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<div class="section" id="torch-func-whirlwind-tour">
<h1>torch.func Whirlwind Tour<a class="headerlink" href="#torch-func-whirlwind-tour" title="Permalink to this heading">¶</a></h1>
<div class="section" id="what-is-torch-func">
<h2>What is torch.func?<a class="headerlink" href="#what-is-torch-func" title="Permalink to this heading">¶</a></h2>
<p>torch.func, previously known as functorch, is a library for
<a class="reference external" href="https://github.com/google/jax">JAX</a>-like composable function transforms in
PyTorch.</p>
<ul class="simple">
<li><p>A “function transform” is a higher-order function that accepts a numerical
function and returns a new function that computes a different quantity.</p></li>
<li><p>torch.func has auto-differentiation transforms (<code class="docutils literal notranslate"><span class="pre">grad(f)</span></code> returns a function
that computes the gradient of <code class="docutils literal notranslate"><span class="pre">f</span></code>), a vectorization/batching transform
(<code class="docutils literal notranslate"><span class="pre">vmap(f)</span></code> returns a function that computes <code class="docutils literal notranslate"><span class="pre">f</span></code> over batches of inputs),
and others.</p></li>
<li><p>These function transforms can compose with each other arbitrarily. For
example, composing <code class="docutils literal notranslate"><span class="pre">vmap(grad(f))</span></code> computes a quantity called
per-sample-gradients that stock PyTorch cannot efficiently compute today.</p></li>
</ul>
</div>
<div class="section" id="why-composable-function-transforms">
<h2>Why composable function transforms?<a class="headerlink" href="#why-composable-function-transforms" title="Permalink to this heading">¶</a></h2>
<p>There are a number of use cases that are tricky to do in PyTorch today:
- computing per-sample-gradients (or other per-sample quantities)</p>
<ul class="simple">
<li><p>running ensembles of models on a single machine</p></li>
<li><p>efficiently batching together tasks in the inner-loop of MAML</p></li>
<li><p>efficiently computing Jacobians and Hessians</p></li>
<li><p>efficiently computing batched Jacobians and Hessians</p></li>
</ul>
<p>Composing <a class="reference internal" href="generated/torch.func.vmap.html#torch.func.vmap" title="torch.func.vmap"><code class="xref py py-func docutils literal notranslate"><span class="pre">vmap()</span></code></a>, <a class="reference internal" href="generated/torch.func.grad.html#torch.func.grad" title="torch.func.grad"><code class="xref py py-func docutils literal notranslate"><span class="pre">grad()</span></code></a>, <a class="reference internal" href="generated/torch.func.vjp.html#torch.func.vjp" title="torch.func.vjp"><code class="xref py py-func docutils literal notranslate"><span class="pre">vjp()</span></code></a>, and <a class="reference internal" href="generated/torch.func.jvp.html#torch.func.jvp" title="torch.func.jvp"><code class="xref py py-func docutils literal notranslate"><span class="pre">jvp()</span></code></a> transforms
allows us to express the above without designing a separate subsystem for each.</p>
</div>
<div class="section" id="what-are-the-transforms">
<h2>What are the transforms?<a class="headerlink" href="#what-are-the-transforms" title="Permalink to this heading">¶</a></h2>
<div class="section" id="grad-gradient-computation">
<h3><a class="reference internal" href="generated/torch.func.grad.html#torch.func.grad" title="torch.func.grad"><code class="xref py py-func docutils literal notranslate"><span class="pre">grad()</span></code></a> (gradient computation)<a class="headerlink" href="#grad-gradient-computation" title="Permalink to this heading">¶</a></h3>
<p><code class="docutils literal notranslate"><span class="pre">grad(func)</span></code> is our gradient computation transform. It returns a new function
that computes the gradients of <code class="docutils literal notranslate"><span class="pre">func</span></code>. It assumes <code class="docutils literal notranslate"><span class="pre">func</span></code> returns a single-element
Tensor and by default it computes the gradients of the output of <code class="docutils literal notranslate"><span class="pre">func</span></code> w.r.t.
to the first input.</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">torch</span>
<span class="kn">from</span> <span class="nn">torch.func</span> <span class="kn">import</span> <span class="n">grad</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">([])</span>
<span class="n">cos_x</span> <span class="o">=</span> <span class="n">grad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">torch</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">))(</span><span class="n">x</span><span class="p">)</span>
<span class="k">assert</span> <span class="n">torch</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">cos_x</span><span class="p">,</span> <span class="n">x</span><span class="o">.</span><span class="n">cos</span><span class="p">())</span>
<span class="c1"># Second-order gradients</span>
<span class="n">neg_sin_x</span> <span class="o">=</span> <span class="n">grad</span><span class="p">(</span><span class="n">grad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">torch</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)))(</span><span class="n">x</span><span class="p">)</span>
<span class="k">assert</span> <span class="n">torch</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">neg_sin_x</span><span class="p">,</span> <span class="o">-</span><span class="n">x</span><span class="o">.</span><span class="n">sin</span><span class="p">())</span>
</pre></div>
</div>
</div>
<div class="section" id="vmap-auto-vectorization">
<h3><a class="reference internal" href="generated/torch.func.vmap.html#torch.func.vmap" title="torch.func.vmap"><code class="xref py py-func docutils literal notranslate"><span class="pre">vmap()</span></code></a> (auto-vectorization)<a class="headerlink" href="#vmap-auto-vectorization" title="Permalink to this heading">¶</a></h3>
<p>Note: <a class="reference internal" href="generated/torch.func.vmap.html#torch.func.vmap" title="torch.func.vmap"><code class="xref py py-func docutils literal notranslate"><span class="pre">vmap()</span></code></a> imposes restrictions on the code that it can be used on. For more
details, please see <a class="reference internal" href="func.ux_limitations.html#ux-limitations"><span class="std std-ref">UX Limitations</span></a>.</p>
<p><code class="docutils literal notranslate"><span class="pre">vmap(func)(*inputs)</span></code> is a transform that adds a dimension to all Tensor
operations in <code class="docutils literal notranslate"><span class="pre">func</span></code>. <code class="docutils literal notranslate"><span class="pre">vmap(func)</span></code> returns a new function that maps <code class="docutils literal notranslate"><span class="pre">func</span></code>
over some dimension (default: 0) of each Tensor in inputs.</p>
<p>vmap is useful for hiding batch dimensions: one can write a function func that
runs on examples and then lift it to a function that can take batches of
examples with <code class="docutils literal notranslate"><span class="pre">vmap(func)</span></code>, leading to a simpler modeling experience:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">torch</span>
<span class="kn">from</span> <span class="nn">torch.func</span> <span class="kn">import</span> <span class="n">vmap</span>
<span class="n">batch_size</span><span class="p">,</span> <span class="n">feature_size</span> <span class="o">=</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span>
<span class="n">weights</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">feature_size</span><span class="p">,</span> <span class="n">requires_grad</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">model</span><span class="p">(</span><span class="n">feature_vec</span><span class="p">):</span>
<span class="c1"># Very simple linear model with activation</span>
<span class="k">assert</span> <span class="n">feature_vec</span><span class="o">.</span><span class="n">dim</span><span class="p">()</span> <span class="o">==</span> <span class="mi">1</span>
<span class="k">return</span> <span class="n">feature_vec</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">weights</span><span class="p">)</span><span class="o">.</span><span class="n">relu</span><span class="p">()</span>
<span class="n">examples</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">batch_size</span><span class="p">,</span> <span class="n">feature_size</span><span class="p">)</span>
<span class="n">result</span> <span class="o">=</span> <span class="n">vmap</span><span class="p">(</span><span class="n">model</span><span class="p">)(</span><span class="n">examples</span><span class="p">)</span>
</pre></div>
</div>
<p>When composed with <a class="reference internal" href="generated/torch.func.grad.html#torch.func.grad" title="torch.func.grad"><code class="xref py py-func docutils literal notranslate"><span class="pre">grad()</span></code></a>, <a class="reference internal" href="generated/torch.func.vmap.html#torch.func.vmap" title="torch.func.vmap"><code class="xref py py-func docutils literal notranslate"><span class="pre">vmap()</span></code></a> can be used to compute per-sample-gradients:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">torch.func</span> <span class="kn">import</span> <span class="n">vmap</span>
<span class="n">batch_size</span><span class="p">,</span> <span class="n">feature_size</span> <span class="o">=</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span>
<span class="k">def</span> <span class="nf">model</span><span class="p">(</span><span class="n">weights</span><span class="p">,</span><span class="n">feature_vec</span><span class="p">):</span>
<span class="c1"># Very simple linear model with activation</span>
<span class="k">assert</span> <span class="n">feature_vec</span><span class="o">.</span><span class="n">dim</span><span class="p">()</span> <span class="o">==</span> <span class="mi">1</span>
<span class="k">return</span> <span class="n">feature_vec</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">weights</span><span class="p">)</span><span class="o">.</span><span class="n">relu</span><span class="p">()</span>
<span class="k">def</span> <span class="nf">compute_loss</span><span class="p">(</span><span class="n">weights</span><span class="p">,</span> <span class="n">example</span><span class="p">,</span> <span class="n">target</span><span class="p">):</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">model</span><span class="p">(</span><span class="n">weights</span><span class="p">,</span> <span class="n">example</span><span class="p">)</span>
<span class="k">return</span> <span class="p">((</span><span class="n">y</span> <span class="o">-</span> <span class="n">target</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span> <span class="c1"># MSELoss</span>
<span class="n">weights</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">feature_size</span><span class="p">,</span> <span class="n">requires_grad</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="n">examples</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">batch_size</span><span class="p">,</span> <span class="n">feature_size</span><span class="p">)</span>
<span class="n">targets</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">batch_size</span><span class="p">)</span>
<span class="n">inputs</span> <span class="o">=</span> <span class="p">(</span><span class="n">weights</span><span class="p">,</span><span class="n">examples</span><span class="p">,</span> <span class="n">targets</span><span class="p">)</span>
<span class="n">grad_weight_per_example</span> <span class="o">=</span> <span class="n">vmap</span><span class="p">(</span><span class="n">grad</span><span class="p">(</span><span class="n">compute_loss</span><span class="p">),</span> <span class="n">in_dims</span><span class="o">=</span><span class="p">(</span><span class="kc">None</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">))(</span><span class="o">*</span><span class="n">inputs</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="section" id="vjp-vector-jacobian-product">
<h3><a class="reference internal" href="generated/torch.func.vjp.html#torch.func.vjp" title="torch.func.vjp"><code class="xref py py-func docutils literal notranslate"><span class="pre">vjp()</span></code></a> (vector-Jacobian product)<a class="headerlink" href="#vjp-vector-jacobian-product" title="Permalink to this heading">¶</a></h3>
<p>The <a class="reference internal" href="generated/torch.func.vjp.html#torch.func.vjp" title="torch.func.vjp"><code class="xref py py-func docutils literal notranslate"><span class="pre">vjp()</span></code></a> transform applies <code class="docutils literal notranslate"><span class="pre">func</span></code> to <code class="docutils literal notranslate"><span class="pre">inputs</span></code> and returns a new function
that computes the vector-Jacobian product (vjp) given some <code class="docutils literal notranslate"><span class="pre">cotangents</span></code> Tensors.</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">torch.func</span> <span class="kn">import</span> <span class="n">vjp</span>
<span class="n">inputs</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
<span class="n">func</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">sin</span>
<span class="n">cotangents</span> <span class="o">=</span> <span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">3</span><span class="p">),)</span>
<span class="n">outputs</span><span class="p">,</span> <span class="n">vjp_fn</span> <span class="o">=</span> <span class="n">vjp</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="n">inputs</span><span class="p">);</span> <span class="n">vjps</span> <span class="o">=</span> <span class="n">vjp_fn</span><span class="p">(</span><span class="o">*</span><span class="n">cotangents</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="section" id="jvp-jacobian-vector-product">
<h3><a class="reference internal" href="generated/torch.func.jvp.html#torch.func.jvp" title="torch.func.jvp"><code class="xref py py-func docutils literal notranslate"><span class="pre">jvp()</span></code></a> (Jacobian-vector product)<a class="headerlink" href="#jvp-jacobian-vector-product" title="Permalink to this heading">¶</a></h3>
<p>The <a class="reference internal" href="generated/torch.func.jvp.html#torch.func.jvp" title="torch.func.jvp"><code class="xref py py-func docutils literal notranslate"><span class="pre">jvp()</span></code></a> transforms computes Jacobian-vector-products and is also known as
“forward-mode AD”. It is not a higher-order function unlike most other transforms,
but it returns the outputs of <code class="docutils literal notranslate"><span class="pre">func(inputs)</span></code> as well as the jvps.</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">torch.func</span> <span class="kn">import</span> <span class="n">jvp</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
<span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">:</span> <span class="p">(</span><span class="n">x</span> <span class="o">*</span> <span class="n">y</span><span class="p">)</span>
<span class="n">_</span><span class="p">,</span> <span class="n">out_tangent</span> <span class="o">=</span> <span class="n">jvp</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">),</span> <span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="mi">5</span><span class="p">),</span> <span class="n">torch</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="mi">5</span><span class="p">)))</span>
<span class="k">assert</span> <span class="n">torch</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">out_tangent</span><span class="p">,</span> <span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="section" id="jacrev-jacfwd-and-hessian">
<h3><a class="reference internal" href="generated/torch.func.jacrev.html#torch.func.jacrev" title="torch.func.jacrev"><code class="xref py py-func docutils literal notranslate"><span class="pre">jacrev()</span></code></a>, <a class="reference internal" href="generated/torch.func.jacfwd.html#torch.func.jacfwd" title="torch.func.jacfwd"><code class="xref py py-func docutils literal notranslate"><span class="pre">jacfwd()</span></code></a>, and <a class="reference internal" href="generated/torch.func.hessian.html#torch.func.hessian" title="torch.func.hessian"><code class="xref py py-func docutils literal notranslate"><span class="pre">hessian()</span></code></a><a class="headerlink" href="#jacrev-jacfwd-and-hessian" title="Permalink to this heading">¶</a></h3>
<p>The <a class="reference internal" href="generated/torch.func.jacrev.html#torch.func.jacrev" title="torch.func.jacrev"><code class="xref py py-func docutils literal notranslate"><span class="pre">jacrev()</span></code></a> transform returns a new function that takes in <code class="docutils literal notranslate"><span class="pre">x</span></code> and returns
the Jacobian of the function with respect to <code class="docutils literal notranslate"><span class="pre">x</span></code> using reverse-mode AD.</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">torch.func</span> <span class="kn">import</span> <span class="n">jacrev</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
<span class="n">jacobian</span> <span class="o">=</span> <span class="n">jacrev</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">sin</span><span class="p">)(</span><span class="n">x</span><span class="p">)</span>
<span class="n">expected</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">diag</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
<span class="k">assert</span> <span class="n">torch</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">jacobian</span><span class="p">,</span> <span class="n">expected</span><span class="p">)</span>
</pre></div>
</div>
<p><a class="reference internal" href="generated/torch.func.jacrev.html#torch.func.jacrev" title="torch.func.jacrev"><code class="xref py py-func docutils literal notranslate"><span class="pre">jacrev()</span></code></a> can be composed with <a class="reference internal" href="generated/torch.func.vmap.html#torch.func.vmap" title="torch.func.vmap"><code class="xref py py-func docutils literal notranslate"><span class="pre">vmap()</span></code></a> to produce batched jacobians:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">x</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">64</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
<span class="n">jacobian</span> <span class="o">=</span> <span class="n">vmap</span><span class="p">(</span><span class="n">jacrev</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">sin</span><span class="p">))(</span><span class="n">x</span><span class="p">)</span>
<span class="k">assert</span> <span class="n">jacobian</span><span class="o">.</span><span class="n">shape</span> <span class="o">==</span> <span class="p">(</span><span class="mi">64</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
</pre></div>
</div>
<p><a class="reference internal" href="generated/torch.func.jacfwd.html#torch.func.jacfwd" title="torch.func.jacfwd"><code class="xref py py-func docutils literal notranslate"><span class="pre">jacfwd()</span></code></a> is a drop-in replacement for jacrev that computes Jacobians using
forward-mode AD:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">torch.func</span> <span class="kn">import</span> <span class="n">jacfwd</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
<span class="n">jacobian</span> <span class="o">=</span> <span class="n">jacfwd</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">sin</span><span class="p">)(</span><span class="n">x</span><span class="p">)</span>
<span class="n">expected</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">diag</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
<span class="k">assert</span> <span class="n">torch</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">jacobian</span><span class="p">,</span> <span class="n">expected</span><span class="p">)</span>
</pre></div>
</div>
<p>Composing <a class="reference internal" href="generated/torch.func.jacrev.html#torch.func.jacrev" title="torch.func.jacrev"><code class="xref py py-func docutils literal notranslate"><span class="pre">jacrev()</span></code></a> with itself or <a class="reference internal" href="generated/torch.func.jacfwd.html#torch.func.jacfwd" title="torch.func.jacfwd"><code class="xref py py-func docutils literal notranslate"><span class="pre">jacfwd()</span></code></a> can produce hessians:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">f</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="k">return</span> <span class="n">x</span><span class="o">.</span><span class="n">sin</span><span class="p">()</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
<span class="n">hessian0</span> <span class="o">=</span> <span class="n">jacrev</span><span class="p">(</span><span class="n">jacrev</span><span class="p">(</span><span class="n">f</span><span class="p">))(</span><span class="n">x</span><span class="p">)</span>
<span class="n">hessian1</span> <span class="o">=</span> <span class="n">jacfwd</span><span class="p">(</span><span class="n">jacrev</span><span class="p">(</span><span class="n">f</span><span class="p">))(</span><span class="n">x</span><span class="p">)</span>
</pre></div>
</div>
<p><a class="reference internal" href="generated/torch.func.hessian.html#torch.func.hessian" title="torch.func.hessian"><code class="xref py py-func docutils literal notranslate"><span class="pre">hessian()</span></code></a> is a convenience function that combines jacfwd and jacrev:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">torch.func</span> <span class="kn">import</span> <span class="n">hessian</span>
<span class="k">def</span> <span class="nf">f</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="k">return</span> <span class="n">x</span><span class="o">.</span><span class="n">sin</span><span class="p">()</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
<span class="n">hess</span> <span class="o">=</span> <span class="n">hessian</span><span class="p">(</span><span class="n">f</span><span class="p">)(</span><span class="n">x</span><span class="p">)</span>
</pre></div>
</div>
</div>
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