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<li class="toctree-l1"><a class="reference internal" href="autograd.html">Autograd mechanics</a></li>
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<div class="section" id="broadcasting-semantics">
<span id="id1"></span><h1>Broadcasting semantics<a class="headerlink" href="#broadcasting-semantics" title="Permalink to this headline">¶</a></h1>
<p>Many PyTorch operations support <a class="reference external" href="https://numpy.org/doc/stable/user/basics.broadcasting.html#module-numpy.doc.broadcasting" title="(in NumPy v1.19)"><code class="xref any docutils literal notranslate"><span class="pre">NumPy</span> <span class="pre">Broadcasting</span> <span class="pre">Semantics</span></code></a>.</p>
<p>In short, if a PyTorch operation supports broadcast, then its Tensor arguments can be
automatically expanded to be of equal sizes (without making copies of the data).</p>
<div class="section" id="general-semantics">
<h2>General semantics<a class="headerlink" href="#general-semantics" title="Permalink to this headline">¶</a></h2>
<p>Two tensors are “broadcastable” if the following rules hold:</p>
<ul class="simple">
<li><p>Each tensor has at least one dimension.</p></li>
<li><p>When iterating over the dimension sizes, starting at the trailing dimension,
the dimension sizes must either be equal, one of them is 1, or one of them
does not exist.</p></li>
</ul>
<p>For 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">empty</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">y</span><span class="o">=</span><span class="n">torch</span><span class="o">.</span><span class="n">empty</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
<span class="go"># same shapes are always broadcastable (i.e. the above rules always hold)</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">empty</span><span class="p">((</span><span class="mi">0</span><span class="p">,))</span>
<span class="gp">>>> </span><span class="n">y</span><span class="o">=</span><span class="n">torch</span><span class="o">.</span><span class="n">empty</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="go"># x and y are not broadcastable, because x does not have at least 1 dimension</span>
<span class="go"># can line up trailing dimensions</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">empty</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">y</span><span class="o">=</span><span class="n">torch</span><span class="o">.</span><span class="n">empty</span><span class="p">(</span> <span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="go"># x and y are broadcastable.</span>
<span class="go"># 1st trailing dimension: both have size 1</span>
<span class="go"># 2nd trailing dimension: y has size 1</span>
<span class="go"># 3rd trailing dimension: x size == y size</span>
<span class="go"># 4th trailing dimension: y dimension doesn't exist</span>
<span class="go"># but:</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">empty</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">y</span><span class="o">=</span><span class="n">torch</span><span class="o">.</span><span class="n">empty</span><span class="p">(</span> <span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="go"># x and y are not broadcastable, because in the 3rd trailing dimension 2 != 3</span>
</pre></div>
</div>
<p>If two tensors <code class="xref py py-attr docutils literal notranslate"><span class="pre">x</span></code>, <code class="xref py py-attr docutils literal notranslate"><span class="pre">y</span></code> are “broadcastable”, the resulting tensor size
is calculated as follows:</p>
<ul class="simple">
<li><p>If the number of dimensions of <code class="xref py py-attr docutils literal notranslate"><span class="pre">x</span></code> and <code class="xref py py-attr docutils literal notranslate"><span class="pre">y</span></code> are not equal, prepend 1
to the dimensions of the tensor with fewer dimensions to make them equal length.</p></li>
<li><p>Then, for each dimension size, the resulting dimension size is the max of the sizes of
<code class="xref py py-attr docutils literal notranslate"><span class="pre">x</span></code> and <code class="xref py py-attr docutils literal notranslate"><span class="pre">y</span></code> along that dimension.</p></li>
</ul>
<p>For Example:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="c1"># can line up trailing dimensions to make reading easier</span>
<span class="o">>>></span> <span class="n">x</span><span class="o">=</span><span class="n">torch</span><span class="o">.</span><span class="n">empty</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="o">>>></span> <span class="n">y</span><span class="o">=</span><span class="n">torch</span><span class="o">.</span><span class="n">empty</span><span class="p">(</span> <span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="o">>>></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="o">.</span><span class="n">size</span><span class="p">()</span>
<span class="n">torch</span><span class="o">.</span><span class="n">Size</span><span class="p">([</span><span class="mi">5</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
<span class="c1"># but not necessary:</span>
<span class="o">>>></span> <span class="n">x</span><span class="o">=</span><span class="n">torch</span><span class="o">.</span><span class="n">empty</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="o">>>></span> <span class="n">y</span><span class="o">=</span><span class="n">torch</span><span class="o">.</span><span class="n">empty</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">7</span><span class="p">)</span>
<span class="o">>>></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="o">.</span><span class="n">size</span><span class="p">()</span>
<span class="n">torch</span><span class="o">.</span><span class="n">Size</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">7</span><span class="p">])</span>
<span class="o">>>></span> <span class="n">x</span><span class="o">=</span><span class="n">torch</span><span class="o">.</span><span class="n">empty</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="o">>>></span> <span class="n">y</span><span class="o">=</span><span class="n">torch</span><span class="o">.</span><span class="n">empty</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="o">>>></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="o">.</span><span class="n">size</span><span class="p">()</span>
<span class="ne">RuntimeError</span><span class="p">:</span> <span class="n">The</span> <span class="n">size</span> <span class="n">of</span> <span class="n">tensor</span> <span class="n">a</span> <span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="n">must</span> <span class="n">match</span> <span class="n">the</span> <span class="n">size</span> <span class="n">of</span> <span class="n">tensor</span> <span class="n">b</span> <span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="n">at</span> <span class="n">non</span><span class="o">-</span><span class="n">singleton</span> <span class="n">dimension</span> <span class="mi">1</span>
</pre></div>
</div>
</div>
<div class="section" id="in-place-semantics">
<h2>In-place semantics<a class="headerlink" href="#in-place-semantics" title="Permalink to this headline">¶</a></h2>
<p>One complication is that in-place operations do not allow the in-place tensor to change shape
as a result of the broadcast.</p>
<p>For 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">empty</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">y</span><span class="o">=</span><span class="n">torch</span><span class="o">.</span><span class="n">empty</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="gp">>>> </span><span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">add_</span><span class="p">(</span><span class="n">y</span><span class="p">))</span><span class="o">.</span><span class="n">size</span><span class="p">()</span>
<span class="go">torch.Size([5, 3, 4, 1])</span>
<span class="go"># but:</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">empty</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">y</span><span class="o">=</span><span class="n">torch</span><span class="o">.</span><span class="n">empty</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">7</span><span class="p">)</span>
<span class="gp">>>> </span><span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">add_</span><span class="p">(</span><span class="n">y</span><span class="p">))</span><span class="o">.</span><span class="n">size</span><span class="p">()</span>
<span class="go">RuntimeError: The expanded size of the tensor (1) must match the existing size (7) at non-singleton dimension 2.</span>
</pre></div>
</div>
</div>
<div class="section" id="backwards-compatibility">
<h2>Backwards compatibility<a class="headerlink" href="#backwards-compatibility" title="Permalink to this headline">¶</a></h2>
<p>Prior versions of PyTorch allowed certain pointwise functions to execute on tensors with different shapes,
as long as the number of elements in each tensor was equal. The pointwise operation would then be carried
out by viewing each tensor as 1-dimensional. PyTorch now supports broadcasting and the “1-dimensional”
pointwise behavior is considered deprecated and will generate a Python warning in cases where tensors are
not broadcastable, but have the same number of elements.</p>
<p>Note that the introduction of broadcasting can cause backwards incompatible changes in the case where
two tensors do not have the same shape, but are broadcastable and have the same number of elements.
For Example:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">add</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">4</span><span class="p">,</span><span class="mi">1</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">4</span><span class="p">))</span>
</pre></div>
</div>
<p>would previously produce a Tensor with size: torch.Size([4,1]), but now produces a Tensor with size: torch.Size([4,4]).
In order to help identify cases in your code where backwards incompatibilities introduced by broadcasting may exist,
you may set <cite>torch.utils.backcompat.broadcast_warning.enabled</cite> to <cite>True</cite>, which will generate a python warning
in such cases.</p>
<p>For Example:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">utils</span><span class="o">.</span><span class="n">backcompat</span><span class="o">.</span><span class="n">broadcast_warning</span><span class="o">.</span><span class="n">enabled</span><span class="o">=</span><span class="kc">True</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">add</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">4</span><span class="p">,</span><span class="mi">1</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">4</span><span class="p">))</span>
<span class="go">__main__:1: UserWarning: self and other do not have the same shape, but are broadcastable, and have the same number of elements.</span>
<span class="go">Changing behavior in a backwards incompatible manner to broadcasting rather than viewing as 1-dimensional.</span>
</pre></div>
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