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<section id="numerical-accuracy">
<span id="id1"></span><h1>Numerical accuracy<a class="headerlink" href="#numerical-accuracy" title="Permalink to this heading">¶</a></h1>
<p>In modern computers, floating point numbers are represented using IEEE 754 standard.
For more details on floating point arithmetics and IEEE 754 standard, please see
<a class="reference external" href="https://en.wikipedia.org/wiki/Floating-point_arithmetic">Floating point arithmetic</a>
In particular, note that floating point provides limited accuracy (about 7 decimal digits
for single precision floating point numbers, about 16 decimal digits for double precision
floating point numbers) and that floating point addition and multiplication are not
associative, so the order of the operations affects the results.
Because of this, PyTorch is not guaranteed
to produce bitwise identical results for floating point computations that are
mathematically identical. Similarly, bitwise identical results are not guaranteed across
PyTorch releases, individual commits, or different platforms. In particular, CPU and GPU
results can be different even for bitwise-identical inputs and even after controlling for
the sources of randomness.</p>
<section id="batched-computations-or-slice-computations">
<h2>Batched computations or slice computations<a class="headerlink" href="#batched-computations-or-slice-computations" title="Permalink to this heading">¶</a></h2>
<p>Many operations in PyTorch support batched computation, where the same operation is performed
for the elements of the batches of inputs. An example of this is <a class="reference internal" href="../generated/torch.mm.html#torch.mm" title="torch.mm"><code class="xref py py-meth docutils literal notranslate"><span class="pre">torch.mm()</span></code></a> and
<a class="reference internal" href="../generated/torch.bmm.html#torch.bmm" title="torch.bmm"><code class="xref py py-meth docutils literal notranslate"><span class="pre">torch.bmm()</span></code></a>. It is possible to implement batched computation as a loop over batch elements,
and apply the necessary math operations to the individual batch elements, for efficiency reasons
we are not doing that, and typically perform computation for the whole batch. The mathematical
libraries that we are calling, and PyTorch internal implementations of operations can produces
slightly different results in this case, compared to non-batched computations. In particular,
let <code class="docutils literal notranslate"><span class="pre">A</span></code> and <code class="docutils literal notranslate"><span class="pre">B</span></code> be 3D tensors with the dimensions suitable for batched matrix multiplication.
Then <code class="docutils literal notranslate"><span class="pre">(A@B)[0]</span></code> (the first element of the batched result) is not guaranteed to be bitwise
identical to <code class="docutils literal notranslate"><span class="pre">A[0]@B[0]</span></code> (the matrix product of the first elements of the input batches)
even though mathematically it’s an identical computation.</p>
<p>Similarly, an operation applied to a tensor slice is not guaranteed to produce results that are
identical to the slice of the result of the same operation applied to the full tensor. E.g. let
<code class="docutils literal notranslate"><span class="pre">A</span></code> be a 2-dimentional tensor. <code class="docutils literal notranslate"><span class="pre">A.sum(-1)[0]</span></code> is not guaranteed to be bitwise equal to
<code class="docutils literal notranslate"><span class="pre">A[:,0].sum()</span></code>.</p>
</section>
<section id="extremal-values">
<h2>Extremal values<a class="headerlink" href="#extremal-values" title="Permalink to this heading">¶</a></h2>
<p>When inputs contain large values such that intermediate results may overflow the range of the
used datatype, the end result may overflow too, even though it is representable in the original
datatype. E.g.:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">torch</span>
<span class="n">a</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="mf">1e20</span><span class="p">,</span> <span class="mf">1e20</span><span class="p">])</span> <span class="c1"># fp32 type by default</span>
<span class="n">a</span><span class="o">.</span><span class="n">norm</span><span class="p">()</span> <span class="c1"># produces tensor(inf)</span>
<span class="n">a</span><span class="o">.</span><span class="n">double</span><span class="p">()</span><span class="o">.</span><span class="n">norm</span><span class="p">()</span> <span class="c1"># produces tensor(1.4142e+20, dtype=torch.float64), representable in fp32</span>
</pre></div>
</div>
</section>
<section id="tensorfloat-32-tf32-on-nvidia-ampere-devices">
<h2>TensorFloat-32(TF32) on Nvidia Ampere devices<a class="headerlink" href="#tensorfloat-32-tf32-on-nvidia-ampere-devices" title="Permalink to this heading">¶</a></h2>
<p>On Ampere Nvidia GPUs, PyTorch by default uses TensorFloat32 (TF32) to speed up mathematically
intensive operations, in particular matrix multiplications and convolutions. When operation is performed
using TF32 tensor cores, only the first 10 bits of the input mantissa are read. This leads to less accurate
results, and surprising results such as multiplying a matrix by identity matrix produces
results that are different from the input.
Most neural network workloads have the same convergence behavior when using tf32 as they have
with fp32, however, if better accuracy is desired, TF32 can be turned off with
<code class="docutils literal notranslate"><span class="pre">torch.backends.cuda.matmul.allow_tf32</span> <span class="pre">=</span> <span class="pre">False</span></code></p>
<p>For more information see <a class="reference internal" href="cuda.html#tf32-on-ampere"><span class="std std-ref">TensorFloat32</span></a></p>
</section>
<section id="reduced-precision-reduction-for-fp16-gemms">
<h2>Reduced Precision Reduction for FP16 GEMMs<a class="headerlink" href="#reduced-precision-reduction-for-fp16-gemms" title="Permalink to this heading">¶</a></h2>
<p>Half-precision GEMM operations are typically done with intermediate accumulations (reduction) in single-precision for numerical accuracy and improved resilience to overflow. For performance, certain GPU architectures, especially more recent ones, allow a few truncations of the intermediate accumulation results to the reduced precision (e.g., half-precision). This change is often benign from the perspective of model convergence, though it may lead to unexpected results (e.g., <code class="docutils literal notranslate"><span class="pre">inf</span></code> values when the final result should be be representable in half-precision).
If reduced-precision reductions are problematic, they can be turned off with
<code class="docutils literal notranslate"><span class="pre">torch.backends.cuda.matmul.allow_fp16_reduced_precision_reduction</span> <span class="pre">=</span> <span class="pre">False</span></code></p>
<p>For more information see <a class="reference internal" href="cuda.html#fp16reducedprecision"><span class="std std-ref">allow_fp16_reduced_precision_reduction</span></a></p>
</section>
<section id="reduced-precision-fp16-and-bf16-gemms-and-convolutions-on-amd-instinct-mi200-devices">
<span id="fp16-on-mi200"></span><h2>Reduced Precision FP16 and BF16 GEMMs and Convolutions on AMD Instinct MI200 devices<a class="headerlink" href="#reduced-precision-fp16-and-bf16-gemms-and-convolutions-on-amd-instinct-mi200-devices" title="Permalink to this heading">¶</a></h2>
<p>On AMD Instinct MI200 GPUs, the FP16 and BF16 V_DOT2 and MFMA matrix instructions flush input and output denormal values to zero. FP32 and FP64 MFMA matrix instructions do not flush input and output denormal values to zero. The affected instructions are only used by rocBLAS (GEMM) and MIOpen (convolution) kernels; all other PyTorch operations will not encounter this behavior. All other supported AMD GPUs will not encounter this behavior.</p>
<p>rocBLAS and MIOpen provide alternate implementations for affected FP16 operations. Alternate implementations for BF16 operations are not provided; BF16 numbers have a larger dynamic range than FP16 numbers and are less likely to encounter denormal values. For the FP16 alternate implementations, FP16 input values are cast to an intermediate BF16 value and then cast back to FP16 output after the accumulate FP32 operations. In this way, the input and output types are unchanged.</p>
<p>When training using FP16 precision, some models may fail to converge with FP16 denorms flushed to zero. Denormal values more frequently occur in the backward pass of training during gradient calculation. PyTorch by default will use the rocBLAS and MIOpen alternate implementations during the backward pass. The default behavior can be overridden using environment variables, ROCBLAS_INTERNAL_FP16_ALT_IMPL and MIOPEN_DEBUG_CONVOLUTION_ATTRIB_FP16_ALT_IMPL. The behavior of these environment variables is as follows:</p>
<table class="docutils align-default">
<thead>
<tr class="row-odd"><th class="head"></th>
<th class="head"><p>forward</p></th>
<th class="head"><p>backward</p></th>
</tr>
</thead>
<tbody>
<tr class="row-even"><td><p>Env unset</p></td>
<td><p>original</p></td>
<td><p>alternate</p></td>
</tr>
<tr class="row-odd"><td><p>Env set to 1</p></td>
<td><p>alternate</p></td>
<td><p>alternate</p></td>
</tr>
<tr class="row-even"><td><p>Env set to 0</p></td>
<td><p>original</p></td>
<td><p>original</p></td>
</tr>
</tbody>
</table>
<p>The following is the list of operations where rocBLAS may be used:</p>
<ul class="simple">
<li><p>torch.addbmm</p></li>
<li><p>torch.addmm</p></li>
<li><p>torch.baddbmm</p></li>
<li><p>torch.bmm</p></li>
<li><p>torch.mm</p></li>
<li><p>torch.nn.GRUCell</p></li>
<li><p>torch.nn.LSTMCell</p></li>
<li><p>torch.nn.Linear</p></li>
<li><p>torch.sparse.addmm</p></li>
<li><p>the following torch._C._ConvBackend implementations:</p>
<ul>
<li><p>slowNd</p></li>
<li><p>slowNd_transposed</p></li>
<li><p>slowNd_dilated</p></li>
<li><p>slowNd_dilated_transposed</p></li>
</ul>
</li>
</ul>
<p>The following is the list of operations where MIOpen may be used:</p>
<ul class="simple">
<li><p>torch.nn.Conv[Transpose]Nd</p></li>
<li><p>the following torch._C._ConvBackend implementations:</p>
<ul>
<li><p>ConvBackend::Miopen</p></li>
<li><p>ConvBackend::MiopenDepthwise</p></li>
<li><p>ConvBackend::MiopenTranspose</p></li>
</ul>
</li>
</ul>
</section>
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