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<div class="section" id="module-torch.autograd">
<span id="automatic-differentiation-package-torch-autograd"></span><h1>Automatic differentiation package - torch.autograd<a class="headerlink" href="#module-torch.autograd" title="Permalink to this headline">¶</a></h1>
<p><code class="docutils literal notranslate"><span class="pre">torch.autograd</span></code> provides classes and functions implementing automatic
differentiation of arbitrary scalar valued functions. It requires minimal
changes to the existing code - you only need to declare <code class="xref py py-class docutils literal notranslate"><span class="pre">Tensor</span></code> s
for which gradients should be computed with the <code class="docutils literal notranslate"><span class="pre">requires_grad=True</span></code> keyword.
As of now, we only support autograd for floating point <code class="xref py py-class docutils literal notranslate"><span class="pre">Tensor</span></code> types (
half, float, double and bfloat16) and complex <code class="xref py py-class docutils literal notranslate"><span class="pre">Tensor</span></code> types (cfloat, cdouble).</p>
<dl class="function">
<dt id="torch.autograd.backward">
<code class="sig-prename descclassname">torch.autograd.</code><code class="sig-name descname">backward</code><span class="sig-paren">(</span><em class="sig-param">tensors: Union[torch.Tensor, Sequence[torch.Tensor]], grad_tensors: Union[torch.Tensor, Sequence[torch.Tensor], None] = None, retain_graph: Optional[bool] = None, create_graph: bool = False, grad_variables: Union[torch.Tensor, Sequence[torch.Tensor], None] = None</em><span class="sig-paren">)</span> → None<a class="reference internal" href="_modules/torch/autograd.html#backward"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.autograd.backward" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the sum of gradients of given tensors w.r.t. graph leaves.</p>
<p>The graph is differentiated using the chain rule. If any of <code class="docutils literal notranslate"><span class="pre">tensors</span></code>
are non-scalar (i.e. their data has more than one element) and require
gradient, then the Jacobian-vector product would be computed, in this
case the function additionally requires specifying <code class="docutils literal notranslate"><span class="pre">grad_tensors</span></code>.
It should be a sequence of matching length, that contains the “vector”
in the Jacobian-vector product, usually the gradient of the differentiated
function w.r.t. corresponding tensors (<code class="docutils literal notranslate"><span class="pre">None</span></code> is an acceptable value for
all tensors that don’t need gradient tensors).</p>
<p>This function accumulates gradients in the leaves - you might need to zero
<code class="docutils literal notranslate"><span class="pre">.grad</span></code> attributes or set them to <code class="docutils literal notranslate"><span class="pre">None</span></code> before calling it.
See <a class="reference internal" href="#default-grad-layouts"><span class="std std-ref">Default gradient layouts</span></a>
for details on the memory layout of accumulated gradients.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>Using this method with <code class="docutils literal notranslate"><span class="pre">create_graph=True</span></code> will create a reference cycle
between the parameter and its gradient which can cause a memory leak.
We recommend using <code class="docutils literal notranslate"><span class="pre">autograd.grad</span></code> when creating the graph to avoid this.
If you have to use this function, make sure to reset the <code class="docutils literal notranslate"><span class="pre">.grad</span></code> fields of your
parameters to <code class="docutils literal notranslate"><span class="pre">None</span></code> after use to break the cycle and avoid the leak.</p>
</div>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>tensors</strong> (<em>sequence of Tensor</em>) – Tensors of which the derivative will be
computed.</p></li>
<li><p><strong>grad_tensors</strong> (<em>sequence of</em><em> (</em><a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a><em> or </em><a class="reference external" href="https://docs.python.org/3/library/constants.html#None" title="(in Python v3.8)"><em>None</em></a><em>)</em>) – The “vector” in the Jacobian-vector
product, usually gradients w.r.t. each element of corresponding tensors.
None values can be specified for scalar Tensors or ones that don’t require
grad. If a None value would be acceptable for all grad_tensors, then this
argument is optional.</p></li>
<li><p><strong>retain_graph</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.8)"><em>bool</em></a><em>, </em><em>optional</em>) – If <code class="docutils literal notranslate"><span class="pre">False</span></code>, the graph used to compute the grad
will be freed. Note that in nearly all cases setting this option to <code class="docutils literal notranslate"><span class="pre">True</span></code>
is not needed and often can be worked around in a much more efficient
way. Defaults to the value of <code class="docutils literal notranslate"><span class="pre">create_graph</span></code>.</p></li>
<li><p><strong>create_graph</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.8)"><em>bool</em></a><em>, </em><em>optional</em>) – If <code class="docutils literal notranslate"><span class="pre">True</span></code>, graph of the derivative will
be constructed, allowing to compute higher order derivative products.
Defaults to <code class="docutils literal notranslate"><span class="pre">False</span></code>.</p></li>
</ul>
</dd>
</dl>
</dd></dl>
<dl class="function">
<dt id="torch.autograd.grad">
<code class="sig-prename descclassname">torch.autograd.</code><code class="sig-name descname">grad</code><span class="sig-paren">(</span><em class="sig-param">outputs: Union[torch.Tensor, Sequence[torch.Tensor]], inputs: Union[torch.Tensor, Sequence[torch.Tensor]], grad_outputs: Union[torch.Tensor, Sequence[torch.Tensor], None] = None, retain_graph: Optional[bool] = None, create_graph: bool = False, only_inputs: bool = True, allow_unused: bool = False</em><span class="sig-paren">)</span> → Tuple[torch.Tensor, ...]<a class="reference internal" href="_modules/torch/autograd.html#grad"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.autograd.grad" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes and returns the sum of gradients of outputs w.r.t. the inputs.</p>
<p><code class="docutils literal notranslate"><span class="pre">grad_outputs</span></code> should be a sequence of length matching <code class="docutils literal notranslate"><span class="pre">output</span></code>
containing the “vector” in Jacobian-vector product, usually the pre-computed
gradients w.r.t. each of the outputs. If an output doesn’t require_grad,
then the gradient can be <code class="docutils literal notranslate"><span class="pre">None</span></code>).</p>
<p>If <code class="docutils literal notranslate"><span class="pre">only_inputs</span></code> is <code class="docutils literal notranslate"><span class="pre">True</span></code>, the function will only return a list of gradients
w.r.t the specified inputs. If it’s <code class="docutils literal notranslate"><span class="pre">False</span></code>, then gradient w.r.t. all remaining
leaves will still be computed, and will be accumulated into their <code class="docutils literal notranslate"><span class="pre">.grad</span></code>
attribute.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>outputs</strong> (<em>sequence of Tensor</em>) – outputs of the differentiated function.</p></li>
<li><p><strong>inputs</strong> (<em>sequence of Tensor</em>) – Inputs w.r.t. which the gradient will be
returned (and not accumulated into <code class="docutils literal notranslate"><span class="pre">.grad</span></code>).</p></li>
<li><p><strong>grad_outputs</strong> (<em>sequence of Tensor</em>) – The “vector” in the Jacobian-vector product.
Usually gradients w.r.t. each output. None values can be specified for scalar
Tensors or ones that don’t require grad. If a None value would be acceptable
for all grad_tensors, then this argument is optional. Default: None.</p></li>
<li><p><strong>retain_graph</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.8)"><em>bool</em></a><em>, </em><em>optional</em>) – If <code class="docutils literal notranslate"><span class="pre">False</span></code>, the graph used to compute the grad
will be freed. Note that in nearly all cases setting this option to <code class="docutils literal notranslate"><span class="pre">True</span></code>
is not needed and often can be worked around in a much more efficient
way. Defaults to the value of <code class="docutils literal notranslate"><span class="pre">create_graph</span></code>.</p></li>
<li><p><strong>create_graph</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.8)"><em>bool</em></a><em>, </em><em>optional</em>) – If <code class="docutils literal notranslate"><span class="pre">True</span></code>, graph of the derivative will
be constructed, allowing to compute higher order derivative products.
Default: <code class="docutils literal notranslate"><span class="pre">False</span></code>.</p></li>
<li><p><strong>allow_unused</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.8)"><em>bool</em></a><em>, </em><em>optional</em>) – If <code class="docutils literal notranslate"><span class="pre">False</span></code>, specifying inputs that were not
used when computing outputs (and therefore their grad is always zero)
is an error. Defaults to <code class="docutils literal notranslate"><span class="pre">False</span></code>.</p></li>
</ul>
</dd>
</dl>
</dd></dl>
<div class="section" id="functional-higher-level-api">
<span id="functional-api"></span><h2>Functional higher level API<a class="headerlink" href="#functional-higher-level-api" title="Permalink to this headline">¶</a></h2>
<div class="admonition warning">
<p class="admonition-title">Warning</p>
<p>This API is in beta. Even though the function signatures are very unlikely to change, major
improvements to performances are planned before we consider this stable.</p>
</div>
<p>This section contains the higher level API for the autograd that builds on the basic API above
and allows you to compute jacobians, hessians, etc.</p>
<p>This API works with user-provided functions that take only Tensors as input and return
only Tensors.
If your function takes other arguments that are not Tensors or Tensors that don’t have requires_grad set,
you can use a lambda to capture them.
For example, for a function <code class="docutils literal notranslate"><span class="pre">f</span></code> that takes three inputs, a Tensor for which we want the jacobian, another
tensor that should be considered constant and a boolean flag as <code class="docutils literal notranslate"><span class="pre">f(input,</span> <span class="pre">constant,</span> <span class="pre">flag=flag)</span></code>
you can use it as <code class="docutils literal notranslate"><span class="pre">functional.jacobian(lambda</span> <span class="pre">x:</span> <span class="pre">f(x,</span> <span class="pre">constant,</span> <span class="pre">flag=flag),</span> <span class="pre">input)</span></code>.</p>
<dl class="function">
<dt id="torch.autograd.functional.jacobian">
<code class="sig-prename descclassname">torch.autograd.functional.</code><code class="sig-name descname">jacobian</code><span class="sig-paren">(</span><em class="sig-param">func</em>, <em class="sig-param">inputs</em>, <em class="sig-param">create_graph=False</em>, <em class="sig-param">strict=False</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/torch/autograd/functional.html#jacobian"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.autograd.functional.jacobian" title="Permalink to this definition">¶</a></dt>
<dd><p>Function that computes the Jacobian of a given function.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>func</strong> (<em>function</em>) – a Python function that takes Tensor inputs and returns
a tuple of Tensors or a Tensor.</p></li>
<li><p><strong>inputs</strong> (<em>tuple of Tensors</em><em> or </em><a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – inputs to the function <code class="docutils literal notranslate"><span class="pre">func</span></code>.</p></li>
<li><p><strong>create_graph</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.8)"><em>bool</em></a><em>, </em><em>optional</em>) – If <code class="docutils literal notranslate"><span class="pre">True</span></code>, the Jacobian will be
computed in a differentiable manner. Note that when <code class="docutils literal notranslate"><span class="pre">strict</span></code> is
<code class="docutils literal notranslate"><span class="pre">False</span></code>, the result can not require gradients or be disconnected
from the inputs. Defaults to <code class="docutils literal notranslate"><span class="pre">False</span></code>.</p></li>
<li><p><strong>strict</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.8)"><em>bool</em></a><em>, </em><em>optional</em>) – If <code class="docutils literal notranslate"><span class="pre">True</span></code>, an error will be raised when we
detect that there exists an input such that all the outputs are
independent of it. If <code class="docutils literal notranslate"><span class="pre">False</span></code>, we return a Tensor of zeros as the
jacobian for said inputs, which is the expected mathematical value.
Defaults to <code class="docutils literal notranslate"><span class="pre">False</span></code>.</p></li>
</ul>
</dd>
<dt class="field-even">Returns</dt>
<dd class="field-even"><p><dl class="simple">
<dt>if there are a single</dt><dd><p>input and output, this will be a single Tensor containing the
Jacobian for the linearized inputs and output. If one of the two is
a tuple, then the Jacobian will be a tuple of Tensors. If both of
them are tuples, then the Jacobian will be a tuple of tuple of
Tensors where <code class="docutils literal notranslate"><span class="pre">Jacobian[i][j]</span></code> will contain the Jacobian of the
<code class="docutils literal notranslate"><span class="pre">i</span></code>th output and <code class="docutils literal notranslate"><span class="pre">j</span></code>th input and will have as size the
concatenation of the sizes of the corresponding output and the
corresponding input.</p>
</dd>
</dl>
</p>
</dd>
<dt class="field-odd">Return type</dt>
<dd class="field-odd"><p>Jacobian (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor">Tensor</a> or nested tuple of Tensors)</p>
</dd>
</dl>
<p class="rubric">Example</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="k">def</span> <span class="nf">exp_reducer</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="gp">... </span> <span class="k">return</span> <span class="n">x</span><span class="o">.</span><span class="n">exp</span><span class="p">()</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">dim</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">inputs</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">jacobian</span><span class="p">(</span><span class="n">exp_reducer</span><span class="p">,</span> <span class="n">inputs</span><span class="p">)</span>
<span class="go">tensor([[[1.4917, 2.4352],</span>
<span class="go"> [0.0000, 0.0000]],</span>
</pre></div>
</div>
<blockquote>
<div><dl class="simple">
<dt>[[0.0000, 0.0000],</dt><dd><p>[2.4369, 2.3799]]])</p>
</dd>
</dl>
</div></blockquote>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">jacobian</span><span class="p">(</span><span class="n">exp_reducer</span><span class="p">,</span> <span class="n">inputs</span><span class="p">,</span> <span class="n">create_graph</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="go">tensor([[[1.4917, 2.4352],</span>
<span class="go"> [0.0000, 0.0000]],</span>
</pre></div>
</div>
<blockquote>
<div><dl class="simple">
<dt>[[0.0000, 0.0000],</dt><dd><p>[2.4369, 2.3799]]], grad_fn=<ViewBackward>)</p>
</dd>
</dl>
</div></blockquote>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="k">def</span> <span class="nf">exp_adder</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="gp">... </span> <span class="k">return</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">x</span><span class="o">.</span><span class="n">exp</span><span class="p">()</span> <span class="o">+</span> <span class="mi">3</span> <span class="o">*</span> <span class="n">y</span>
<span class="gp">>>> </span><span class="n">inputs</span> <span class="o">=</span> <span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="n">torch</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span>
<span class="gp">>>> </span><span class="n">jacobian</span><span class="p">(</span><span class="n">exp_adder</span><span class="p">,</span> <span class="n">inputs</span><span class="p">)</span>
<span class="go">(tensor([[2.8052, 0.0000],</span>
<span class="go"> [0.0000, 3.3963]]),</span>
<span class="go"> tensor([[3., 0.],</span>
<span class="go"> [0., 3.]]))</span>
</pre></div>
</div>
</dd></dl>
<dl class="function">
<dt id="torch.autograd.functional.hessian">
<code class="sig-prename descclassname">torch.autograd.functional.</code><code class="sig-name descname">hessian</code><span class="sig-paren">(</span><em class="sig-param">func</em>, <em class="sig-param">inputs</em>, <em class="sig-param">create_graph=False</em>, <em class="sig-param">strict=False</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/torch/autograd/functional.html#hessian"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.autograd.functional.hessian" title="Permalink to this definition">¶</a></dt>
<dd><p>Function that computes the Hessian of a given scalar function.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>func</strong> (<em>function</em>) – a Python function that takes Tensor inputs and returns
a Tensor with a single element.</p></li>
<li><p><strong>inputs</strong> (<em>tuple of Tensors</em><em> or </em><a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – inputs to the function <code class="docutils literal notranslate"><span class="pre">func</span></code>.</p></li>
<li><p><strong>create_graph</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.8)"><em>bool</em></a><em>, </em><em>optional</em>) – If <code class="docutils literal notranslate"><span class="pre">True</span></code>, the Hessian will be computed in
a differentiable manner. Note that when <code class="docutils literal notranslate"><span class="pre">strict</span></code> is <code class="docutils literal notranslate"><span class="pre">False</span></code>, the result can not
require gradients or be disconnected from the inputs.
Defaults to <code class="docutils literal notranslate"><span class="pre">False</span></code>.</p></li>
<li><p><strong>strict</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.8)"><em>bool</em></a><em>, </em><em>optional</em>) – If <code class="docutils literal notranslate"><span class="pre">True</span></code>, an error will be raised when we detect that there exists an input
such that all the outputs are independent of it. If <code class="docutils literal notranslate"><span class="pre">False</span></code>, we return a Tensor of zeros as the
hessian for said inputs, which is the expected mathematical value.
Defaults to <code class="docutils literal notranslate"><span class="pre">False</span></code>.</p></li>
</ul>
</dd>
<dt class="field-even">Returns</dt>
<dd class="field-even"><p><dl class="simple">
<dt>Hessian (Tensor or a tuple of tuple of Tensors) if there are a single input,</dt><dd><p>this will be a single Tensor containing the Hessian for the input.
If it is a tuple, then the Hessian will be a tuple of tuples where
<code class="docutils literal notranslate"><span class="pre">Hessian[i][j]</span></code> will contain the Hessian of the <code class="docutils literal notranslate"><span class="pre">i</span></code>th input
and <code class="docutils literal notranslate"><span class="pre">j</span></code>th input with size the sum of the size of the <code class="docutils literal notranslate"><span class="pre">i</span></code>th input plus
the size of the <code class="docutils literal notranslate"><span class="pre">j</span></code>th input.</p>
</dd>
</dl>
</p>
</dd>
</dl>
<p class="rubric">Example</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="k">def</span> <span class="nf">pow_reducer</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="gp">... </span> <span class="k">return</span> <span class="n">x</span><span class="o">.</span><span class="n">pow</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span>
<span class="gp">>>> </span><span class="n">inputs</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">hessian</span><span class="p">(</span><span class="n">pow_reducer</span><span class="p">,</span> <span class="n">inputs</span><span class="p">)</span>
<span class="go">tensor([[[[5.2265, 0.0000],</span>
<span class="go"> [0.0000, 0.0000]],</span>
</pre></div>
</div>
<blockquote>
<div><blockquote>
<div><dl class="simple">
<dt>[[0.0000, 4.8221],</dt><dd><p>[0.0000, 0.0000]]],</p>
</dd>
</dl>
</div></blockquote>
<dl>
<dt>[[[0.0000, 0.0000],</dt><dd><blockquote>
<div><p>[1.9456, 0.0000]],</p>
</div></blockquote>
<dl class="simple">
<dt>[[0.0000, 0.0000],</dt><dd><p>[0.0000, 3.2550]]]])</p>
</dd>
</dl>
</dd>
</dl>
</div></blockquote>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">hessian</span><span class="p">(</span><span class="n">pow_reducer</span><span class="p">,</span> <span class="n">inputs</span><span class="p">,</span> <span class="n">create_graph</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="go">tensor([[[[5.2265, 0.0000],</span>
<span class="go"> [0.0000, 0.0000]],</span>
</pre></div>
</div>
<blockquote>
<div><blockquote>
<div><dl class="simple">
<dt>[[0.0000, 4.8221],</dt><dd><p>[0.0000, 0.0000]]],</p>
</dd>
</dl>
</div></blockquote>
<dl>
<dt>[[[0.0000, 0.0000],</dt><dd><blockquote>
<div><p>[1.9456, 0.0000]],</p>
</div></blockquote>
<dl class="simple">
<dt>[[0.0000, 0.0000],</dt><dd><p>[0.0000, 3.2550]]]], grad_fn=<ViewBackward>)</p>
</dd>
</dl>
</dd>
</dl>
</div></blockquote>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="k">def</span> <span class="nf">pow_adder_reducer</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="gp">... </span> <span class="k">return</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">x</span><span class="o">.</span><span class="n">pow</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="mi">3</span> <span class="o">*</span> <span class="n">y</span><span class="o">.</span><span class="n">pow</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span>
<span class="gp">>>> </span><span class="n">inputs</span> <span class="o">=</span> <span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="n">torch</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span>
<span class="gp">>>> </span><span class="n">hessian</span><span class="p">(</span><span class="n">pow_adder_reducer</span><span class="p">,</span> <span class="n">inputs</span><span class="p">)</span>
<span class="go">((tensor([[4., 0.],</span>
<span class="go"> [0., 4.]]),</span>
<span class="go"> tensor([[0., 0.],</span>
<span class="go"> [0., 0.]])),</span>
<span class="go"> (tensor([[0., 0.],</span>
<span class="go"> [0., 0.]]),</span>
<span class="go"> tensor([[6., 0.],</span>
<span class="go"> [0., 6.]])))</span>
</pre></div>
</div>
</dd></dl>
<dl class="function">
<dt id="torch.autograd.functional.vjp">
<code class="sig-prename descclassname">torch.autograd.functional.</code><code class="sig-name descname">vjp</code><span class="sig-paren">(</span><em class="sig-param">func</em>, <em class="sig-param">inputs</em>, <em class="sig-param">v=None</em>, <em class="sig-param">create_graph=False</em>, <em class="sig-param">strict=False</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/torch/autograd/functional.html#vjp"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.autograd.functional.vjp" title="Permalink to this definition">¶</a></dt>
<dd><p>Function that computes the dot product between a vector <code class="docutils literal notranslate"><span class="pre">v</span></code> and the
Jacobian of the given function at the point given by the inputs.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>func</strong> (<em>function</em>) – a Python function that takes Tensor inputs and returns
a tuple of Tensors or a Tensor.</p></li>
<li><p><strong>inputs</strong> (<em>tuple of Tensors</em><em> or </em><a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – inputs to the function <code class="docutils literal notranslate"><span class="pre">func</span></code>.</p></li>
<li><p><strong>v</strong> (<em>tuple of Tensors</em><em> or </em><a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – The vector for which the vector
Jacobian product is computed. Must be the same size as the output
of <code class="docutils literal notranslate"><span class="pre">func</span></code>. This argument is optional when the output of <code class="docutils literal notranslate"><span class="pre">func</span></code>
contains a single element and (if it is not provided) will be set
as a Tensor containing a single <code class="docutils literal notranslate"><span class="pre">1</span></code>.</p></li>
<li><p><strong>create_graph</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.8)"><em>bool</em></a><em>, </em><em>optional</em>) – If <code class="docutils literal notranslate"><span class="pre">True</span></code>, both the output and result
will be computed in a differentiable way. Note that when <code class="docutils literal notranslate"><span class="pre">strict</span></code>
is <code class="docutils literal notranslate"><span class="pre">False</span></code>, the result can not require gradients or be
disconnected from the inputs. Defaults to <code class="docutils literal notranslate"><span class="pre">False</span></code>.</p></li>
<li><p><strong>strict</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.8)"><em>bool</em></a><em>, </em><em>optional</em>) – If <code class="docutils literal notranslate"><span class="pre">True</span></code>, an error will be raised when we
detect that there exists an input such that all the outputs are
independent of it. If <code class="docutils literal notranslate"><span class="pre">False</span></code>, we return a Tensor of zeros as the
vjp for said inputs, which is the expected mathematical value.
Defaults to <code class="docutils literal notranslate"><span class="pre">False</span></code>.</p></li>
</ul>
</dd>
<dt class="field-even">Returns</dt>
<dd class="field-even"><p>result of the dot product with
the same shape as the inputs.</p>
</dd>
<dt class="field-odd">Return type</dt>
<dd class="field-odd"><p>vjp (tuple of Tensors or <a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor">Tensor</a>)</p>
</dd>
</dl>
<p class="rubric">Example</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="k">def</span> <span class="nf">exp_reducer</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="gp">... </span> <span class="k">return</span> <span class="n">x</span><span class="o">.</span><span class="n">exp</span><span class="p">()</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">dim</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">inputs</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">v</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">vjp</span><span class="p">(</span><span class="n">exp_reducer</span><span class="p">,</span> <span class="n">inputs</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span>
<span class="go">(tensor([5.7817, 7.2458, 5.7830, 6.7782]),</span>
<span class="go"> tensor([[1.4458, 1.3962, 1.3042, 1.6354],</span>
<span class="go"> [2.1288, 1.0652, 1.5483, 2.5035],</span>
<span class="go"> [2.2046, 1.1292, 1.1432, 1.3059],</span>
<span class="go"> [1.3225, 1.6652, 1.7753, 2.0152]]))</span>
</pre></div>
</div>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">vjp</span><span class="p">(</span><span class="n">exp_reducer</span><span class="p">,</span> <span class="n">inputs</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">create_graph</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="go">(tensor([5.7817, 7.2458, 5.7830, 6.7782], grad_fn=<SumBackward1>),</span>
<span class="go"> tensor([[1.4458, 1.3962, 1.3042, 1.6354],</span>
<span class="go"> [2.1288, 1.0652, 1.5483, 2.5035],</span>
<span class="go"> [2.2046, 1.1292, 1.1432, 1.3059],</span>
<span class="go"> [1.3225, 1.6652, 1.7753, 2.0152]], grad_fn=<MulBackward0>))</span>
</pre></div>
</div>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="k">def</span> <span class="nf">adder</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="gp">... </span> <span class="k">return</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">x</span> <span class="o">+</span> <span class="mi">3</span> <span class="o">*</span> <span class="n">y</span>
<span class="gp">>>> </span><span class="n">inputs</span> <span class="o">=</span> <span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="n">torch</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span>
<span class="gp">>>> </span><span class="n">v</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">vjp</span><span class="p">(</span><span class="n">adder</span><span class="p">,</span> <span class="n">inputs</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span>
<span class="go">(tensor([2.4225, 2.3340]),</span>
<span class="go"> (tensor([2., 2.]), tensor([3., 3.])))</span>
</pre></div>
</div>
</dd></dl>
<dl class="function">
<dt id="torch.autograd.functional.jvp">
<code class="sig-prename descclassname">torch.autograd.functional.</code><code class="sig-name descname">jvp</code><span class="sig-paren">(</span><em class="sig-param">func</em>, <em class="sig-param">inputs</em>, <em class="sig-param">v=None</em>, <em class="sig-param">create_graph=False</em>, <em class="sig-param">strict=False</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/torch/autograd/functional.html#jvp"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.autograd.functional.jvp" title="Permalink to this definition">¶</a></dt>
<dd><p>Function that computes the dot product between the Jacobian of
the given function at the point given by the inputs and a vector <code class="docutils literal notranslate"><span class="pre">v</span></code>.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>func</strong> (<em>function</em>) – a Python function that takes Tensor inputs and returns
a tuple of Tensors or a Tensor.</p></li>
<li><p><strong>inputs</strong> (<em>tuple of Tensors</em><em> or </em><a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – inputs to the function <code class="docutils literal notranslate"><span class="pre">func</span></code>.</p></li>
<li><p><strong>v</strong> (<em>tuple of Tensors</em><em> or </em><a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – The vector for which the Jacobian
vector product is computed. Must be the same size as the input of
<code class="docutils literal notranslate"><span class="pre">func</span></code>. This argument is optional when the input to <code class="docutils literal notranslate"><span class="pre">func</span></code>
contains a single element and (if it is not provided) will be set
as a Tensor containing a single <code class="docutils literal notranslate"><span class="pre">1</span></code>.</p></li>
<li><p><strong>create_graph</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.8)"><em>bool</em></a><em>, </em><em>optional</em>) – If <code class="docutils literal notranslate"><span class="pre">True</span></code>, both the output and result
will be computed in a differentiable way. Note that when <code class="docutils literal notranslate"><span class="pre">strict</span></code>
is <code class="docutils literal notranslate"><span class="pre">False</span></code>, the result can not require gradients or be
disconnected from the inputs. Defaults to <code class="docutils literal notranslate"><span class="pre">False</span></code>.</p></li>
<li><p><strong>strict</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.8)"><em>bool</em></a><em>, </em><em>optional</em>) – If <code class="docutils literal notranslate"><span class="pre">True</span></code>, an error will be raised when we
detect that there exists an input such that all the outputs are
independent of it. If <code class="docutils literal notranslate"><span class="pre">False</span></code>, we return a Tensor of zeros as the
jvp for said inputs, which is the expected mathematical value.
Defaults to <code class="docutils literal notranslate"><span class="pre">False</span></code>.</p></li>
</ul>
</dd>
<dt class="field-even">Returns</dt>
<dd class="field-even"><p>result of the dot product with
the same shape as the output.</p>
</dd>
<dt class="field-odd">Return type</dt>
<dd class="field-odd"><p>jvp (tuple of Tensors or <a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor">Tensor</a>)</p>
</dd>
</dl>
<p class="rubric">Example</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="k">def</span> <span class="nf">exp_reducer</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="gp">... </span> <span class="k">return</span> <span class="n">x</span><span class="o">.</span><span class="n">exp</span><span class="p">()</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">dim</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">inputs</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">v</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">jvp</span><span class="p">(</span><span class="n">exp_reducer</span><span class="p">,</span> <span class="n">inputs</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span>
<span class="go">(tensor([6.3090, 4.6742, 7.9114, 8.2106]),</span>
<span class="go"> tensor([6.3090, 4.6742, 7.9114, 8.2106]))</span>
</pre></div>
</div>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">jvp</span><span class="p">(</span><span class="n">exp_reducer</span><span class="p">,</span> <span class="n">inputs</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">create_graph</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="go">(tensor([6.3090, 4.6742, 7.9114, 8.2106], grad_fn=<SumBackward1>),</span>
<span class="go"> tensor([6.3090, 4.6742, 7.9114, 8.2106], grad_fn=<SqueezeBackward1>))</span>
</pre></div>
</div>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="k">def</span> <span class="nf">adder</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="gp">... </span> <span class="k">return</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">x</span> <span class="o">+</span> <span class="mi">3</span> <span class="o">*</span> <span class="n">y</span>
<span class="gp">>>> </span><span class="n">inputs</span> <span class="o">=</span> <span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="n">torch</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span>
<span class="gp">>>> </span><span class="n">v</span> <span class="o">=</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">2</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">2</span><span class="p">))</span>
<span class="gp">>>> </span><span class="n">jvp</span><span class="p">(</span><span class="n">adder</span><span class="p">,</span> <span class="n">inputs</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span>
<span class="go">(tensor([2.2399, 2.5005]),</span>
<span class="go"> tensor([5., 5.]))</span>
</pre></div>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>The jvp is currently computed by using the backward of the backward
(sometimes called the double backwards trick) as we don’t have support
for forward mode AD in PyTorch at the moment.</p>
</div>
</dd></dl>
<dl class="function">
<dt id="torch.autograd.functional.vhp">
<code class="sig-prename descclassname">torch.autograd.functional.</code><code class="sig-name descname">vhp</code><span class="sig-paren">(</span><em class="sig-param">func</em>, <em class="sig-param">inputs</em>, <em class="sig-param">v=None</em>, <em class="sig-param">create_graph=False</em>, <em class="sig-param">strict=False</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/torch/autograd/functional.html#vhp"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.autograd.functional.vhp" title="Permalink to this definition">¶</a></dt>
<dd><p>Function that computes the dot product between a vector <code class="docutils literal notranslate"><span class="pre">v</span></code> and the
Hessian of a given scalar function at the point given by the inputs.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>func</strong> (<em>function</em>) – a Python function that takes Tensor inputs and returns
a Tensor with a single element.</p></li>
<li><p><strong>inputs</strong> (<em>tuple of Tensors</em><em> or </em><a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – inputs to the function <code class="docutils literal notranslate"><span class="pre">func</span></code>.</p></li>
<li><p><strong>v</strong> (<em>tuple of Tensors</em><em> or </em><a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – The vector for which the vector Hessian
product is computed. Must be the same size as the input of
<code class="docutils literal notranslate"><span class="pre">func</span></code>. This argument is optional when <code class="docutils literal notranslate"><span class="pre">func</span></code>’s input contains
a single element and (if it is not provided) will be set as a
Tensor containing a single <code class="docutils literal notranslate"><span class="pre">1</span></code>.</p></li>
<li><p><strong>create_graph</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.8)"><em>bool</em></a><em>, </em><em>optional</em>) – If <code class="docutils literal notranslate"><span class="pre">True</span></code>, both the output and result
will be computed in a differentiable way. Note that when <code class="docutils literal notranslate"><span class="pre">strict</span></code>
is <code class="docutils literal notranslate"><span class="pre">False</span></code>, the result can not require gradients or be
disconnected from the inputs.
Defaults to <code class="docutils literal notranslate"><span class="pre">False</span></code>.</p></li>
<li><p><strong>strict</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.8)"><em>bool</em></a><em>, </em><em>optional</em>) – If <code class="docutils literal notranslate"><span class="pre">True</span></code>, an error will be raised when we
detect that there exists an input such that all the outputs are
independent of it. If <code class="docutils literal notranslate"><span class="pre">False</span></code>, we return a Tensor of zeros as the
vhp for said inputs, which is the expected mathematical value.
Defaults to <code class="docutils literal notranslate"><span class="pre">False</span></code>.</p></li>
</ul>
</dd>
<dt class="field-even">Returns</dt>
<dd class="field-even"><p><dl>
<dt>tuple with:</dt><dd><p>func_output (tuple of Tensors or Tensor): output of <code class="docutils literal notranslate"><span class="pre">func(inputs)</span></code></p>
<p>vhp (tuple of Tensors or Tensor): result of the dot product with the
same shape as the inputs.</p>
</dd>
</dl>
</p>
</dd>
<dt class="field-odd">Return type</dt>
<dd class="field-odd"><p>output (<a class="reference external" href="https://docs.python.org/3/library/stdtypes.html#tuple" title="(in Python v3.8)">tuple</a>)</p>
</dd>
</dl>
<p class="rubric">Example</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="k">def</span> <span class="nf">pow_reducer</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="gp">... </span> <span class="k">return</span> <span class="n">x</span><span class="o">.</span><span class="n">pow</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span>
<span class="gp">>>> </span><span class="n">inputs</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">v</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">vhp</span><span class="p">(</span><span class="n">pow_reducer</span><span class="p">,</span> <span class="n">inputs</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span>
<span class="go">(tensor(0.5591),</span>
<span class="go"> tensor([[1.0689, 1.2431],</span>
<span class="go"> [3.0989, 4.4456]]))</span>
<span class="gp">>>> </span><span class="n">vhp</span><span class="p">(</span><span class="n">pow_reducer</span><span class="p">,</span> <span class="n">inputs</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">create_graph</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="go">(tensor(0.5591, grad_fn=<SumBackward0>),</span>
<span class="go"> tensor([[1.0689, 1.2431],</span>
<span class="go"> [3.0989, 4.4456]], grad_fn=<MulBackward0>))</span>
<span class="gp">>>> </span><span class="k">def</span> <span class="nf">pow_adder_reducer</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="gp">... </span> <span class="k">return</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">x</span><span class="o">.</span><span class="n">pow</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="mi">3</span> <span class="o">*</span> <span class="n">y</span><span class="o">.</span><span class="n">pow</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span>
<span class="gp">>>> </span><span class="n">inputs</span> <span class="o">=</span> <span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="n">torch</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span>
<span class="gp">>>> </span><span class="n">v</span> <span class="o">=</span> <span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="mi">2</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">2</span><span class="p">))</span>
<span class="gp">>>> </span><span class="n">vhp</span><span class="p">(</span><span class="n">pow_adder_reducer</span><span class="p">,</span> <span class="n">inputs</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span>
<span class="go">(tensor(4.8053),</span>
<span class="go"> (tensor([0., 0.]),</span>
<span class="go"> tensor([6., 6.])))</span>
</pre></div>
</div>
</dd></dl>
<dl class="function">
<dt id="torch.autograd.functional.hvp">
<code class="sig-prename descclassname">torch.autograd.functional.</code><code class="sig-name descname">hvp</code><span class="sig-paren">(</span><em class="sig-param">func</em>, <em class="sig-param">inputs</em>, <em class="sig-param">v=None</em>, <em class="sig-param">create_graph=False</em>, <em class="sig-param">strict=False</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/torch/autograd/functional.html#hvp"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.autograd.functional.hvp" title="Permalink to this definition">¶</a></dt>
<dd><p>Function that computes the dot product between the Hessian of a given scalar
function and a vector <code class="docutils literal notranslate"><span class="pre">v</span></code> at the point given by the inputs.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>func</strong> (<em>function</em>) – a Python function that takes Tensor inputs and returns
a Tensor with a single element.</p></li>
<li><p><strong>inputs</strong> (<em>tuple of Tensors</em><em> or </em><a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – inputs to the function <code class="docutils literal notranslate"><span class="pre">func</span></code>.</p></li>
<li><p><strong>v</strong> (<em>tuple of Tensors</em><em> or </em><a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – The vector for which the Hessian vector
product is computed. Must be the same size as the input of
<code class="docutils literal notranslate"><span class="pre">func</span></code>. This argument is optional when <code class="docutils literal notranslate"><span class="pre">func</span></code>’s input contains
a single element and (if it is not provided) will be set as a
Tensor containing a single <code class="docutils literal notranslate"><span class="pre">1</span></code>.</p></li>
<li><p><strong>create_graph</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.8)"><em>bool</em></a><em>, </em><em>optional</em>) – If <code class="docutils literal notranslate"><span class="pre">True</span></code>, both the output and result will be
computed in a differentiable way. Note that when <code class="docutils literal notranslate"><span class="pre">strict</span></code> is
<code class="docutils literal notranslate"><span class="pre">False</span></code>, the result can not require gradients or be disconnected
from the inputs. Defaults to <code class="docutils literal notranslate"><span class="pre">False</span></code>.</p></li>
<li><p><strong>strict</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.8)"><em>bool</em></a><em>, </em><em>optional</em>) – If <code class="docutils literal notranslate"><span class="pre">True</span></code>, an error will be raised when we
detect that there exists an input such that all the outputs are
independent of it. If <code class="docutils literal notranslate"><span class="pre">False</span></code>, we return a Tensor of zeros as the
hvp for said inputs, which is the expected mathematical value.
Defaults to <code class="docutils literal notranslate"><span class="pre">False</span></code>.</p></li>
</ul>
</dd>
<dt class="field-even">Returns</dt>
<dd class="field-even"><p><dl class="simple">
<dt>output of <code class="docutils literal notranslate"><span class="pre">func(inputs)</span></code></dt><dd><p>hvp (tuple of Tensors or Tensor): result of the dot product with
the same shape as the inputs.</p>
</dd>
</dl>
</p>
</dd>
<dt class="field-odd">Return type</dt>
<dd class="field-odd"><p>func_output (tuple of Tensors or <a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor">Tensor</a>)</p>
</dd>
</dl>
<p class="rubric">Example</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="k">def</span> <span class="nf">pow_reducer</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="gp">... </span> <span class="k">return</span> <span class="n">x</span><span class="o">.</span><span class="n">pow</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span>
<span class="gp">>>> </span><span class="n">inputs</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">v</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">hvp</span><span class="p">(</span><span class="n">pow_reducer</span><span class="p">,</span> <span class="n">inputs</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span>
<span class="go">(tensor(0.1448),</span>
<span class="go"> tensor([[2.0239, 1.6456],</span>
<span class="go"> [2.4988, 1.4310]]))</span>
</pre></div>
</div>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">hvp</span><span class="p">(</span><span class="n">pow_reducer</span><span class="p">,</span> <span class="n">inputs</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">create_graph</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="go">(tensor(0.1448, grad_fn=<SumBackward0>),</span>
<span class="go"> tensor([[2.0239, 1.6456],</span>
<span class="go"> [2.4988, 1.4310]], grad_fn=<MulBackward0>))</span>
</pre></div>
</div>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="k">def</span> <span class="nf">pow_adder_reducer</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="gp">... </span> <span class="k">return</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">x</span><span class="o">.</span><span class="n">pow</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="mi">3</span> <span class="o">*</span> <span class="n">y</span><span class="o">.</span><span class="n">pow</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span>
<span class="gp">>>> </span><span class="n">inputs</span> <span class="o">=</span> <span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="n">torch</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span>
<span class="gp">>>> </span><span class="n">v</span> <span class="o">=</span> <span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="mi">2</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">2</span><span class="p">))</span>
<span class="gp">>>> </span><span class="n">hvp</span><span class="p">(</span><span class="n">pow_adder_reducer</span><span class="p">,</span> <span class="n">inputs</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span>
<span class="go">(tensor(2.3030),</span>
<span class="go"> (tensor([0., 0.]),</span>
<span class="go"> tensor([6., 6.])))</span>
</pre></div>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>This function is significantly slower than <cite>vhp</cite> due to backward mode AD constraints.
If your functions is twice continuously differentiable, then hvp = vhp.t(). So if you
know that your function satisfies this condition, you should use vhp instead that is
much faster with the current implementation.</p>
</div>
</dd></dl>
</div>
<div class="section" id="locally-disabling-gradient-computation">
<span id="locally-disable-grad"></span><h2>Locally disabling gradient computation<a class="headerlink" href="#locally-disabling-gradient-computation" title="Permalink to this headline">¶</a></h2>
<dl class="class">
<dt id="torch.autograd.no_grad">
<em class="property">class </em><code class="sig-prename descclassname">torch.autograd.</code><code class="sig-name descname">no_grad</code><a class="reference internal" href="_modules/torch/autograd/grad_mode.html#no_grad"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.autograd.no_grad" title="Permalink to this definition">¶</a></dt>
<dd><p>Context-manager that disabled gradient calculation.</p>
<p>Disabling gradient calculation is useful for inference, when you are sure
that you will not call <code class="xref py py-meth docutils literal notranslate"><span class="pre">Tensor.backward()</span></code>. It will reduce memory
consumption for computations that would otherwise have <cite>requires_grad=True</cite>.</p>
<p>In this mode, the result of every computation will have
<cite>requires_grad=False</cite>, even when the inputs have <cite>requires_grad=True</cite>.</p>
<p>This context manager is thread local; it will not affect computation
in other threads.</p>
<p>Also functions as a decorator. (Make sure to instantiate with parenthesis.)</p>
<p>Example:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">x</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span><span class="mi">1</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="gp">>>> </span><span class="k">with</span> <span class="n">torch</span><span class="o">.</span><span class="n">no_grad</span><span class="p">():</span>
<span class="gp">... </span> <span class="n">y</span> <span class="o">=</span> <span class="n">x</span> <span class="o">*</span> <span class="mi">2</span>
<span class="gp">>>> </span><span class="n">y</span><span class="o">.</span><span class="n">requires_grad</span>
<span class="go">False</span>
<span class="gp">>>> </span><span class="nd">@torch</span><span class="o">.</span><span class="n">no_grad</span><span class="p">()</span>
<span class="gp">... </span><span class="k">def</span> <span class="nf">doubler</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="gp">... </span> <span class="k">return</span> <span class="n">x</span> <span class="o">*</span> <span class="mi">2</span>
<span class="gp">>>> </span><span class="n">z</span> <span class="o">=</span> <span class="n">doubler</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">z</span><span class="o">.</span><span class="n">requires_grad</span>
<span class="go">False</span>
</pre></div>
</div>
</dd></dl>
<dl class="class">
<dt id="torch.autograd.enable_grad">
<em class="property">class </em><code class="sig-prename descclassname">torch.autograd.</code><code class="sig-name descname">enable_grad</code><a class="reference internal" href="_modules/torch/autograd/grad_mode.html#enable_grad"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.autograd.enable_grad" title="Permalink to this definition">¶</a></dt>
<dd><p>Context-manager that enables gradient calculation.</p>
<p>Enables gradient calculation, if it has been disabled via <a class="reference internal" href="#torch.autograd.no_grad" title="torch.autograd.no_grad"><code class="xref py py-class docutils literal notranslate"><span class="pre">no_grad</span></code></a>
or <a class="reference internal" href="#torch.autograd.set_grad_enabled" title="torch.autograd.set_grad_enabled"><code class="xref py py-class docutils literal notranslate"><span class="pre">set_grad_enabled</span></code></a>.</p>
<p>This context manager is thread local; it will not affect computation
in other threads.</p>
<p>Also functions as a decorator. (Make sure to instantiate with parenthesis.)</p>
<p>Example:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">x</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span><span class="mi">1</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="gp">>>> </span><span class="k">with</span> <span class="n">torch</span><span class="o">.</span><span class="n">no_grad</span><span class="p">():</span>
<span class="gp">... </span> <span class="k">with</span> <span class="n">torch</span><span class="o">.</span><span class="n">enable_grad</span><span class="p">():</span>
<span class="gp">... </span> <span class="n">y</span> <span class="o">=</span> <span class="n">x</span> <span class="o">*</span> <span class="mi">2</span>
<span class="gp">>>> </span><span class="n">y</span><span class="o">.</span><span class="n">requires_grad</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">y</span><span class="o">.</span><span class="n">backward</span><span class="p">()</span>
<span class="gp">>>> </span><span class="n">x</span><span class="o">.</span><span class="n">grad</span>
<span class="gp">>>> </span><span class="nd">@torch</span><span class="o">.</span><span class="n">enable_grad</span><span class="p">()</span>
<span class="gp">... </span><span class="k">def</span> <span class="nf">doubler</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="gp">... </span> <span class="k">return</span> <span class="n">x</span> <span class="o">*</span> <span class="mi">2</span>
<span class="gp">>>> </span><span class="k">with</span> <span class="n">torch</span><span class="o">.</span><span class="n">no_grad</span><span class="p">():</span>
<span class="gp">... </span> <span class="n">z</span> <span class="o">=</span> <span class="n">doubler</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">z</span><span class="o">.</span><span class="n">requires_grad</span>
<span class="go">True</span>
</pre></div>
</div>
</dd></dl>
<dl class="class">
<dt id="torch.autograd.set_grad_enabled">
<em class="property">class </em><code class="sig-prename descclassname">torch.autograd.</code><code class="sig-name descname">set_grad_enabled</code><span class="sig-paren">(</span><em class="sig-param">mode</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/torch/autograd/grad_mode.html#set_grad_enabled"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.autograd.set_grad_enabled" title="Permalink to this definition">¶</a></dt>
<dd><p>Context-manager that sets gradient calculation to on or off.</p>
<p><code class="docutils literal notranslate"><span class="pre">set_grad_enabled</span></code> will enable or disable grads based on its argument <code class="xref py py-attr docutils literal notranslate"><span class="pre">mode</span></code>.
It can be used as a context-manager or as a function.</p>
<p>This context manager is thread local; it will not affect computation
in other threads.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><p><strong>mode</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.8)"><em>bool</em></a>) – Flag whether to enable grad (<code class="docutils literal notranslate"><span class="pre">True</span></code>), or disable
(<code class="docutils literal notranslate"><span class="pre">False</span></code>). This can be used to conditionally enable
gradients.</p>
</dd>
</dl>
<p>Example:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">x</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span><span class="mi">1</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="gp">>>> </span><span class="n">is_train</span> <span class="o">=</span> <span class="kc">False</span>
<span class="gp">>>> </span><span class="k">with</span> <span class="n">torch</span><span class="o">.</span><span class="n">set_grad_enabled</span><span class="p">(</span><span class="n">is_train</span><span class="p">):</span>
<span class="gp">... </span> <span class="n">y</span> <span class="o">=</span> <span class="n">x</span> <span class="o">*</span> <span class="mi">2</span>
<span class="gp">>>> </span><span class="n">y</span><span class="o">.</span><span class="n">requires_grad</span>
<span class="go">False</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">set_grad_enabled</span><span class="p">(</span><span class="kc">True</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">y</span> <span class="o">=</span> <span class="n">x</span> <span class="o">*</span> <span class="mi">2</span>
<span class="gp">>>> </span><span class="n">y</span><span class="o">.</span><span class="n">requires_grad</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">set_grad_enabled</span><span class="p">(</span><span class="kc">False</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">y</span> <span class="o">=</span> <span class="n">x</span> <span class="o">*</span> <span class="mi">2</span>
<span class="gp">>>> </span><span class="n">y</span><span class="o">.</span><span class="n">requires_grad</span>
<span class="go">False</span>
</pre></div>
</div>
</dd></dl>
</div>
<div class="section" id="default-gradient-layouts">
<span id="default-grad-layouts"></span><h2>Default gradient layouts<a class="headerlink" href="#default-gradient-layouts" title="Permalink to this headline">¶</a></h2>
<p>When a non-sparse <code class="docutils literal notranslate"><span class="pre">param</span></code> receives a non-sparse gradient during
<a class="reference internal" href="#torch.autograd.backward" title="torch.autograd.backward"><code class="xref py py-func docutils literal notranslate"><span class="pre">torch.autograd.backward()</span></code></a> or <a class="reference internal" href="#torch.Tensor.backward" title="torch.Tensor.backward"><code class="xref py py-func docutils literal notranslate"><span class="pre">torch.Tensor.backward()</span></code></a>