span class="o">.</span><span class="py">size</span> <span class="o">-</span> <span class="mi">1</span><span class="o">)),</span> |
| <span class="nf">weightsWithIntercept</span><span class="o">(</span><span class="nv">weightsWithIntercept</span><span class="o">.</span><span class="py">size</span> <span class="o">-</span> <span class="mi">1</span><span class="o">))</span> |
| |
| <span class="c1">// Clear the default threshold.</span> |
| <span class="nv">model</span><span class="o">.</span><span class="py">clearThreshold</span><span class="o">()</span> |
| |
| <span class="c1">// Compute raw scores on the test set.</span> |
| <span class="k">val</span> <span class="nv">scoreAndLabels</span> <span class="k">=</span> <span class="nv">test</span><span class="o">.</span><span class="py">map</span> <span class="o">{</span> <span class="n">point</span> <span class="k">=></span> |
| <span class="k">val</span> <span class="nv">score</span> <span class="k">=</span> <span class="nv">model</span><span class="o">.</span><span class="py">predict</span><span class="o">(</span><span class="nv">point</span><span class="o">.</span><span class="py">features</span><span class="o">)</span> |
| <span class="o">(</span><span class="n">score</span><span class="o">,</span> <span class="nv">point</span><span class="o">.</span><span class="py">label</span><span class="o">)</span> |
| <span class="o">}</span> |
| |
| <span class="c1">// Get evaluation metrics.</span> |
| <span class="k">val</span> <span class="nv">metrics</span> <span class="k">=</span> <span class="k">new</span> <span class="nc">BinaryClassificationMetrics</span><span class="o">(</span><span class="n">scoreAndLabels</span><span class="o">)</span> |
| <span class="k">val</span> <span class="nv">auROC</span> <span class="k">=</span> <span class="nv">metrics</span><span class="o">.</span><span class="py">areaUnderROC</span><span class="o">()</span> |
| |
| <span class="nf">println</span><span class="o">(</span><span class="s">"Loss of each step in training process"</span><span class="o">)</span> |
| <span class="nv">loss</span><span class="o">.</span><span class="py">foreach</span><span class="o">(</span><span class="n">println</span><span class="o">)</span> |
| <span class="nf">println</span><span class="o">(</span><span class="n">s</span><span class="s">"Area under ROC = $auROC"</span><span class="o">)</span></code></pre></div> |
| <div><small>Find full example code at "examples/src/main/scala/org/apache/spark/examples/mllib/LBFGSExample.scala" in the Spark repo.</small></div> |
| </div> |
| |
| <div data-lang="java"> |
| <p>Refer to the <a href="api/java/org/apache/spark/mllib/optimization/LBFGS.html"><code class="language-plaintext highlighter-rouge">LBFGS</code> Java docs</a> and <a href="api/java/org/apache/spark/mllib/optimization/SquaredL2Updater.html"><code class="language-plaintext highlighter-rouge">SquaredL2Updater</code> Java docs</a> for details on the API.</p> |
| |
| <div class="highlight"><pre class="codehilite"><code><span class="kn">import</span> <span class="nn">java.util.Arrays</span><span class="o">;</span> |
| |
| <span class="kn">import</span> <span class="nn">scala.Tuple2</span><span class="o">;</span> |
| |
| <span class="kn">import</span> <span class="nn">org.apache.spark.api.java.*</span><span class="o">;</span> |
| <span class="kn">import</span> <span class="nn">org.apache.spark.mllib.classification.LogisticRegressionModel</span><span class="o">;</span> |
| <span class="kn">import</span> <span class="nn">org.apache.spark.mllib.evaluation.BinaryClassificationMetrics</span><span class="o">;</span> |
| <span class="kn">import</span> <span class="nn">org.apache.spark.mllib.linalg.Vector</span><span class="o">;</span> |
| <span class="kn">import</span> <span class="nn">org.apache.spark.mllib.linalg.Vectors</span><span class="o">;</span> |
| <span class="kn">import</span> <span class="nn">org.apache.spark.mllib.optimization.*</span><span class="o">;</span> |
| <span class="kn">import</span> <span class="nn">org.apache.spark.mllib.regression.LabeledPoint</span><span class="o">;</span> |
| <span class="kn">import</span> <span class="nn">org.apache.spark.mllib.util.MLUtils</span><span class="o">;</span> |
| <span class="kn">import</span> <span class="nn">org.apache.spark.SparkConf</span><span class="o">;</span> |
| <span class="kn">import</span> <span class="nn">org.apache.spark.SparkContext</span><span class="o">;</span> |
| |
| <span class="nc">String</span> <span class="n">path</span> <span class="o">=</span> <span class="s">"data/mllib/sample_libsvm_data.txt"</span><span class="o">;</span> |
| <span class="nc">JavaRDD</span><span class="o"><</span><span class="nc">LabeledPoint</span><span class="o">></span> <span class="n">data</span> <span class="o">=</span> <span class="nc">MLUtils</span><span class="o">.</span><span class="na">loadLibSVMFile</span><span class="o">(</span><span class="n">sc</span><span class="o">,</span> <span class="n">path</span><span class="o">).</span><span class="na">toJavaRDD</span><span class="o">();</span> |
| <span class="kt">int</span> <span class="n">numFeatures</span> <span class="o">=</span> <span class="n">data</span><span class="o">.</span><span class="na">take</span><span class="o">(</span><span class="mi">1</span><span class="o">).</span><span class="na">get</span><span class="o">(</span><span class="mi">0</span><span class="o">).</span><span class="na">features</span><span class="o">().</span><span class="na">size</span><span class="o">();</span> |
| |
| <span class="c1">// Split initial RDD into two... [60% training data, 40% testing data].</span> |
| <span class="nc">JavaRDD</span><span class="o"><</span><span class="nc">LabeledPoint</span><span class="o">></span> <span class="n">trainingInit</span> <span class="o">=</span> <span class="n">data</span><span class="o">.</span><span class="na">sample</span><span class="o">(</span><span class="kc">false</span><span class="o">,</span> <span class="mf">0.6</span><span class="o">,</span> <span class="mi">11L</span><span class="o">);</span> |
| <span class="nc">JavaRDD</span><span class="o"><</span><span class="nc">LabeledPoint</span><span class="o">></span> <span class="n">test</span> <span class="o">=</span> <span class="n">data</span><span class="o">.</span><span class="na">subtract</span><span class="o">(</span><span class="n">trainingInit</span><span class="o">);</span> |
| |
| <span class="c1">// Append 1 into the training data as intercept.</span> |
| <span class="nc">JavaPairRDD</span><span class="o"><</span><span class="nc">Object</span><span class="o">,</span> <span class="nc">Vector</span><span class="o">></span> <span class="n">training</span> <span class="o">=</span> <span class="n">data</span><span class="o">.</span><span class="na">mapToPair</span><span class="o">(</span><span class="n">p</span> <span class="o">-></span> |
| <span class="k">new</span> <span class="nc">Tuple2</span><span class="o"><>(</span><span class="n">p</span><span class="o">.</span><span class="na">label</span><span class="o">(),</span> <span class="nc">MLUtils</span><span class="o">.</span><span class="na">appendBias</span><span class="o">(</span><span class="n">p</span><span class="o">.</span><span class="na">features</span><span class="o">())));</span> |
| <span class="n">training</span><span class="o">.</span><span class="na">cache</span><span class="o">();</span> |
| |
| <span class="c1">// Run training algorithm to build the model.</span> |
| <span class="kt">int</span> <span class="n">numCorrections</span> <span class="o">=</span> <span class="mi">10</span><span class="o">;</span> |
| <span class="kt">double</span> <span class="n">convergenceTol</span> <span class="o">=</span> <span class="mi">1</span><span class="n">e</span><span class="o">-</span><span class="mi">4</span><span class="o">;</span> |
| <span class="kt">int</span> <span class="n">maxNumIterations</span> <span class="o">=</span> <span class="mi">20</span><span class="o">;</span> |
| <span class="kt">double</span> <span class="n">regParam</span> <span class="o">=</span> <span class="mf">0.1</span><span class="o">;</span> |
| <span class="nc">Vector</span> <span class="n">initialWeightsWithIntercept</span> <span class="o">=</span> <span class="nc">Vectors</span><span class="o">.</span><span class="na">dense</span><span class="o">(</span><span class="k">new</span> <span class="kt">double</span><span class="o">[</span><span class="n">numFeatures</span> <span class="o">+</span> <span class="mi">1</span><span class="o">]);</span> |
| |
| <span class="nc">Tuple2</span><span class="o"><</span><span class="nc">Vector</span><span class="o">,</span> <span class="kt">double</span><span class="o">[]></span> <span class="n">result</span> <span class="o">=</span> <span class="no">LBFGS</span><span class="o">.</span><span class="na">runLBFGS</span><span class="o">(</span> |
| <span class="n">training</span><span class="o">.</span><span class="na">rdd</span><span class="o">(),</span> |
| <span class="k">new</span> <span class="nf">LogisticGradient</span><span class="o">(),</span> |
| <span class="k">new</span> <span class="nf">SquaredL2Updater</span><span class="o">(),</span> |
| <span class="n">numCorrections</span><span class="o">,</span> |
| <span class="n">convergenceTol</span><span class="o">,</span> |
| <span class="n">maxNumIterations</span><span class="o">,</span> |
| <span class="n">regParam</span><span class="o">,</span> |
| <span class="n">initialWeightsWithIntercept</span><span class="o">);</span> |
| <span class="nc">Vector</span> <span class="n">weightsWithIntercept</span> <span class="o">=</span> <span class="n">result</span><span class="o">.</span><span class="na">_1</span><span class="o">();</span> |
| <span class="kt">double</span><span class="o">[]</span> <span class="n">loss</span> <span class="o">=</span> <span class="n">result</span><span class="o">.</span><span class="na">_2</span><span class="o">();</span> |
| |
| <span class="nc">LogisticRegressionModel</span> <span class="n">model</span> <span class="o">=</span> <span class="k">new</span> <span class="nc">LogisticRegressionModel</span><span class="o">(</span> |
| <span class="nc">Vectors</span><span class="o">.</span><span class="na">dense</span><span class="o">(</span><span class="nc">Arrays</span><span class="o">.</span><span class="na">copyOf</span><span class="o">(</span><span class="n">weightsWithIntercept</span><span class="o">.</span><span class="na">toArray</span><span class="o">(),</span> <span class="n">weightsWithIntercept</span><span class="o">.</span><span class="na">size</span><span class="o">()</span> <span class="o">-</span> <span class="mi">1</span><span class="o">)),</span> |
| <span class="o">(</span><span class="n">weightsWithIntercept</span><span class="o">.</span><span class="na">toArray</span><span class="o">())[</span><span class="n">weightsWithIntercept</span><span class="o">.</span><span class="na">size</span><span class="o">()</span> <span class="o">-</span> <span class="mi">1</span><span class="o">]);</span> |
| |
| <span class="c1">// Clear the default threshold.</span> |
| <span class="n">model</span><span class="o">.</span><span class="na">clearThreshold</span><span class="o">();</span> |
| |
| <span class="c1">// Compute raw scores on the test set.</span> |
| <span class="nc">JavaPairRDD</span><span class="o"><</span><span class="nc">Object</span><span class="o">,</span> <span class="nc">Object</span><span class="o">></span> <span class="n">scoreAndLabels</span> <span class="o">=</span> <span class="n">test</span><span class="o">.</span><span class="na">mapToPair</span><span class="o">(</span><span class="n">p</span> <span class="o">-></span> |
| <span class="k">new</span> <span class="nc">Tuple2</span><span class="o"><>(</span><span class="n">model</span><span class="o">.</span><span class="na">predict</span><span class="o">(</span><span class="n">p</span><span class="o">.</span><span class="na">features</span><span class="o">()),</span> <span class="n">p</span><span class="o">.</span><span class="na">label</span><span class="o">()));</span> |
| |
| <span class="c1">// Get evaluation metrics.</span> |
| <span class="nc">BinaryClassificationMetrics</span> <span class="n">metrics</span> <span class="o">=</span> |
| <span class="k">new</span> <span class="nf">BinaryClassificationMetrics</span><span class="o">(</span><span class="n">scoreAndLabels</span><span class="o">.</span><span class="na">rdd</span><span class="o">());</span> |
| <span class="kt">double</span> <span class="n">auROC</span> <span class="o">=</span> <span class="n">metrics</span><span class="o">.</span><span class="na">areaUnderROC</span><span class="o">();</span> |
| |
| <span class="nc">System</span><span class="o">.</span><span class="na">out</span><span class="o">.</span><span class="na">println</span><span class="o">(</span><span class="s">"Loss of each step in training process"</span><span class="o">);</span> |
| <span class="k">for</span> <span class="o">(</span><span class="kt">double</span> <span class="n">l</span> <span class="o">:</span> <span class="n">loss</span><span class="o">)</span> <span class="o">{</span> |
| <span class="nc">System</span><span class="o">.</span><span class="na">out</span><span class="o">.</span><span class="na">println</span><span class="o">(</span><span class="n">l</span><span class="o">);</span> |
| <span class="o">}</span> |
| <span class="nc">System</span><span class="o">.</span><span class="na">out</span><span class="o">.</span><span class="na">println</span><span class="o">(</span><span class="s">"Area under ROC = "</span> <span class="o">+</span> <span class="n">auROC</span><span class="o">);</span></code></pre></div> |
| <div><small>Find full example code at "examples/src/main/java/org/apache/spark/examples/mllib/JavaLBFGSExample.java" in the Spark repo.</small></div> |
| </div> |
| </div> |
| |
| <h2 id="developers-notes">Developer’s notes</h2> |
| |
| <p>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.</p> |
| |
| <p><code class="language-plaintext highlighter-rouge">Updater</code> 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.</p> |
| |
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