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<div class="section" id="matrix-factorization">
<span id="matrix-factorization"></span><h1>Matrix Factorization<a class="headerlink" href="#matrix-factorization" title="Permalink to this headline"></a></h1>
<p>In a recommendation system, there is a group of users and a set of items. Given
that each users have rated some items in the system, we would like to predict
how the users would rate the items that they have not yet rated, such that we
can make recommendations to the users.</p>
<p>Matrix factorization is one of the mainly used algorithm in recommendation
systems. It can be used to discover latent features underlying the interactions
between two different kinds of entities.</p>
<p>Assume we assign a k-dimensional vector to each user and a k-dimensional vector
to each item such that the dot product of these two vectors gives the user’s
rating of that item. We can learn the user and item vectors directly, which is
essentially performing SVD on the user-item matrix. We can also try to learn the
latent features using multi-layer neural networks.</p>
<p>In this tutorial, we will work though the steps to implement these ideas in
MXNet.</p>
<div class="section" id="prepare-data">
<span id="prepare-data"></span><h2>Prepare Data<a class="headerlink" href="#prepare-data" title="Permalink to this headline"></a></h2>
<p>We use the <a class="reference external" href="http://grouplens.org/datasets/movielens/">MovieLens</a> data here, but
it can apply to other datasets as well. Each row of this dataset contains a
tuple of user id, movie id, rating, and time stamp, we will only use the first
three items. We first define the a batch which contains n tuples. It also
provides name and shape information to MXNet about the data and label.</p>
<div class="highlight-python"><div class="highlight"><pre><span></span><span class="k">class</span> <span class="nc">Batch</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
<span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">data_names</span><span class="p">,</span> <span class="n">data</span><span class="p">,</span> <span class="n">label_names</span><span class="p">,</span> <span class="n">label</span><span class="p">):</span>
<span class="bp">self</span><span class="o">.</span><span class="n">data</span> <span class="o">=</span> <span class="n">data</span>
<span class="bp">self</span><span class="o">.</span><span class="n">label</span> <span class="o">=</span> <span class="n">label</span>
<span class="bp">self</span><span class="o">.</span><span class="n">data_names</span> <span class="o">=</span> <span class="n">data_names</span>
<span class="bp">self</span><span class="o">.</span><span class="n">label_names</span> <span class="o">=</span> <span class="n">label_names</span>
<span class="nd">@property</span>
<span class="k">def</span> <span class="nf">provide_data</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="k">return</span> <span class="p">[(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span> <span class="k">for</span> <span class="n">n</span><span class="p">,</span> <span class="n">x</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">data_names</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">data</span><span class="p">)]</span>
<span class="nd">@property</span>
<span class="k">def</span> <span class="nf">provide_label</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="k">return</span> <span class="p">[(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span> <span class="k">for</span> <span class="n">n</span><span class="p">,</span> <span class="n">x</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">label_names</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">label</span><span class="p">)]</span>
</pre></div>
</div>
<p>Then we define a data iterator, which returns a batch of tuples each time.</p>
<div class="highlight-python"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">mxnet</span> <span class="kn">as</span> <span class="nn">mx</span>
<span class="kn">import</span> <span class="nn">random</span>
<span class="k">class</span> <span class="nc">Batch</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
<span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">data_names</span><span class="p">,</span> <span class="n">data</span><span class="p">,</span> <span class="n">label_names</span><span class="p">,</span> <span class="n">label</span><span class="p">):</span>
<span class="bp">self</span><span class="o">.</span><span class="n">data</span> <span class="o">=</span> <span class="n">data</span>
<span class="bp">self</span><span class="o">.</span><span class="n">label</span> <span class="o">=</span> <span class="n">label</span>
<span class="bp">self</span><span class="o">.</span><span class="n">data_names</span> <span class="o">=</span> <span class="n">data_names</span>
<span class="bp">self</span><span class="o">.</span><span class="n">label_names</span> <span class="o">=</span> <span class="n">label_names</span>
<span class="nd">@property</span>
<span class="k">def</span> <span class="nf">provide_data</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="k">return</span> <span class="p">[(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span> <span class="k">for</span> <span class="n">n</span><span class="p">,</span> <span class="n">x</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">data_names</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">data</span><span class="p">)]</span>
<span class="nd">@property</span>
<span class="k">def</span> <span class="nf">provide_label</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="k">return</span> <span class="p">[(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span> <span class="k">for</span> <span class="n">n</span><span class="p">,</span> <span class="n">x</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">label_names</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">label</span><span class="p">)]</span>
<span class="k">class</span> <span class="nc">DataIter</span><span class="p">(</span><span class="n">mx</span><span class="o">.</span><span class="n">io</span><span class="o">.</span><span class="n">DataIter</span><span class="p">):</span>
<span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">fname</span><span class="p">,</span> <span class="n">batch_size</span><span class="p">):</span>
<span class="nb">super</span><span class="p">(</span><span class="n">DataIter</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
<span class="bp">self</span><span class="o">.</span><span class="n">batch_size</span> <span class="o">=</span> <span class="n">batch_size</span>
<span class="bp">self</span><span class="o">.</span><span class="n">data</span> <span class="o">=</span> <span class="p">[]</span>
<span class="k">for</span> <span class="n">line</span> <span class="ow">in</span> <span class="nb">file</span><span class="p">(</span><span class="n">fname</span><span class="p">):</span>
<span class="n">tks</span> <span class="o">=</span> <span class="n">line</span><span class="o">.</span><span class="n">strip</span><span class="p">()</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="s1">'</span><span class="se">\t</span><span class="s1">'</span><span class="p">)</span>
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">tks</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">4</span><span class="p">:</span>
<span class="k">continue</span>
<span class="bp">self</span><span class="o">.</span><span class="n">data</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="nb">int</span><span class="p">(</span><span class="n">tks</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="nb">int</span><span class="p">(</span><span class="n">tks</span><span class="p">[</span><span class="mi">1</span><span class="p">]),</span> <span class="nb">float</span><span class="p">(</span><span class="n">tks</span><span class="p">[</span><span class="mi">2</span><span class="p">])))</span>
<span class="bp">self</span><span class="o">.</span><span class="n">provide_data</span> <span class="o">=</span> <span class="p">[(</span><span class="s1">'user'</span><span class="p">,</span> <span class="p">(</span><span class="n">batch_size</span><span class="p">,</span> <span class="p">)),</span> <span class="p">(</span><span class="s1">'item'</span><span class="p">,</span> <span class="p">(</span><span class="n">batch_size</span><span class="p">,</span> <span class="p">))]</span>
<span class="bp">self</span><span class="o">.</span><span class="n">provide_label</span> <span class="o">=</span> <span class="p">[(</span><span class="s1">'score'</span><span class="p">,</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">batch_size</span><span class="p">,</span> <span class="p">))]</span>
<span class="k">def</span> <span class="fm">__iter__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">data</span><span class="p">)</span> <span class="o">/</span> <span class="bp">self</span><span class="o">.</span><span class="n">batch_size</span><span class="p">):</span>
<span class="n">users</span> <span class="o">=</span> <span class="p">[]</span>
<span class="n">items</span> <span class="o">=</span> <span class="p">[]</span>
<span class="n">scores</span> <span class="o">=</span> <span class="p">[]</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">batch_size</span><span class="p">):</span>
<span class="n">j</span> <span class="o">=</span> <span class="n">k</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">batch_size</span> <span class="o">+</span> <span class="n">i</span>
<span class="n">user</span><span class="p">,</span> <span class="n">item</span><span class="p">,</span> <span class="n">score</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">data</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
<span class="n">users</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">user</span><span class="p">)</span>
<span class="n">items</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">item</span><span class="p">)</span>
<span class="n">scores</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">score</span><span class="p">)</span>
<span class="n">data_all</span> <span class="o">=</span> <span class="p">[</span><span class="n">mx</span><span class="o">.</span><span class="n">nd</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">users</span><span class="p">),</span> <span class="n">mx</span><span class="o">.</span><span class="n">nd</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">items</span><span class="p">)]</span>
<span class="n">label_all</span> <span class="o">=</span> <span class="p">[</span><span class="n">mx</span><span class="o">.</span><span class="n">nd</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">scores</span><span class="p">)]</span>
<span class="n">data_names</span> <span class="o">=</span> <span class="p">[</span><span class="s1">'user'</span><span class="p">,</span> <span class="s1">'item'</span><span class="p">]</span>
<span class="n">label_names</span> <span class="o">=</span> <span class="p">[</span><span class="s1">'score'</span><span class="p">]</span>
<span class="n">data_batch</span> <span class="o">=</span> <span class="n">Batch</span><span class="p">(</span><span class="n">data_names</span><span class="p">,</span> <span class="n">data_all</span><span class="p">,</span> <span class="n">label_names</span><span class="p">,</span> <span class="n">label_all</span><span class="p">)</span>
<span class="k">yield</span> <span class="n">data_batch</span>
<span class="k">def</span> <span class="nf">reset</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="n">random</span><span class="o">.</span><span class="n">shuffle</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">data</span><span class="p">)</span>
</pre></div>
</div>
<p>Now we download the data and provide a function to obtain the data iterator:</p>
<div class="highlight-python"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">os</span>
<span class="kn">import</span> <span class="nn">urllib</span>
<span class="kn">import</span> <span class="nn">zipfile</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">exists</span><span class="p">(</span><span class="s1">'ml-100k.zip'</span><span class="p">):</span>
<span class="n">urllib</span><span class="o">.</span><span class="n">urlretrieve</span><span class="p">(</span><span class="s1">'http://files.grouplens.org/datasets/movielens/ml-100k.zip'</span><span class="p">,</span> <span class="s1">'ml-100k.zip'</span><span class="p">)</span>
<span class="k">with</span> <span class="n">zipfile</span><span class="o">.</span><span class="n">ZipFile</span><span class="p">(</span><span class="s2">"ml-100k.zip"</span><span class="p">,</span><span class="s2">"r"</span><span class="p">)</span> <span class="k">as</span> <span class="n">f</span><span class="p">:</span>
<span class="n">f</span><span class="o">.</span><span class="n">extractall</span><span class="p">(</span><span class="s2">"./"</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">get_data</span><span class="p">(</span><span class="n">batch_size</span><span class="p">):</span>
<span class="k">return</span> <span class="p">(</span><span class="n">DataIter</span><span class="p">(</span><span class="s1">'./ml-100k/u1.base'</span><span class="p">,</span> <span class="n">batch_size</span><span class="p">),</span> <span class="n">DataIter</span><span class="p">(</span><span class="s1">'./ml-100k/u1.test'</span><span class="p">,</span> <span class="n">batch_size</span><span class="p">))</span>
</pre></div>
</div>
<p>Finally we calculate the numbers of users and items for later use.</p>
<div class="highlight-python"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">max_id</span><span class="p">(</span><span class="n">fname</span><span class="p">):</span>
<span class="n">mu</span> <span class="o">=</span> <span class="mi">0</span>
<span class="n">mi</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">for</span> <span class="n">line</span> <span class="ow">in</span> <span class="nb">file</span><span class="p">(</span><span class="n">fname</span><span class="p">):</span>
<span class="n">tks</span> <span class="o">=</span> <span class="n">line</span><span class="o">.</span><span class="n">strip</span><span class="p">()</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="s1">'</span><span class="se">\t</span><span class="s1">'</span><span class="p">)</span>
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">tks</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">4</span><span class="p">:</span>
<span class="k">continue</span>
<span class="n">mu</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">mu</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">tks</span><span class="p">[</span><span class="mi">0</span><span class="p">]))</span>
<span class="n">mi</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">mi</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">tks</span><span class="p">[</span><span class="mi">1</span><span class="p">]))</span>
<span class="k">return</span> <span class="n">mu</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">mi</span> <span class="o">+</span> <span class="mi">1</span>
<span class="n">max_user</span><span class="p">,</span> <span class="n">max_item</span> <span class="o">=</span> <span class="n">max_id</span><span class="p">(</span><span class="s1">'./ml-100k/u.data'</span><span class="p">)</span>
<span class="p">(</span><span class="n">max_user</span><span class="p">,</span> <span class="n">max_item</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="section" id="optimization">
<span id="optimization"></span><h2>Optimization<a class="headerlink" href="#optimization" title="Permalink to this headline"></a></h2>
<p>We first implement the RMSE (root-mean-square error) measurement, which is
commonly used by matrix factorization.</p>
<div class="highlight-python"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">math</span>
<span class="k">def</span> <span class="nf">RMSE</span><span class="p">(</span><span class="n">label</span><span class="p">,</span> <span class="n">pred</span><span class="p">):</span>
<span class="n">ret</span> <span class="o">=</span> <span class="mf">0.0</span>
<span class="n">n</span> <span class="o">=</span> <span class="mf">0.0</span>
<span class="n">pred</span> <span class="o">=</span> <span class="n">pred</span><span class="o">.</span><span class="n">flatten</span><span class="p">()</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">label</span><span class="p">)):</span>
<span class="n">ret</span> <span class="o">+=</span> <span class="p">(</span><span class="n">label</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">-</span> <span class="n">pred</span><span class="p">[</span><span class="n">i</span><span class="p">])</span> <span class="o">*</span> <span class="p">(</span><span class="n">label</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">-</span> <span class="n">pred</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
<span class="n">n</span> <span class="o">+=</span> <span class="mf">1.0</span>
<span class="k">return</span> <span class="n">math</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">ret</span> <span class="o">/</span> <span class="n">n</span><span class="p">)</span>
</pre></div>
</div>
<p>Then we define a general training module, which is borrowed from the image
classification application.</p>
<div class="highlight-python"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">train</span><span class="p">(</span><span class="n">network</span><span class="p">,</span> <span class="n">batch_size</span><span class="p">,</span> <span class="n">num_epoch</span><span class="p">,</span> <span class="n">learning_rate</span><span class="p">):</span>
<span class="n">model</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">model</span><span class="o">.</span><span class="n">FeedForward</span><span class="p">(</span>
<span class="n">ctx</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">gpu</span><span class="p">(</span><span class="mi">0</span><span class="p">),</span>
<span class="n">symbol</span> <span class="o">=</span> <span class="n">network</span><span class="p">,</span>
<span class="n">num_epoch</span> <span class="o">=</span> <span class="n">num_epoch</span><span class="p">,</span>
<span class="n">learning_rate</span> <span class="o">=</span> <span class="n">learning_rate</span><span class="p">,</span>
<span class="n">wd</span> <span class="o">=</span> <span class="mf">0.0001</span><span class="p">,</span>
<span class="n">momentum</span> <span class="o">=</span> <span class="mf">0.9</span><span class="p">)</span>
<span class="n">batch_size</span> <span class="o">=</span> <span class="mi">64</span>
<span class="n">train</span><span class="p">,</span> <span class="n">test</span> <span class="o">=</span> <span class="n">get_data</span><span class="p">(</span><span class="n">batch_size</span><span class="p">)</span>
<span class="kn">import</span> <span class="nn">logging</span>
<span class="n">head</span> <span class="o">=</span> <span class="s1">'</span><span class="si">%(asctime)-15s</span><span class="s1"> </span><span class="si">%(message)s</span><span class="s1">'</span>
<span class="n">logging</span><span class="o">.</span><span class="n">basicConfig</span><span class="p">(</span><span class="n">level</span><span class="o">=</span><span class="n">logging</span><span class="o">.</span><span class="n">DEBUG</span><span class="p">)</span>
<span class="n">model</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span> <span class="o">=</span> <span class="n">train</span><span class="p">,</span>
<span class="n">eval_data</span> <span class="o">=</span> <span class="n">test</span><span class="p">,</span>
<span class="n">eval_metric</span> <span class="o">=</span> <span class="n">RMSE</span><span class="p">,</span>
<span class="n">batch_end_callback</span><span class="o">=</span><span class="n">mx</span><span class="o">.</span><span class="n">callback</span><span class="o">.</span><span class="n">Speedometer</span><span class="p">(</span><span class="n">batch_size</span><span class="p">,</span> <span class="mi">20000</span><span class="o">/</span><span class="n">batch_size</span><span class="p">),)</span>
</pre></div>
</div>
</div>
<div class="section" id="networks">
<span id="networks"></span><h2>Networks<a class="headerlink" href="#networks" title="Permalink to this headline"></a></h2>
<p>Now we try various networks. We first learn the latent vectors directly.</p>
<div class="highlight-python"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">plain_net</span><span class="p">(</span><span class="n">k</span><span class="p">):</span>
<span class="c1"># input</span>
<span class="n">user</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">symbol</span><span class="o">.</span><span class="n">Variable</span><span class="p">(</span><span class="s1">'user'</span><span class="p">)</span>
<span class="n">item</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">symbol</span><span class="o">.</span><span class="n">Variable</span><span class="p">(</span><span class="s1">'item'</span><span class="p">)</span>
<span class="n">score</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">symbol</span><span class="o">.</span><span class="n">Variable</span><span class="p">(</span><span class="s1">'score'</span><span class="p">)</span>
<span class="c1"># user feature lookup</span>
<span class="n">user</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">symbol</span><span class="o">.</span><span class="n">Embedding</span><span class="p">(</span><span class="n">data</span> <span class="o">=</span> <span class="n">user</span><span class="p">,</span> <span class="n">input_dim</span> <span class="o">=</span> <span class="n">max_user</span><span class="p">,</span> <span class="n">output_dim</span> <span class="o">=</span> <span class="n">k</span><span class="p">)</span>
<span class="c1"># item feature lookup</span>
<span class="n">item</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">symbol</span><span class="o">.</span><span class="n">Embedding</span><span class="p">(</span><span class="n">data</span> <span class="o">=</span> <span class="n">item</span><span class="p">,</span> <span class="n">input_dim</span> <span class="o">=</span> <span class="n">max_item</span><span class="p">,</span> <span class="n">output_dim</span> <span class="o">=</span> <span class="n">k</span><span class="p">)</span>
<span class="c1"># predict by the inner product, which is elementwise product and then sum</span>
<span class="n">pred</span> <span class="o">=</span> <span class="n">user</span> <span class="o">*</span> <span class="n">item</span>
<span class="n">pred</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">symbol</span><span class="o">.</span><span class="n">sum_axis</span><span class="p">(</span><span class="n">data</span> <span class="o">=</span> <span class="n">pred</span><span class="p">,</span> <span class="n">axis</span> <span class="o">=</span> <span class="mi">1</span><span class="p">)</span>
<span class="n">pred</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">symbol</span><span class="o">.</span><span class="n">Flatten</span><span class="p">(</span><span class="n">data</span> <span class="o">=</span> <span class="n">pred</span><span class="p">)</span>
<span class="c1"># loss layer</span>
<span class="n">pred</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">symbol</span><span class="o">.</span><span class="n">LinearRegressionOutput</span><span class="p">(</span><span class="n">data</span> <span class="o">=</span> <span class="n">pred</span><span class="p">,</span> <span class="n">label</span> <span class="o">=</span> <span class="n">score</span><span class="p">)</span>
<span class="k">return</span> <span class="n">pred</span>
<span class="n">train</span><span class="p">(</span><span class="n">plain_net</span><span class="p">(</span><span class="mi">64</span><span class="p">),</span> <span class="n">batch_size</span><span class="o">=</span><span class="mi">64</span><span class="p">,</span> <span class="n">num_epoch</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span> <span class="n">learning_rate</span><span class="o">=.</span><span class="mo">05</span><span class="p">)</span>
</pre></div>
</div>
<p>Next we try to use 2 layers neural network to learn the latent variables, which stack a fully connected layer above the embedding layers:</p>
<div class="highlight-python"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">get_one_layer_mlp</span><span class="p">(</span><span class="n">hidden</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
<span class="c1"># input</span>
<span class="n">user</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">symbol</span><span class="o">.</span><span class="n">Variable</span><span class="p">(</span><span class="s1">'user'</span><span class="p">)</span>
<span class="n">item</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">symbol</span><span class="o">.</span><span class="n">Variable</span><span class="p">(</span><span class="s1">'item'</span><span class="p">)</span>
<span class="n">score</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">symbol</span><span class="o">.</span><span class="n">Variable</span><span class="p">(</span><span class="s1">'score'</span><span class="p">)</span>
<span class="c1"># user latent features</span>
<span class="n">user</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">symbol</span><span class="o">.</span><span class="n">Embedding</span><span class="p">(</span><span class="n">data</span> <span class="o">=</span> <span class="n">user</span><span class="p">,</span> <span class="n">input_dim</span> <span class="o">=</span> <span class="n">max_user</span><span class="p">,</span> <span class="n">output_dim</span> <span class="o">=</span> <span class="n">k</span><span class="p">)</span>
<span class="n">user</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">symbol</span><span class="o">.</span><span class="n">Activation</span><span class="p">(</span><span class="n">data</span> <span class="o">=</span> <span class="n">user</span><span class="p">,</span> <span class="n">act_type</span><span class="o">=</span><span class="s2">"relu"</span><span class="p">)</span>
<span class="n">user</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">symbol</span><span class="o">.</span><span class="n">FullyConnected</span><span class="p">(</span><span class="n">data</span> <span class="o">=</span> <span class="n">user</span><span class="p">,</span> <span class="n">num_hidden</span> <span class="o">=</span> <span class="n">hidden</span><span class="p">)</span>
<span class="c1"># item latent features</span>
<span class="n">item</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">symbol</span><span class="o">.</span><span class="n">Embedding</span><span class="p">(</span><span class="n">data</span> <span class="o">=</span> <span class="n">item</span><span class="p">,</span> <span class="n">input_dim</span> <span class="o">=</span> <span class="n">max_item</span><span class="p">,</span> <span class="n">output_dim</span> <span class="o">=</span> <span class="n">k</span><span class="p">)</span>
<span class="n">item</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">symbol</span><span class="o">.</span><span class="n">Activation</span><span class="p">(</span><span class="n">data</span> <span class="o">=</span> <span class="n">item</span><span class="p">,</span> <span class="n">act_type</span><span class="o">=</span><span class="s2">"relu"</span><span class="p">)</span>
<span class="n">item</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">symbol</span><span class="o">.</span><span class="n">FullyConnected</span><span class="p">(</span><span class="n">data</span> <span class="o">=</span> <span class="n">item</span><span class="p">,</span> <span class="n">num_hidden</span> <span class="o">=</span> <span class="n">hidden</span><span class="p">)</span>
<span class="c1"># predict by the inner product</span>
<span class="n">pred</span> <span class="o">=</span> <span class="n">user</span> <span class="o">*</span> <span class="n">item</span>
<span class="n">pred</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">symbol</span><span class="o">.</span><span class="n">sum_axis</span><span class="p">(</span><span class="n">data</span> <span class="o">=</span> <span class="n">pred</span><span class="p">,</span> <span class="n">axis</span> <span class="o">=</span> <span class="mi">1</span><span class="p">)</span>
<span class="n">pred</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">symbol</span><span class="o">.</span><span class="n">Flatten</span><span class="p">(</span><span class="n">data</span> <span class="o">=</span> <span class="n">pred</span><span class="p">)</span>
<span class="c1"># loss layer</span>
<span class="n">pred</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">symbol</span><span class="o">.</span><span class="n">LinearRegressionOutput</span><span class="p">(</span><span class="n">data</span> <span class="o">=</span> <span class="n">pred</span><span class="p">,</span> <span class="n">label</span> <span class="o">=</span> <span class="n">score</span><span class="p">)</span>
<span class="k">return</span> <span class="n">pred</span>
<span class="n">train</span><span class="p">(</span><span class="n">get_one_layer_mlp</span><span class="p">(</span><span class="mi">64</span><span class="p">,</span> <span class="mi">64</span><span class="p">),</span> <span class="n">batch_size</span><span class="o">=</span><span class="mi">64</span><span class="p">,</span> <span class="n">num_epoch</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span> <span class="n">learning_rate</span><span class="o">=.</span><span class="mo">05</span><span class="p">)</span>
</pre></div>
</div>
<p>Adding dropout layers to relief the over-fitting.</p>
<div class="highlight-python"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">get_one_layer_dropout_mlp</span><span class="p">(</span><span class="n">hidden</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
<span class="c1"># input</span>
<span class="n">user</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">symbol</span><span class="o">.</span><span class="n">Variable</span><span class="p">(</span><span class="s1">'user'</span><span class="p">)</span>
<span class="n">item</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">symbol</span><span class="o">.</span><span class="n">Variable</span><span class="p">(</span><span class="s1">'item'</span><span class="p">)</span>
<span class="n">score</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">symbol</span><span class="o">.</span><span class="n">Variable</span><span class="p">(</span><span class="s1">'score'</span><span class="p">)</span>
<span class="c1"># user latent features</span>
<span class="n">user</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">symbol</span><span class="o">.</span><span class="n">Embedding</span><span class="p">(</span><span class="n">data</span> <span class="o">=</span> <span class="n">user</span><span class="p">,</span> <span class="n">input_dim</span> <span class="o">=</span> <span class="n">max_user</span><span class="p">,</span> <span class="n">output_dim</span> <span class="o">=</span> <span class="n">k</span><span class="p">)</span>
<span class="n">user</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">symbol</span><span class="o">.</span><span class="n">Activation</span><span class="p">(</span><span class="n">data</span> <span class="o">=</span> <span class="n">user</span><span class="p">,</span> <span class="n">act_type</span><span class="o">=</span><span class="s2">"relu"</span><span class="p">)</span>
<span class="n">user</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">symbol</span><span class="o">.</span><span class="n">FullyConnected</span><span class="p">(</span><span class="n">data</span> <span class="o">=</span> <span class="n">user</span><span class="p">,</span> <span class="n">num_hidden</span> <span class="o">=</span> <span class="n">hidden</span><span class="p">)</span>
<span class="n">user</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">symbol</span><span class="o">.</span><span class="n">Dropout</span><span class="p">(</span><span class="n">data</span><span class="o">=</span><span class="n">user</span><span class="p">,</span> <span class="n">p</span><span class="o">=</span><span class="mf">0.5</span><span class="p">)</span>
<span class="c1"># item latent features</span>
<span class="n">item</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">symbol</span><span class="o">.</span><span class="n">Embedding</span><span class="p">(</span><span class="n">data</span> <span class="o">=</span> <span class="n">item</span><span class="p">,</span> <span class="n">input_dim</span> <span class="o">=</span> <span class="n">max_item</span><span class="p">,</span> <span class="n">output_dim</span> <span class="o">=</span> <span class="n">k</span><span class="p">)</span>
<span class="n">item</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">symbol</span><span class="o">.</span><span class="n">Activation</span><span class="p">(</span><span class="n">data</span> <span class="o">=</span> <span class="n">item</span><span class="p">,</span> <span class="n">act_type</span><span class="o">=</span><span class="s2">"relu"</span><span class="p">)</span>
<span class="n">item</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">symbol</span><span class="o">.</span><span class="n">FullyConnected</span><span class="p">(</span><span class="n">data</span> <span class="o">=</span> <span class="n">item</span><span class="p">,</span> <span class="n">num_hidden</span> <span class="o">=</span> <span class="n">hidden</span><span class="p">)</span>
<span class="n">item</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">symbol</span><span class="o">.</span><span class="n">Dropout</span><span class="p">(</span><span class="n">data</span><span class="o">=</span><span class="n">item</span><span class="p">,</span> <span class="n">p</span><span class="o">=</span><span class="mf">0.5</span><span class="p">)</span>
<span class="c1"># predict by the inner product</span>
<span class="n">pred</span> <span class="o">=</span> <span class="n">user</span> <span class="o">*</span> <span class="n">item</span>
<span class="n">pred</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">symbol</span><span class="o">.</span><span class="n">sum_axis</span><span class="p">(</span><span class="n">data</span> <span class="o">=</span> <span class="n">pred</span><span class="p">,</span> <span class="n">axis</span> <span class="o">=</span> <span class="mi">1</span><span class="p">)</span>
<span class="n">pred</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">symbol</span><span class="o">.</span><span class="n">Flatten</span><span class="p">(</span><span class="n">data</span> <span class="o">=</span> <span class="n">pred</span><span class="p">)</span>
<span class="c1"># loss layer</span>
<span class="n">pred</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">symbol</span><span class="o">.</span><span class="n">LinearRegressionOutput</span><span class="p">(</span><span class="n">data</span> <span class="o">=</span> <span class="n">pred</span><span class="p">,</span> <span class="n">label</span> <span class="o">=</span> <span class="n">score</span><span class="p">)</span>
<span class="k">return</span> <span class="n">pred</span>
<span class="n">train</span><span class="p">(</span><span class="n">get_one_layer_mlp</span><span class="p">(</span><span class="mi">256</span><span class="p">,</span> <span class="mi">512</span><span class="p">),</span> <span class="n">batch_size</span><span class="o">=</span><span class="mi">64</span><span class="p">,</span> <span class="n">num_epoch</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span> <span class="n">learning_rate</span><span class="o">=.</span><span class="mo">05</span><span class="p">)</span>
</pre></div>
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