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<h1>Linear Regression</h1>
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<section id="linear-regression">
<span id="id1"></span><h1>Linear Regression<a class="headerlink" href="#linear-regression" title="Link to this heading">#</a></h1>
<p>Let’s implement a basic linear regression model as a starting point to
learn MLX. First import the core package and setup some problem metadata:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span><span class="w"> </span><span class="nn">mlx.core</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">mx</span>
<span class="n">num_features</span> <span class="o">=</span> <span class="mi">100</span>
<span class="n">num_examples</span> <span class="o">=</span> <span class="mi">1_000</span>
<span class="n">num_iters</span> <span class="o">=</span> <span class="mi">10_000</span> <span class="c1"># iterations of SGD</span>
<span class="n">lr</span> <span class="o">=</span> <span class="mf">0.01</span> <span class="c1"># learning rate for SGD</span>
</pre></div>
</div>
<p>We’ll generate a synthetic dataset by:</p>
<ol class="arabic simple">
<li><p>Sampling the design matrix <code class="docutils literal notranslate"><span class="pre">X</span></code>.</p></li>
<li><p>Sampling a ground truth parameter vector <code class="docutils literal notranslate"><span class="pre">w_star</span></code>.</p></li>
<li><p>Compute the dependent values <code class="docutils literal notranslate"><span class="pre">y</span></code> by adding Gaussian noise to <code class="docutils literal notranslate"><span class="pre">X</span> <span class="pre">&#64;</span> <span class="pre">w_star</span></code>.</p></li>
</ol>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># True parameters</span>
<span class="n">w_star</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">((</span><span class="n">num_features</span><span class="p">,))</span>
<span class="c1"># Input examples (design matrix)</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">((</span><span class="n">num_examples</span><span class="p">,</span> <span class="n">num_features</span><span class="p">))</span>
<span class="c1"># Noisy labels</span>
<span class="n">eps</span> <span class="o">=</span> <span class="mf">1e-2</span> <span class="o">*</span> <span class="n">mx</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">((</span><span class="n">num_examples</span><span class="p">,))</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">X</span> <span class="o">@</span> <span class="n">w_star</span> <span class="o">+</span> <span class="n">eps</span>
</pre></div>
</div>
<p>We will use SGD to find the optimal weights. To start, define the squared loss
and get the gradient function of the loss with respect to the parameters.</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span><span class="w"> </span><span class="nf">loss_fn</span><span class="p">(</span><span class="n">w</span><span class="p">):</span>
<span class="k">return</span> <span class="mf">0.5</span> <span class="o">*</span> <span class="n">mx</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">mx</span><span class="o">.</span><span class="n">square</span><span class="p">(</span><span class="n">X</span> <span class="o">@</span> <span class="n">w</span> <span class="o">-</span> <span class="n">y</span><span class="p">))</span>
<span class="n">grad_fn</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">grad</span><span class="p">(</span><span class="n">loss_fn</span><span class="p">)</span>
</pre></div>
</div>
<p>Start the optimization by initializing the parameters <code class="docutils literal notranslate"><span class="pre">w</span></code> randomly. Then
repeatedly update the parameters for <code class="docutils literal notranslate"><span class="pre">num_iters</span></code> iterations.</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">w</span> <span class="o">=</span> <span class="mf">1e-2</span> <span class="o">*</span> <span class="n">mx</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">((</span><span class="n">num_features</span><span class="p">,))</span>
<span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">num_iters</span><span class="p">):</span>
<span class="n">grad</span> <span class="o">=</span> <span class="n">grad_fn</span><span class="p">(</span><span class="n">w</span><span class="p">)</span>
<span class="n">w</span> <span class="o">=</span> <span class="n">w</span> <span class="o">-</span> <span class="n">lr</span> <span class="o">*</span> <span class="n">grad</span>
<span class="n">mx</span><span class="o">.</span><span class="n">eval</span><span class="p">(</span><span class="n">w</span><span class="p">)</span>
</pre></div>
</div>
<p>Finally, compute the loss of the learned parameters and verify that they are
close to the ground truth parameters.</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">loss</span> <span class="o">=</span> <span class="n">loss_fn</span><span class="p">(</span><span class="n">w</span><span class="p">)</span>
<span class="n">error_norm</span> <span class="o">=</span> <span class="n">mx</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">mx</span><span class="o">.</span><span class="n">square</span><span class="p">(</span><span class="n">w</span> <span class="o">-</span> <span class="n">w_star</span><span class="p">))</span><span class="o">.</span><span class="n">item</span><span class="p">()</span> <span class="o">**</span> <span class="mf">0.5</span>
<span class="nb">print</span><span class="p">(</span>
<span class="sa">f</span><span class="s2">&quot;Loss </span><span class="si">{</span><span class="n">loss</span><span class="o">.</span><span class="n">item</span><span class="p">()</span><span class="si">:</span><span class="s2">.5f</span><span class="si">}</span><span class="s2">, |w-w*| = </span><span class="si">{</span><span class="n">error_norm</span><span class="si">:</span><span class="s2">.5f</span><span class="si">}</span><span class="s2">, &quot;</span>
<span class="p">)</span>
<span class="c1"># Should print something close to: Loss 0.00005, |w-w*| = 0.00364</span>
</pre></div>
</div>
<p>Complete <a class="reference external" href="https://github.com/ml-explore/mlx/tree/main/examples/python/linear_regression.py">linear regression</a>
and <a class="reference external" href="https://github.com/ml-explore/mlx/tree/main/examples/python/logistic_regression.py">logistic regression</a>
examples are available in the MLX GitHub repo.</p>
</section>
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