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| <h1><a href="../ml">FlinkML</a> - Alternating Least Squares</h1> |
| |
| |
| |
| <ul id="markdown-toc"> |
| <li><a href="#description" id="markdown-toc-description">Description</a></li> |
| <li><a href="#operations" id="markdown-toc-operations">Operations</a> <ul> |
| <li><a href="#fit" id="markdown-toc-fit">Fit</a></li> |
| <li><a href="#predict" id="markdown-toc-predict">Predict</a></li> |
| </ul> |
| </li> |
| <li><a href="#parameters" id="markdown-toc-parameters">Parameters</a></li> |
| <li><a href="#examples" id="markdown-toc-examples">Examples</a></li> |
| </ul> |
| |
| <h2 id="description">Description</h2> |
| |
| <p>The alternating least squares (ALS) algorithm factorizes a given matrix $R$ into two factors $U$ and $V$ such that $R \approx U^TV$. |
| The unknown row dimension is given as a parameter to the algorithm and is called latent factors. |
| Since matrix factorization can be used in the context of recommendation, the matrices $U$ and $V$ can be called user and item matrix, respectively. |
| The $i$th column of the user matrix is denoted by $u_i$ and the $i$th column of the item matrix is $v_i$. |
| The matrix $R$ can be called the ratings matrix with <script type="math/tex">(R)_{i,j} = r_{i,j}</script>.</p> |
| |
| <p>In order to find the user and item matrix, the following problem is solved:</p> |
| |
| <script type="math/tex; mode=display">\arg\min_{U,V} \sum_{\{i,j\mid r_{i,j} \not= 0\}} \left(r_{i,j} - u_{i}^Tv_{j}\right)^2 + |
| \lambda \left(\sum_{i} n_{u_i} \left\lVert u_i \right\rVert^2 + \sum_{j} n_{v_j} \left\lVert v_j \right\rVert^2 \right)</script> |
| |
| <p>with $\lambda$ being the regularization factor, <script type="math/tex">n_{u_i}</script> being the number of items the user $i$ has rated and <script type="math/tex">n_{v_j}</script> being the number of times the item $j$ has been rated. |
| This regularization scheme to avoid overfitting is called weighted-$\lambda$-regularization. |
| Details can be found in the work of <a href="http://dx.doi.org/10.1007/978-3-540-68880-8_32">Zhou et al.</a>.</p> |
| |
| <p>By fixing one of the matrices $U$ or $V$, we obtain a quadratic form which can be solved directly. |
| The solution of the modified problem is guaranteed to monotonically decrease the overall cost function. |
| By applying this step alternately to the matrices $U$ and $V$, we can iteratively improve the matrix factorization.</p> |
| |
| <p>The matrix $R$ is given in its sparse representation as a tuple of $(i, j, r)$ where $i$ denotes the row index, $j$ the column index and $r$ is the matrix value at position $(i,j)$.</p> |
| |
| <h2 id="operations">Operations</h2> |
| |
| <p><code>ALS</code> is a <code>Predictor</code>. |
| As such, it supports the <code>fit</code> and <code>predict</code> operation.</p> |
| |
| <h3 id="fit">Fit</h3> |
| |
| <p>ALS is trained on the sparse representation of the rating matrix:</p> |
| |
| <ul> |
| <li><code>fit: DataSet[(Int, Int, Double)] => Unit</code></li> |
| </ul> |
| |
| <h3 id="predict">Predict</h3> |
| |
| <p>ALS predicts for each tuple of row and column index the rating:</p> |
| |
| <ul> |
| <li><code>predict: DataSet[(Int, Int)] => DataSet[(Int, Int, Double)]</code></li> |
| </ul> |
| |
| <h2 id="parameters">Parameters</h2> |
| |
| <p>The alternating least squares implementation can be controlled by the following parameters:</p> |
| |
| <table class="table table-bordered"> |
| <thead> |
| <tr> |
| <th class="text-left" style="width: 20%">Parameters</th> |
| <th class="text-center">Description</th> |
| </tr> |
| </thead> |
| |
| <tbody> |
| <tr> |
| <td><strong>NumFactors</strong></td> |
| <td> |
| <p> |
| The number of latent factors to use for the underlying model. |
| It is equivalent to the dimension of the calculated user and item vectors. |
| (Default value: <strong>10</strong>) |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td><strong>Lambda</strong></td> |
| <td> |
| <p> |
| Regularization factor. Tune this value in order to avoid overfitting or poor performance due to strong generalization. |
| (Default value: <strong>1</strong>) |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td><strong>Iterations</strong></td> |
| <td> |
| <p> |
| The maximum number of iterations. |
| (Default value: <strong>10</strong>) |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td><strong>Blocks</strong></td> |
| <td> |
| <p> |
| The number of blocks into which the user and item matrix are grouped. |
| The fewer blocks one uses, the less data is sent redundantly. |
| However, bigger blocks entail bigger update messages which have to be stored on the heap. |
| If the algorithm fails because of an OutOfMemoryException, then try to increase the number of blocks. |
| (Default value: <strong>None</strong>) |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td><strong>Seed</strong></td> |
| <td> |
| <p> |
| Random seed used to generate the initial item matrix for the algorithm. |
| (Default value: <strong>0</strong>) |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td><strong>TemporaryPath</strong></td> |
| <td> |
| <p> |
| Path to a temporary directory into which intermediate results are stored. |
| If this value is set, then the algorithm is split into two preprocessing steps, the ALS iteration and a post-processing step which calculates a last ALS half-step. |
| The preprocessing steps calculate the <code>OutBlockInformation</code> and <code>InBlockInformation</code> for the given rating matrix. |
| The results of the individual steps are stored in the specified directory. |
| By splitting the algorithm into multiple smaller steps, Flink does not have to split the available memory amongst too many operators. |
| This allows the system to process bigger individual messages and improves the overall performance. |
| (Default value: <strong>None</strong>) |
| </p> |
| </td> |
| </tr> |
| </tbody> |
| </table> |
| |
| <h2 id="examples">Examples</h2> |
| |
| <div class="highlight"><pre><code class="language-scala" data-lang="scala"><span class="c1">// Read input data set from a csv file</span> |
| <span class="k">val</span> <span class="n">inputDS</span><span class="k">:</span> <span class="kt">DataSet</span><span class="o">[(</span><span class="kt">Int</span>, <span class="kt">Int</span>, <span class="kt">Double</span><span class="o">)]</span> <span class="k">=</span> <span class="n">env</span><span class="o">.</span><span class="n">readCsvFile</span><span class="o">[(</span><span class="kt">Int</span>, <span class="kt">Int</span>, <span class="kt">Double</span><span class="o">)](</span> |
| <span class="n">pathToTrainingFile</span><span class="o">)</span> |
| |
| <span class="c1">// Setup the ALS learner</span> |
| <span class="k">val</span> <span class="n">als</span> <span class="k">=</span> <span class="nc">ALS</span><span class="o">()</span> |
| <span class="o">.</span><span class="n">setIterations</span><span class="o">(</span><span class="mi">10</span><span class="o">)</span> |
| <span class="o">.</span><span class="n">setNumFactors</span><span class="o">(</span><span class="mi">10</span><span class="o">)</span> |
| <span class="o">.</span><span class="n">setBlocks</span><span class="o">(</span><span class="mi">100</span><span class="o">)</span> |
| <span class="o">.</span><span class="n">setTemporaryPath</span><span class="o">(</span><span class="s">"hdfs://tempPath"</span><span class="o">)</span> |
| |
| <span class="c1">// Set the other parameters via a parameter map</span> |
| <span class="k">val</span> <span class="n">parameters</span> <span class="k">=</span> <span class="nc">ParameterMap</span><span class="o">()</span> |
| <span class="o">.</span><span class="n">add</span><span class="o">(</span><span class="nc">ALS</span><span class="o">.</span><span class="nc">Lambda</span><span class="o">,</span> <span class="mf">0.9</span><span class="o">)</span> |
| <span class="o">.</span><span class="n">add</span><span class="o">(</span><span class="nc">ALS</span><span class="o">.</span><span class="nc">Seed</span><span class="o">,</span> <span class="mi">42L</span><span class="o">)</span> |
| |
| <span class="c1">// Calculate the factorization</span> |
| <span class="n">als</span><span class="o">.</span><span class="n">fit</span><span class="o">(</span><span class="n">inputDS</span><span class="o">,</span> <span class="n">parameters</span><span class="o">)</span> |
| |
| <span class="c1">// Read the testing data set from a csv file</span> |
| <span class="k">val</span> <span class="n">testingDS</span><span class="k">:</span> <span class="kt">DataSet</span><span class="o">[(</span><span class="kt">Int</span>, <span class="kt">Int</span><span class="o">)]</span> <span class="k">=</span> <span class="n">env</span><span class="o">.</span><span class="n">readCsvFile</span><span class="o">[(</span><span class="kt">Int</span>, <span class="kt">Int</span><span class="o">)](</span><span class="n">pathToData</span><span class="o">)</span> |
| |
| <span class="c1">// Calculate the ratings according to the matrix factorization</span> |
| <span class="k">val</span> <span class="n">predictedRatings</span> <span class="k">=</span> <span class="n">als</span><span class="o">.</span><span class="n">predict</span><span class="o">(</span><span class="n">testingDS</span><span class="o">)</span></code></pre></div> |
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