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<p class="admonition-title">Note</p>
<p>Click <a class="reference internal" href="#sphx-glr-download-getting-started-tutorials-05-layer-norm-py"><span class="std std-ref">here</span></a>
to download the full example code</p>
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<div class="sphx-glr-example-title section" id="layer-normalization">
<span id="sphx-glr-getting-started-tutorials-05-layer-norm-py"></span><h1>Layer Normalization<a class="headerlink" href="#layer-normalization" title="Permalink to this headline"></a></h1>
<img alt="05 layer norm" class="sphx-glr-single-img" src="../../_images/sphx_glr_05-layer-norm_001.png" />
<p class="sphx-glr-script-out">Out:</p>
<div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>layer-norm:
N Triton Torch Apex
[GH-PAGES] Updated website
2022-06-19 00:46:49 +00:00
0 1024.0 585.142849 277.694907 468.114273
[GH-PAGES] Updated website
2022-06-05 21:05:02 +00:00
1 1536.0 630.153868 323.368435 511.999982
[GH-PAGES] Updated website
2022-06-19 00:46:49 +00:00
2 2048.0 682.666643 334.367358 520.126988
[GH-PAGES] Updated website
2022-06-22 00:49:37 +00:00
3 2560.0 694.237267 362.477870 512.000013
4 3072.0 712.347810 378.092307 501.551037
5 3584.0 725.873439 384.859062 451.527536
[GH-PAGES] Updated website
2022-06-21 00:46:27 +00:00
6 4096.0 728.177767 381.023256 451.972420
[GH-PAGES] Updated website
2022-06-22 00:49:37 +00:00
7 4608.0 676.403666 396.387087 431.157877
8 5120.0 688.403381 397.669909 422.268057
9 5632.0 704.000002 395.228063 415.262685
[GH-PAGES] Updated website
2022-06-13 00:48:38 +00:00
10 6144.0 702.171410 402.885254 411.313806
[GH-PAGES] Updated website
2022-06-22 00:49:37 +00:00
11 6656.0 700.631610 398.861429 400.360920
12 7168.0 686.754468 395.475867 386.154893
13 7680.0 682.666656 392.587863 386.415087
14 8192.0 642.509816 387.786968 368.179771
15 8704.0 630.153861 387.922008 378.434774
16 9216.0 609.322328 407.337026 383.002605
17 9728.0 589.575753 408.524944 383.369452
18 10240.0 566.920437 409.600010 382.803739
19 10752.0 549.623009 410.577576 381.445676
20 11264.0 534.789310 406.826188 374.686074
21 11776.0 523.377770 409.599991 376.831982
[GH-PAGES] Updated website
2022-06-21 00:46:27 +00:00
22 12288.0 516.031509 413.911572 383.251457
[GH-PAGES] Updated website
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23 12800.0 505.679014 409.599981 376.470582
24 13312.0 494.180982 405.699062 376.976995
25 13824.0 481.882350 412.656711 379.389355
26 14336.0 471.967074 405.975225 372.969090
27 14848.0 461.297068 406.794504 375.304904
28 15360.0 454.269882 406.887417 377.511515
[GH-PAGES] Updated website
2022-06-21 00:46:27 +00:00
29 15872.0 447.887117 406.974373 376.225175
[GH-PAGES] Updated website
2022-06-05 21:05:02 +00:00
</pre></div>
</div>
<div class="line-block">
<div class="line"><br /></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">torch</span>
<span class="kn">import</span> <span class="nn">triton</span>
<span class="kn">import</span> <span class="nn">triton.language</span> <span class="k">as</span> <span class="nn">tl</span>
<span class="k">try</span><span class="p">:</span>
<span class="c1"># This is https://github.com/NVIDIA/apex, NOT the apex on PyPi, so it</span>
<span class="c1"># should not be added to extras_require in setup.py.</span>
<span class="kn">import</span> <span class="nn">apex</span>
<span class="n">HAS_APEX</span> <span class="o">=</span> <span class="kc">True</span>
<span class="k">except</span> <span class="ne">ModuleNotFoundError</span><span class="p">:</span>
<span class="n">HAS_APEX</span> <span class="o">=</span> <span class="kc">False</span>
<span class="nd">@triton</span><span class="o">.</span><span class="n">jit</span>
<span class="k">def</span> <span class="nf">_layer_norm_fwd_fused</span><span class="p">(</span>
<span class="n">Out</span><span class="p">,</span>
<span class="n">A</span><span class="p">,</span>
<span class="n">Weight</span><span class="p">,</span>
<span class="n">Bias</span><span class="p">,</span>
<span class="n">Mean</span><span class="p">,</span> <span class="n">Rstd</span><span class="p">,</span>
<span class="n">stride</span><span class="p">,</span> <span class="n">N</span><span class="p">,</span> <span class="n">eps</span><span class="p">,</span>
<span class="n">BLOCK_SIZE</span><span class="p">:</span> <span class="n">tl</span><span class="o">.</span><span class="n">constexpr</span><span class="p">,</span>
<span class="p">):</span>
<span class="c1"># position of elements processed by this program</span>
<span class="n">row</span> <span class="o">=</span> <span class="n">tl</span><span class="o">.</span><span class="n">program_id</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="n">Out</span> <span class="o">+=</span> <span class="n">row</span> <span class="o">*</span> <span class="n">stride</span>
<span class="n">A</span> <span class="o">+=</span> <span class="n">row</span> <span class="o">*</span> <span class="n">stride</span>
<span class="c1"># compute mean</span>
<span class="n">mean</span> <span class="o">=</span> <span class="mi">0</span>
<span class="n">_mean</span> <span class="o">=</span> <span class="n">tl</span><span class="o">.</span><span class="n">zeros</span><span class="p">([</span><span class="n">BLOCK_SIZE</span><span class="p">],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">tl</span><span class="o">.</span><span class="n">float32</span><span class="p">)</span>
<span class="k">for</span> <span class="n">off</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">N</span><span class="p">,</span> <span class="n">BLOCK_SIZE</span><span class="p">):</span>
<span class="n">cols</span> <span class="o">=</span> <span class="n">off</span> <span class="o">+</span> <span class="n">tl</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">BLOCK_SIZE</span><span class="p">)</span>
<span class="n">a</span> <span class="o">=</span> <span class="n">tl</span><span class="o">.</span><span class="n">load</span><span class="p">(</span><span class="n">A</span> <span class="o">+</span> <span class="n">cols</span><span class="p">,</span> <span class="n">mask</span><span class="o">=</span><span class="n">cols</span> <span class="o">&lt;</span> <span class="n">N</span><span class="p">,</span> <span class="n">other</span><span class="o">=</span><span class="mf">0.</span><span class="p">,</span> <span class="n">eviction_policy</span><span class="o">=</span><span class="s2">&quot;evict_last&quot;</span><span class="p">)</span><span class="o">.</span><span class="n">to</span><span class="p">(</span><span class="n">tl</span><span class="o">.</span><span class="n">float32</span><span class="p">)</span>
<span class="n">_mean</span> <span class="o">+=</span> <span class="n">a</span>
<span class="n">mean</span> <span class="o">=</span> <span class="n">tl</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">_mean</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span> <span class="o">/</span> <span class="n">N</span>
<span class="c1"># compute variance</span>
<span class="n">_var</span> <span class="o">=</span> <span class="n">tl</span><span class="o">.</span><span class="n">zeros</span><span class="p">([</span><span class="n">BLOCK_SIZE</span><span class="p">],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">tl</span><span class="o">.</span><span class="n">float32</span><span class="p">)</span>
<span class="k">for</span> <span class="n">off</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">N</span><span class="p">,</span> <span class="n">BLOCK_SIZE</span><span class="p">):</span>
<span class="n">cols</span> <span class="o">=</span> <span class="n">off</span> <span class="o">+</span> <span class="n">tl</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">BLOCK_SIZE</span><span class="p">)</span>
<span class="n">a</span> <span class="o">=</span> <span class="n">tl</span><span class="o">.</span><span class="n">load</span><span class="p">(</span><span class="n">A</span> <span class="o">+</span> <span class="n">cols</span><span class="p">,</span> <span class="n">mask</span><span class="o">=</span><span class="n">cols</span> <span class="o">&lt;</span> <span class="n">N</span><span class="p">,</span> <span class="n">other</span><span class="o">=</span><span class="mf">0.</span><span class="p">,</span> <span class="n">eviction_policy</span><span class="o">=</span><span class="s2">&quot;evict_last&quot;</span><span class="p">)</span><span class="o">.</span><span class="n">to</span><span class="p">(</span><span class="n">tl</span><span class="o">.</span><span class="n">float32</span><span class="p">)</span>
<span class="n">a</span> <span class="o">=</span> <span class="n">tl</span><span class="o">.</span><span class="n">where</span><span class="p">(</span><span class="n">cols</span> <span class="o">&lt;</span> <span class="n">N</span><span class="p">,</span> <span class="n">a</span> <span class="o">-</span> <span class="n">mean</span><span class="p">,</span> <span class="mf">0.</span><span class="p">)</span>
<span class="n">_var</span> <span class="o">+=</span> <span class="n">a</span> <span class="o">*</span> <span class="n">a</span>
<span class="n">var</span> <span class="o">=</span> <span class="n">tl</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">_var</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span> <span class="o">/</span> <span class="n">N</span>
<span class="n">rstd</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">/</span> <span class="n">tl</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">var</span> <span class="o">+</span> <span class="n">eps</span><span class="p">)</span>
<span class="c1"># write-back mean/rstd</span>
<span class="n">tl</span><span class="o">.</span><span class="n">store</span><span class="p">(</span><span class="n">Mean</span> <span class="o">+</span> <span class="n">row</span><span class="p">,</span> <span class="n">mean</span><span class="p">)</span>
<span class="n">tl</span><span class="o">.</span><span class="n">store</span><span class="p">(</span><span class="n">Rstd</span> <span class="o">+</span> <span class="n">row</span><span class="p">,</span> <span class="n">rstd</span><span class="p">)</span>
<span class="c1"># multiply by weight and add bias</span>
<span class="k">for</span> <span class="n">off</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">N</span><span class="p">,</span> <span class="n">BLOCK_SIZE</span><span class="p">):</span>
<span class="n">cols</span> <span class="o">=</span> <span class="n">off</span> <span class="o">+</span> <span class="n">tl</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">BLOCK_SIZE</span><span class="p">)</span>
<span class="n">mask</span> <span class="o">=</span> <span class="n">cols</span> <span class="o">&lt;</span> <span class="n">N</span>
<span class="n">weight</span> <span class="o">=</span> <span class="n">tl</span><span class="o">.</span><span class="n">load</span><span class="p">(</span><span class="n">Weight</span> <span class="o">+</span> <span class="n">cols</span><span class="p">,</span> <span class="n">mask</span><span class="o">=</span><span class="n">mask</span><span class="p">)</span>
<span class="n">bias</span> <span class="o">=</span> <span class="n">tl</span><span class="o">.</span><span class="n">load</span><span class="p">(</span><span class="n">Bias</span> <span class="o">+</span> <span class="n">cols</span><span class="p">,</span> <span class="n">mask</span><span class="o">=</span><span class="n">mask</span><span class="p">)</span>
<span class="n">a</span> <span class="o">=</span> <span class="n">tl</span><span class="o">.</span><span class="n">load</span><span class="p">(</span><span class="n">A</span> <span class="o">+</span> <span class="n">cols</span><span class="p">,</span> <span class="n">mask</span><span class="o">=</span><span class="n">mask</span><span class="p">,</span> <span class="n">other</span><span class="o">=</span><span class="mf">0.</span><span class="p">,</span> <span class="n">eviction_policy</span><span class="o">=</span><span class="s2">&quot;evict_first&quot;</span><span class="p">)</span><span class="o">.</span><span class="n">to</span><span class="p">(</span><span class="n">tl</span><span class="o">.</span><span class="n">float32</span><span class="p">)</span>
<span class="n">a_hat</span> <span class="o">=</span> <span class="p">(</span><span class="n">a</span> <span class="o">-</span> <span class="n">mean</span><span class="p">)</span> <span class="o">*</span> <span class="n">rstd</span>
<span class="n">out</span> <span class="o">=</span> <span class="n">a_hat</span> <span class="o">*</span> <span class="n">weight</span> <span class="o">+</span> <span class="n">bias</span>
<span class="c1"># # write-back</span>
<span class="n">tl</span><span class="o">.</span><span class="n">store</span><span class="p">(</span><span class="n">Out</span> <span class="o">+</span> <span class="n">cols</span><span class="p">,</span> <span class="n">out</span><span class="p">,</span> <span class="n">mask</span><span class="o">=</span><span class="n">mask</span><span class="p">)</span>
<span class="c1"># Backward pass (DA + partial DW + partial DB)</span>
<span class="nd">@triton</span><span class="o">.</span><span class="n">jit</span>
<span class="k">def</span> <span class="nf">_layer_norm_bwd_dx_fused</span><span class="p">(</span>
<span class="n">_DA</span><span class="p">,</span>
<span class="n">_DOut</span><span class="p">,</span>
<span class="n">_A</span><span class="p">,</span>
<span class="n">Weight</span><span class="p">,</span>
<span class="n">Mean</span><span class="p">,</span> <span class="n">Rstd</span><span class="p">,</span>
<span class="n">stride</span><span class="p">,</span> <span class="n">NumRows</span><span class="p">,</span> <span class="n">NumCols</span><span class="p">,</span> <span class="n">eps</span><span class="p">,</span>
<span class="n">BLOCK_SIZE_N</span><span class="p">:</span> <span class="n">tl</span><span class="o">.</span><span class="n">constexpr</span><span class="p">,</span>
<span class="p">):</span>
<span class="c1"># position of elements processed by this program</span>
<span class="n">pid</span> <span class="o">=</span> <span class="n">tl</span><span class="o">.</span><span class="n">program_id</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="n">row</span> <span class="o">=</span> <span class="n">pid</span>
<span class="n">A</span> <span class="o">=</span> <span class="n">_A</span> <span class="o">+</span> <span class="n">row</span> <span class="o">*</span> <span class="n">stride</span>
<span class="n">DOut</span> <span class="o">=</span> <span class="n">_DOut</span> <span class="o">+</span> <span class="n">row</span> <span class="o">*</span> <span class="n">stride</span>
<span class="n">DA</span> <span class="o">=</span> <span class="n">_DA</span> <span class="o">+</span> <span class="n">row</span> <span class="o">*</span> <span class="n">stride</span>
<span class="n">mean</span> <span class="o">=</span> <span class="n">tl</span><span class="o">.</span><span class="n">load</span><span class="p">(</span><span class="n">Mean</span> <span class="o">+</span> <span class="n">row</span><span class="p">)</span>
<span class="n">rstd</span> <span class="o">=</span> <span class="n">tl</span><span class="o">.</span><span class="n">load</span><span class="p">(</span><span class="n">Rstd</span> <span class="o">+</span> <span class="n">row</span><span class="p">)</span>
<span class="c1"># load data to SRAM</span>
<span class="n">_mean1</span> <span class="o">=</span> <span class="n">tl</span><span class="o">.</span><span class="n">zeros</span><span class="p">([</span><span class="n">BLOCK_SIZE_N</span><span class="p">],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">tl</span><span class="o">.</span><span class="n">float32</span><span class="p">)</span>
<span class="n">_mean2</span> <span class="o">=</span> <span class="n">tl</span><span class="o">.</span><span class="n">zeros</span><span class="p">([</span><span class="n">BLOCK_SIZE_N</span><span class="p">],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">tl</span><span class="o">.</span><span class="n">float32</span><span class="p">)</span>
<span class="k">for</span> <span class="n">off</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">NumCols</span><span class="p">,</span> <span class="n">BLOCK_SIZE_N</span><span class="p">):</span>
<span class="n">cols</span> <span class="o">=</span> <span class="n">off</span> <span class="o">+</span> <span class="n">tl</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">BLOCK_SIZE_N</span><span class="p">)</span>
<span class="n">mask</span> <span class="o">=</span> <span class="n">cols</span> <span class="o">&lt;</span> <span class="n">NumCols</span>
<span class="n">a</span> <span class="o">=</span> <span class="n">tl</span><span class="o">.</span><span class="n">load</span><span class="p">(</span><span class="n">A</span> <span class="o">+</span> <span class="n">cols</span><span class="p">,</span> <span class="n">mask</span><span class="o">=</span><span class="n">mask</span><span class="p">,</span> <span class="n">other</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span><span class="o">.</span><span class="n">to</span><span class="p">(</span><span class="n">tl</span><span class="o">.</span><span class="n">float32</span><span class="p">)</span>
<span class="n">dout</span> <span class="o">=</span> <span class="n">tl</span><span class="o">.</span><span class="n">load</span><span class="p">(</span><span class="n">DOut</span> <span class="o">+</span> <span class="n">cols</span><span class="p">,</span> <span class="n">mask</span><span class="o">=</span><span class="n">mask</span><span class="p">,</span> <span class="n">other</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span><span class="o">.</span><span class="n">to</span><span class="p">(</span><span class="n">tl</span><span class="o">.</span><span class="n">float32</span><span class="p">)</span>
<span class="n">weight</span> <span class="o">=</span> <span class="n">tl</span><span class="o">.</span><span class="n">load</span><span class="p">(</span><span class="n">Weight</span> <span class="o">+</span> <span class="n">cols</span><span class="p">,</span> <span class="n">mask</span><span class="o">=</span><span class="n">mask</span><span class="p">,</span> <span class="n">other</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span><span class="o">.</span><span class="n">to</span><span class="p">(</span><span class="n">tl</span><span class="o">.</span><span class="n">float32</span><span class="p">)</span>
<span class="n">a_hat</span> <span class="o">=</span> <span class="p">(</span><span class="n">a</span> <span class="o">-</span> <span class="n">mean</span><span class="p">)</span> <span class="o">*</span> <span class="n">rstd</span>
<span class="n">wdout</span> <span class="o">=</span> <span class="n">weight</span> <span class="o">*</span> <span class="n">dout</span>
<span class="n">_mean1</span> <span class="o">+=</span> <span class="n">a_hat</span> <span class="o">*</span> <span class="n">wdout</span>
<span class="n">_mean2</span> <span class="o">+=</span> <span class="n">wdout</span>
<span class="n">mean1</span> <span class="o">=</span> <span class="n">tl</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">_mean1</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span> <span class="o">/</span> <span class="n">NumCols</span>
<span class="n">mean2</span> <span class="o">=</span> <span class="mf">0.</span>
<span class="n">mean2</span> <span class="o">=</span> <span class="n">tl</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">_mean2</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span> <span class="o">/</span> <span class="n">NumCols</span>
<span class="k">for</span> <span class="n">off</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">NumCols</span><span class="p">,</span> <span class="n">BLOCK_SIZE_N</span><span class="p">):</span>
<span class="n">cols</span> <span class="o">=</span> <span class="n">off</span> <span class="o">+</span> <span class="n">tl</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">BLOCK_SIZE_N</span><span class="p">)</span>
<span class="n">mask</span> <span class="o">=</span> <span class="n">cols</span> <span class="o">&lt;</span> <span class="n">NumCols</span>
<span class="n">a</span> <span class="o">=</span> <span class="n">tl</span><span class="o">.</span><span class="n">load</span><span class="p">(</span><span class="n">A</span> <span class="o">+</span> <span class="n">cols</span><span class="p">,</span> <span class="n">mask</span><span class="o">=</span><span class="n">mask</span><span class="p">,</span> <span class="n">other</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span><span class="o">.</span><span class="n">to</span><span class="p">(</span><span class="n">tl</span><span class="o">.</span><span class="n">float32</span><span class="p">)</span>
<span class="n">dout</span> <span class="o">=</span> <span class="n">tl</span><span class="o">.</span><span class="n">load</span><span class="p">(</span><span class="n">DOut</span> <span class="o">+</span> <span class="n">cols</span><span class="p">,</span> <span class="n">mask</span><span class="o">=</span><span class="n">mask</span><span class="p">,</span> <span class="n">other</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span><span class="o">.</span><span class="n">to</span><span class="p">(</span><span class="n">tl</span><span class="o">.</span><span class="n">float32</span><span class="p">)</span>
<span class="n">weight</span> <span class="o">=</span> <span class="n">tl</span><span class="o">.</span><span class="n">load</span><span class="p">(</span><span class="n">Weight</span> <span class="o">+</span> <span class="n">cols</span><span class="p">,</span> <span class="n">mask</span><span class="o">=</span><span class="n">mask</span><span class="p">,</span> <span class="n">other</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span><span class="o">.</span><span class="n">to</span><span class="p">(</span><span class="n">tl</span><span class="o">.</span><span class="n">float32</span><span class="p">)</span>
<span class="n">a_hat</span> <span class="o">=</span> <span class="p">(</span><span class="n">a</span> <span class="o">-</span> <span class="n">mean</span><span class="p">)</span> <span class="o">*</span> <span class="n">rstd</span>
<span class="n">wdout</span> <span class="o">=</span> <span class="n">weight</span> <span class="o">*</span> <span class="n">dout</span>
<span class="n">da</span> <span class="o">=</span> <span class="p">(</span><span class="n">wdout</span> <span class="o">-</span> <span class="p">(</span><span class="n">a_hat</span> <span class="o">*</span> <span class="n">mean1</span> <span class="o">+</span> <span class="n">mean2</span><span class="p">))</span> <span class="o">*</span> <span class="n">rstd</span>
<span class="c1"># write-back dx</span>
<span class="n">tl</span><span class="o">.</span><span class="n">store</span><span class="p">(</span><span class="n">DA</span> <span class="o">+</span> <span class="n">cols</span><span class="p">,</span> <span class="n">da</span><span class="p">,</span> <span class="n">mask</span><span class="o">=</span><span class="n">mask</span><span class="p">)</span>
<span class="c1"># Backward pass (total DW + total DB)</span>
<span class="nd">@triton</span><span class="o">.</span><span class="n">jit</span>
<span class="k">def</span> <span class="nf">_layer_norm_bwd_dwdb</span><span class="p">(</span>
<span class="n">A</span><span class="p">,</span> <span class="n">DOut</span><span class="p">,</span>
<span class="n">Mean</span><span class="p">,</span> <span class="n">Var</span><span class="p">,</span>
<span class="n">DW</span><span class="p">,</span>
<span class="n">DB</span><span class="p">,</span>
<span class="n">M</span><span class="p">,</span> <span class="n">N</span><span class="p">,</span>
<span class="n">BLOCK_SIZE_M</span><span class="p">:</span> <span class="n">tl</span><span class="o">.</span><span class="n">constexpr</span><span class="p">,</span>
<span class="n">BLOCK_SIZE_N</span><span class="p">:</span> <span class="n">tl</span><span class="o">.</span><span class="n">constexpr</span><span class="p">,</span>
<span class="p">):</span>
<span class="n">pid</span> <span class="o">=</span> <span class="n">tl</span><span class="o">.</span><span class="n">program_id</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="n">cols</span> <span class="o">=</span> <span class="n">pid</span> <span class="o">*</span> <span class="n">BLOCK_SIZE_N</span> <span class="o">+</span> <span class="n">tl</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">BLOCK_SIZE_N</span><span class="p">)</span>
<span class="n">dw</span> <span class="o">=</span> <span class="n">tl</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">BLOCK_SIZE_M</span><span class="p">,</span> <span class="n">BLOCK_SIZE_N</span><span class="p">),</span> <span class="n">dtype</span><span class="o">=</span><span class="n">tl</span><span class="o">.</span><span class="n">float32</span><span class="p">)</span>
<span class="n">db</span> <span class="o">=</span> <span class="n">tl</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">BLOCK_SIZE_M</span><span class="p">,</span> <span class="n">BLOCK_SIZE_N</span><span class="p">),</span> <span class="n">dtype</span><span class="o">=</span><span class="n">tl</span><span class="o">.</span><span class="n">float32</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">M</span><span class="p">,</span> <span class="n">BLOCK_SIZE_M</span><span class="p">):</span>
<span class="n">rows</span> <span class="o">=</span> <span class="n">i</span> <span class="o">+</span> <span class="n">tl</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">BLOCK_SIZE_M</span><span class="p">)</span>
<span class="n">mask</span> <span class="o">=</span> <span class="p">(</span><span class="n">rows</span><span class="p">[:,</span> <span class="kc">None</span><span class="p">]</span> <span class="o">&lt;</span> <span class="n">M</span><span class="p">)</span> <span class="o">&amp;</span> <span class="p">(</span><span class="n">cols</span><span class="p">[</span><span class="kc">None</span><span class="p">,</span> <span class="p">:]</span> <span class="o">&lt;</span> <span class="n">N</span><span class="p">)</span>
<span class="n">offs</span> <span class="o">=</span> <span class="n">rows</span><span class="p">[:,</span> <span class="kc">None</span><span class="p">]</span> <span class="o">*</span> <span class="n">N</span> <span class="o">+</span> <span class="n">cols</span><span class="p">[</span><span class="kc">None</span><span class="p">,</span> <span class="p">:]</span>
<span class="n">a</span> <span class="o">=</span> <span class="n">tl</span><span class="o">.</span><span class="n">load</span><span class="p">(</span><span class="n">A</span> <span class="o">+</span> <span class="n">offs</span><span class="p">,</span> <span class="n">mask</span><span class="o">=</span><span class="n">mask</span><span class="p">,</span> <span class="n">other</span><span class="o">=</span><span class="mf">0.</span><span class="p">)</span><span class="o">.</span><span class="n">to</span><span class="p">(</span><span class="n">tl</span><span class="o">.</span><span class="n">float32</span><span class="p">)</span>
<span class="n">dout</span> <span class="o">=</span> <span class="n">tl</span><span class="o">.</span><span class="n">load</span><span class="p">(</span><span class="n">DOut</span> <span class="o">+</span> <span class="n">offs</span><span class="p">,</span> <span class="n">mask</span><span class="o">=</span><span class="n">mask</span><span class="p">,</span> <span class="n">other</span><span class="o">=</span><span class="mf">0.</span><span class="p">)</span><span class="o">.</span><span class="n">to</span><span class="p">(</span><span class="n">tl</span><span class="o">.</span><span class="n">float32</span><span class="p">)</span>
<span class="n">mean</span> <span class="o">=</span> <span class="n">tl</span><span class="o">.</span><span class="n">load</span><span class="p">(</span><span class="n">Mean</span> <span class="o">+</span> <span class="n">rows</span><span class="p">,</span> <span class="n">mask</span><span class="o">=</span><span class="n">rows</span> <span class="o">&lt;</span> <span class="n">M</span><span class="p">,</span> <span class="n">other</span><span class="o">=</span><span class="mf">0.</span><span class="p">)</span>
<span class="n">rstd</span> <span class="o">=</span> <span class="n">tl</span><span class="o">.</span><span class="n">load</span><span class="p">(</span><span class="n">Var</span> <span class="o">+</span> <span class="n">rows</span><span class="p">,</span> <span class="n">mask</span><span class="o">=</span><span class="n">rows</span> <span class="o">&lt;</span> <span class="n">M</span><span class="p">,</span> <span class="n">other</span><span class="o">=</span><span class="mf">0.</span><span class="p">)</span>
<span class="n">a_hat</span> <span class="o">=</span> <span class="p">(</span><span class="n">a</span> <span class="o">-</span> <span class="n">mean</span><span class="p">[:,</span> <span class="kc">None</span><span class="p">])</span> <span class="o">*</span> <span class="n">rstd</span><span class="p">[:,</span> <span class="kc">None</span><span class="p">]</span>
<span class="n">dw</span> <span class="o">+=</span> <span class="n">dout</span> <span class="o">*</span> <span class="n">a_hat</span>
<span class="n">db</span> <span class="o">+=</span> <span class="n">dout</span>
<span class="n">sum_dw</span> <span class="o">=</span> <span class="n">tl</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">dw</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="n">sum_db</span> <span class="o">=</span> <span class="n">tl</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">db</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="n">tl</span><span class="o">.</span><span class="n">store</span><span class="p">(</span><span class="n">DW</span> <span class="o">+</span> <span class="n">cols</span><span class="p">,</span> <span class="n">sum_dw</span><span class="p">,</span> <span class="n">mask</span><span class="o">=</span><span class="n">cols</span> <span class="o">&lt;</span> <span class="n">N</span><span class="p">)</span>
<span class="n">tl</span><span class="o">.</span><span class="n">store</span><span class="p">(</span><span class="n">DB</span> <span class="o">+</span> <span class="n">cols</span><span class="p">,</span> <span class="n">sum_db</span><span class="p">,</span> <span class="n">mask</span><span class="o">=</span><span class="n">cols</span> <span class="o">&lt;</span> <span class="n">N</span><span class="p">)</span>
<span class="k">class</span> <span class="nc">LayerNorm</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">autograd</span><span class="o">.</span><span class="n">Function</span><span class="p">):</span>
<span class="nd">@staticmethod</span>
<span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="n">ctx</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">normalized_shape</span><span class="p">,</span> <span class="n">weight</span><span class="p">,</span> <span class="n">bias</span><span class="p">,</span> <span class="n">eps</span><span class="p">):</span>
<span class="c1"># allocate output</span>
<span class="n">out</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">empty_like</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
<span class="c1"># reshape input data into 2D tensor</span>
<span class="n">a_arg</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">a</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
<span class="n">M</span><span class="p">,</span> <span class="n">N</span> <span class="o">=</span> <span class="n">a_arg</span><span class="o">.</span><span class="n">shape</span>
<span class="n">mean</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="n">M</span><span class="p">,),</span> <span class="n">dtype</span><span class="o">=</span><span class="n">torch</span><span class="o">.</span><span class="n">float32</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="s2">&quot;cuda&quot;</span><span class="p">)</span>
<span class="n">rstd</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="n">M</span><span class="p">,),</span> <span class="n">dtype</span><span class="o">=</span><span class="n">torch</span><span class="o">.</span><span class="n">float32</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="s2">&quot;cuda&quot;</span><span class="p">)</span>
<span class="c1"># Less than 64KB per feature: enqueue fused kernel</span>
<span class="n">MAX_FUSED_SIZE</span> <span class="o">=</span> <span class="mi">65536</span> <span class="o">//</span> <span class="n">a</span><span class="o">.</span><span class="n">element_size</span><span class="p">()</span>
<span class="n">BLOCK_SIZE</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">MAX_FUSED_SIZE</span><span class="p">,</span> <span class="n">triton</span><span class="o">.</span><span class="n">next_power_of_2</span><span class="p">(</span><span class="n">N</span><span class="p">))</span>
<span class="n">BLOCK_SIZE</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">BLOCK_SIZE</span><span class="p">,</span> <span class="mi">128</span><span class="p">)</span>
<span class="n">BLOCK_SIZE</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">BLOCK_SIZE</span><span class="p">,</span> <span class="mi">4096</span><span class="p">)</span>
<span class="c1"># heuristics for number of warps</span>
<span class="n">num_warps</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="nb">max</span><span class="p">(</span><span class="n">BLOCK_SIZE</span> <span class="o">//</span> <span class="mi">256</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="mi">8</span><span class="p">)</span>
<span class="n">_layer_norm_fwd_fused</span><span class="p">[(</span><span class="n">M</span><span class="p">,)](</span>
<span class="n">out</span><span class="p">,</span>
<span class="n">a_arg</span><span class="p">,</span>
<span class="n">weight</span><span class="p">,</span>
<span class="n">bias</span><span class="p">,</span>
<span class="n">mean</span><span class="p">,</span> <span class="n">rstd</span><span class="p">,</span>
<span class="n">a_arg</span><span class="o">.</span><span class="n">stride</span><span class="p">(</span><span class="mi">0</span><span class="p">),</span> <span class="n">N</span><span class="p">,</span> <span class="n">eps</span><span class="p">,</span>
<span class="n">BLOCK_SIZE</span><span class="o">=</span><span class="n">BLOCK_SIZE</span><span class="p">,</span>
<span class="n">num_warps</span><span class="o">=</span><span class="n">num_warps</span><span class="p">,</span>
<span class="p">)</span>
<span class="n">ctx</span><span class="o">.</span><span class="n">save_for_backward</span><span class="p">(</span>
<span class="n">a</span><span class="p">,</span> <span class="n">weight</span><span class="p">,</span> <span class="n">bias</span><span class="p">,</span> <span class="n">mean</span><span class="p">,</span> <span class="n">rstd</span><span class="p">,</span>
<span class="p">)</span>
<span class="n">ctx</span><span class="o">.</span><span class="n">BLOCK_SIZE</span> <span class="o">=</span> <span class="n">BLOCK_SIZE</span>
<span class="n">ctx</span><span class="o">.</span><span class="n">num_warps</span> <span class="o">=</span> <span class="n">num_warps</span>
<span class="n">ctx</span><span class="o">.</span><span class="n">eps</span> <span class="o">=</span> <span class="n">eps</span>
<span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">bias</span><span class="p">,</span> <span class="s2">&quot;config&quot;</span><span class="p">):</span>
<span class="k">assert</span> <span class="n">bias</span><span class="o">.</span><span class="n">config</span><span class="o">.</span><span class="n">grad_scale_name</span> <span class="o">==</span> <span class="n">weight</span><span class="o">.</span><span class="n">config</span><span class="o">.</span><span class="n">grad_scale_name</span>
<span class="n">grad_scale_name</span> <span class="o">=</span> <span class="n">bias</span><span class="o">.</span><span class="n">config</span><span class="o">.</span><span class="n">grad_scale_name</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">grad_scale_name</span> <span class="o">=</span> <span class="kc">None</span>
<span class="n">ctx</span><span class="o">.</span><span class="n">grad_scale_gain_bias_name</span> <span class="o">=</span> <span class="n">grad_scale_name</span>
<span class="k">return</span> <span class="n">out</span>
<span class="nd">@staticmethod</span>
<span class="k">def</span> <span class="nf">backward</span><span class="p">(</span><span class="n">ctx</span><span class="p">,</span> <span class="n">dout</span><span class="p">):</span>
<span class="k">assert</span> <span class="n">dout</span><span class="o">.</span><span class="n">is_contiguous</span><span class="p">()</span>
<span class="n">a</span><span class="p">,</span> <span class="n">weight</span><span class="p">,</span> <span class="n">bias</span><span class="p">,</span> <span class="n">mean</span><span class="p">,</span> <span class="n">var</span> <span class="o">=</span> <span class="n">ctx</span><span class="o">.</span><span class="n">saved_tensors</span>
<span class="c1"># heuristics for amount of parallel reduction stream for DG/DB</span>
<span class="n">N</span> <span class="o">=</span> <span class="n">weight</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
<span class="c1"># allocate output</span>
<span class="n">da</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">empty_like</span><span class="p">(</span><span class="n">dout</span><span class="p">)</span>
<span class="c1"># enqueue kernel using forward pass heuristics</span>
<span class="c1"># also compute partial sums for DW and DB</span>
<span class="n">x_arg</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">a</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
<span class="n">M</span><span class="p">,</span> <span class="n">N</span> <span class="o">=</span> <span class="n">x_arg</span><span class="o">.</span><span class="n">shape</span>
<span class="n">dweight</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="n">weight</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">],),</span> <span class="n">dtype</span><span class="o">=</span><span class="n">weight</span><span class="o">.</span><span class="n">dtype</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="n">weight</span><span class="o">.</span><span class="n">device</span><span class="p">)</span>
<span class="n">dbias</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="n">weight</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">],),</span> <span class="n">dtype</span><span class="o">=</span><span class="n">weight</span><span class="o">.</span><span class="n">dtype</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="n">weight</span><span class="o">.</span><span class="n">device</span><span class="p">)</span>
<span class="n">_layer_norm_bwd_dx_fused</span><span class="p">[(</span><span class="n">M</span><span class="p">,)](</span>
<span class="n">da</span><span class="p">,</span>
<span class="n">dout</span><span class="p">,</span>
<span class="n">a</span><span class="p">,</span>
<span class="n">weight</span><span class="p">,</span>
<span class="n">mean</span><span class="p">,</span> <span class="n">var</span><span class="p">,</span>
<span class="n">x_arg</span><span class="o">.</span><span class="n">stride</span><span class="p">(</span><span class="mi">0</span><span class="p">),</span> <span class="n">M</span><span class="p">,</span> <span class="n">N</span><span class="p">,</span>
<span class="n">ctx</span><span class="o">.</span><span class="n">eps</span><span class="p">,</span>
<span class="n">BLOCK_SIZE_N</span><span class="o">=</span><span class="n">ctx</span><span class="o">.</span><span class="n">BLOCK_SIZE</span><span class="p">,</span>
<span class="n">num_warps</span><span class="o">=</span><span class="n">ctx</span><span class="o">.</span><span class="n">num_warps</span><span class="p">,</span>
<span class="p">)</span>
<span class="c1"># accumulate partial sums in separate kernel</span>
<span class="n">grid</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">meta</span><span class="p">:</span> <span class="p">[</span><span class="n">triton</span><span class="o">.</span><span class="n">cdiv</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">meta</span><span class="p">[</span><span class="s2">&quot;BLOCK_SIZE_N&quot;</span><span class="p">])]</span>
<span class="n">_layer_norm_bwd_dwdb</span><span class="p">[</span><span class="n">grid</span><span class="p">](</span>
<span class="n">a</span><span class="p">,</span> <span class="n">dout</span><span class="p">,</span>
<span class="n">mean</span><span class="p">,</span> <span class="n">var</span><span class="p">,</span>
<span class="n">dweight</span><span class="p">,</span>
<span class="n">dbias</span><span class="p">,</span>
<span class="n">M</span><span class="p">,</span>
<span class="n">N</span><span class="p">,</span>
<span class="n">BLOCK_SIZE_M</span><span class="o">=</span><span class="mi">32</span><span class="p">,</span>
<span class="n">BLOCK_SIZE_N</span><span class="o">=</span><span class="mi">128</span><span class="p">,</span>
<span class="p">)</span>
<span class="k">return</span> <span class="p">(</span><span class="n">da</span><span class="p">,</span> <span class="kc">None</span><span class="p">,</span> <span class="n">dweight</span><span class="p">,</span> <span class="n">dbias</span><span class="p">,</span> <span class="kc">None</span><span class="p">,</span> <span class="kc">None</span><span class="p">,</span>
<span class="kc">None</span><span class="p">,</span> <span class="kc">None</span><span class="p">,</span> <span class="kc">None</span><span class="p">,</span> <span class="kc">None</span><span class="p">,</span>
<span class="kc">None</span><span class="p">,</span>
<span class="kc">None</span><span class="p">,</span> <span class="kc">None</span><span class="p">,</span> <span class="kc">None</span><span class="p">,</span>
<span class="kc">None</span><span class="p">,</span>
<span class="kc">None</span><span class="p">,</span> <span class="kc">None</span><span class="p">,</span> <span class="kc">None</span><span class="p">,</span>
<span class="kc">None</span><span class="p">,</span> <span class="kc">None</span><span class="p">,</span> <span class="kc">None</span><span class="p">,</span>
<span class="kc">None</span><span class="p">,</span> <span class="kc">None</span><span class="p">,</span> <span class="kc">None</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">layer_norm</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">normalized_shape</span><span class="p">,</span> <span class="n">weight</span><span class="p">,</span> <span class="n">bias</span><span class="p">,</span> <span class="n">eps</span><span class="p">):</span>
<span class="k">return</span> <span class="n">LayerNorm</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">normalized_shape</span><span class="p">,</span> <span class="n">weight</span><span class="p">,</span> <span class="n">bias</span><span class="p">,</span> <span class="n">eps</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">test_layer_norm</span><span class="p">(</span><span class="n">M</span><span class="p">,</span> <span class="n">N</span><span class="p">,</span> <span class="n">dtype</span><span class="p">,</span> <span class="n">eps</span><span class="o">=</span><span class="mf">1e-5</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="s1">&#39;cuda&#39;</span><span class="p">):</span>
<span class="n">torch</span><span class="o">.</span><span class="n">manual_seed</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="c1"># create data</span>
<span class="n">x_shape</span> <span class="o">=</span> <span class="p">(</span><span class="n">M</span><span class="p">,</span> <span class="n">N</span><span class="p">)</span>
<span class="n">w_shape</span> <span class="o">=</span> <span class="p">(</span><span class="n">x_shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="p">)</span>
<span class="n">weight</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="n">w_shape</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">dtype</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="s1">&#39;cuda&#39;</span><span class="p">,</span> <span class="n">requires_grad</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="n">bias</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="n">w_shape</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">dtype</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="s1">&#39;cuda&#39;</span><span class="p">,</span> <span class="n">requires_grad</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="n">x</span> <span class="o">=</span> <span class="o">-</span><span class="mf">2.3</span> <span class="o">+</span> <span class="mf">0.5</span> <span class="o">*</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">x_shape</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">dtype</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="s1">&#39;cuda&#39;</span><span class="p">)</span>
<span class="n">dy</span> <span class="o">=</span> <span class="mf">.1</span> <span class="o">*</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn_like</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="n">x</span><span class="o">.</span><span class="n">requires_grad_</span><span class="p">(</span><span class="kc">True</span><span class="p">)</span>
<span class="c1"># forward pass</span>
<span class="n">y_tri</span> <span class="o">=</span> <span class="n">layer_norm</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">w_shape</span><span class="p">,</span> <span class="n">weight</span><span class="p">,</span> <span class="n">bias</span><span class="p">,</span> <span class="n">eps</span><span class="p">)</span>
<span class="n">y_ref</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">nn</span><span class="o">.</span><span class="n">functional</span><span class="o">.</span><span class="n">layer_norm</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">w_shape</span><span class="p">,</span> <span class="n">weight</span><span class="p">,</span> <span class="n">bias</span><span class="p">,</span> <span class="n">eps</span><span class="p">)</span><span class="o">.</span><span class="n">to</span><span class="p">(</span><span class="n">dtype</span><span class="p">)</span>
<span class="c1"># backward pass (triton)</span>
<span class="n">y_tri</span><span class="o">.</span><span class="n">backward</span><span class="p">(</span><span class="n">dy</span><span class="p">,</span> <span class="n">retain_graph</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="n">dx_tri</span><span class="p">,</span> <span class="n">dw_tri</span><span class="p">,</span> <span class="n">db_tri</span> <span class="o">=</span> <span class="p">[</span><span class="n">_</span><span class="o">.</span><span class="n">grad</span><span class="o">.</span><span class="n">clone</span><span class="p">()</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="n">weight</span><span class="p">,</span> <span class="n">bias</span><span class="p">]]</span>
<span class="n">x</span><span class="o">.</span><span class="n">grad</span><span class="p">,</span> <span class="n">weight</span><span class="o">.</span><span class="n">grad</span><span class="p">,</span> <span class="n">bias</span><span class="o">.</span><span class="n">grad</span> <span class="o">=</span> <span class="kc">None</span><span class="p">,</span> <span class="kc">None</span><span class="p">,</span> <span class="kc">None</span>
<span class="c1"># backward pass (torch)</span>
<span class="n">y_ref</span><span class="o">.</span><span class="n">backward</span><span class="p">(</span><span class="n">dy</span><span class="p">,</span> <span class="n">retain_graph</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="n">dx_ref</span><span class="p">,</span> <span class="n">dw_ref</span><span class="p">,</span> <span class="n">db_ref</span> <span class="o">=</span> <span class="p">[</span><span class="n">_</span><span class="o">.</span><span class="n">grad</span><span class="o">.</span><span class="n">clone</span><span class="p">()</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="n">weight</span><span class="p">,</span> <span class="n">bias</span><span class="p">]]</span>
<span class="c1"># compare</span>
<span class="n">triton</span><span class="o">.</span><span class="n">testing</span><span class="o">.</span><span class="n">assert_almost_equal</span><span class="p">(</span><span class="n">y_tri</span><span class="p">,</span> <span class="n">y_ref</span><span class="p">)</span>
<span class="n">triton</span><span class="o">.</span><span class="n">testing</span><span class="o">.</span><span class="n">assert_almost_equal</span><span class="p">(</span><span class="n">dx_tri</span><span class="p">,</span> <span class="n">dx_ref</span><span class="p">)</span>
<span class="n">triton</span><span class="o">.</span><span class="n">testing</span><span class="o">.</span><span class="n">assert_almost_equal</span><span class="p">(</span><span class="n">db_tri</span><span class="p">,</span> <span class="n">db_ref</span><span class="p">,</span> <span class="n">decimal</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="n">triton</span><span class="o">.</span><span class="n">testing</span><span class="o">.</span><span class="n">assert_almost_equal</span><span class="p">(</span><span class="n">dw_tri</span><span class="p">,</span> <span class="n">dw_ref</span><span class="p">,</span> <span class="n">decimal</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="nd">@triton</span><span class="o">.</span><span class="n">testing</span><span class="o">.</span><span class="n">perf_report</span><span class="p">(</span>
<span class="n">triton</span><span class="o">.</span><span class="n">testing</span><span class="o">.</span><span class="n">Benchmark</span><span class="p">(</span>
<span class="n">x_names</span><span class="o">=</span><span class="p">[</span><span class="s1">&#39;N&#39;</span><span class="p">],</span>
<span class="n">x_vals</span><span class="o">=</span><span class="p">[</span><span class="mi">512</span> <span class="o">*</span> <span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">32</span><span class="p">)],</span>
<span class="n">line_arg</span><span class="o">=</span><span class="s1">&#39;provider&#39;</span><span class="p">,</span>
<span class="n">line_vals</span><span class="o">=</span><span class="p">[</span><span class="s1">&#39;triton&#39;</span><span class="p">,</span> <span class="s1">&#39;torch&#39;</span><span class="p">]</span> <span class="o">+</span> <span class="p">([</span><span class="s1">&#39;apex&#39;</span><span class="p">]</span> <span class="k">if</span> <span class="n">HAS_APEX</span> <span class="k">else</span> <span class="p">[]),</span>
<span class="n">line_names</span><span class="o">=</span><span class="p">[</span><span class="s1">&#39;Triton&#39;</span><span class="p">,</span> <span class="s1">&#39;Torch&#39;</span><span class="p">]</span> <span class="o">+</span> <span class="p">([</span><span class="s1">&#39;Apex&#39;</span><span class="p">]</span> <span class="k">if</span> <span class="n">HAS_APEX</span> <span class="k">else</span> <span class="p">[]),</span>
<span class="n">styles</span><span class="o">=</span><span class="p">[(</span><span class="s1">&#39;blue&#39;</span><span class="p">,</span> <span class="s1">&#39;-&#39;</span><span class="p">),</span> <span class="p">(</span><span class="s1">&#39;green&#39;</span><span class="p">,</span> <span class="s1">&#39;-&#39;</span><span class="p">),</span> <span class="p">(</span><span class="s1">&#39;orange&#39;</span><span class="p">,</span> <span class="s1">&#39;-&#39;</span><span class="p">)],</span>
<span class="n">ylabel</span><span class="o">=</span><span class="s1">&#39;GB/s&#39;</span><span class="p">,</span>
<span class="n">plot_name</span><span class="o">=</span><span class="s1">&#39;layer-norm&#39;</span><span class="p">,</span>
<span class="n">args</span><span class="o">=</span><span class="p">{</span><span class="s1">&#39;M&#39;</span><span class="p">:</span> <span class="mi">4096</span><span class="p">,</span> <span class="s1">&#39;dtype&#39;</span><span class="p">:</span> <span class="n">torch</span><span class="o">.</span><span class="n">float16</span><span class="p">,</span> <span class="s1">&#39;mode&#39;</span><span class="p">:</span> <span class="s1">&#39;forward&#39;</span><span class="p">}</span>
<span class="p">)</span>
<span class="p">)</span>
<span class="k">def</span> <span class="nf">bench_layer_norm</span><span class="p">(</span><span class="n">M</span><span class="p">,</span> <span class="n">N</span><span class="p">,</span> <span class="n">dtype</span><span class="p">,</span> <span class="n">provider</span><span class="p">,</span> <span class="n">mode</span><span class="p">,</span> <span class="n">eps</span><span class="o">=</span><span class="mf">1e-5</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="s1">&#39;cuda&#39;</span><span class="p">):</span>
<span class="c1"># create data</span>
<span class="n">x_shape</span> <span class="o">=</span> <span class="p">(</span><span class="n">M</span><span class="p">,</span> <span class="n">N</span><span class="p">)</span>
<span class="n">w_shape</span> <span class="o">=</span> <span class="p">(</span><span class="n">x_shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="p">)</span>
<span class="n">weight</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="n">w_shape</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">dtype</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="s1">&#39;cuda&#39;</span><span class="p">,</span> <span class="n">requires_grad</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="n">bias</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="n">w_shape</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">dtype</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="s1">&#39;cuda&#39;</span><span class="p">,</span> <span class="n">requires_grad</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="n">x</span> <span class="o">=</span> <span class="o">-</span><span class="mf">2.3</span> <span class="o">+</span> <span class="mf">0.5</span> <span class="o">*</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">x_shape</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">dtype</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="s1">&#39;cuda&#39;</span><span class="p">)</span>
<span class="n">dy</span> <span class="o">=</span> <span class="mf">.1</span> <span class="o">*</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn_like</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="n">x</span><span class="o">.</span><span class="n">requires_grad_</span><span class="p">(</span><span class="kc">True</span><span class="p">)</span>
<span class="c1"># utility functions</span>
<span class="k">if</span> <span class="n">provider</span> <span class="o">==</span> <span class="s1">&#39;triton&#39;</span><span class="p">:</span>
<span class="n">y_fwd</span> <span class="o">=</span> <span class="k">lambda</span><span class="p">:</span> <span class="n">layer_norm</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">w_shape</span><span class="p">,</span> <span class="n">weight</span><span class="p">,</span> <span class="n">bias</span><span class="p">,</span> <span class="n">eps</span><span class="p">)</span>
<span class="k">if</span> <span class="n">provider</span> <span class="o">==</span> <span class="s1">&#39;torch&#39;</span><span class="p">:</span>
<span class="n">y_fwd</span> <span class="o">=</span> <span class="k">lambda</span><span class="p">:</span> <span class="n">torch</span><span class="o">.</span><span class="n">nn</span><span class="o">.</span><span class="n">functional</span><span class="o">.</span><span class="n">layer_norm</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">w_shape</span><span class="p">,</span> <span class="n">weight</span><span class="p">,</span> <span class="n">bias</span><span class="p">,</span> <span class="n">eps</span><span class="p">)</span>
<span class="k">if</span> <span class="n">provider</span> <span class="o">==</span> <span class="s1">&#39;apex&#39;</span><span class="p">:</span>
<span class="n">apex_layer_norm</span> <span class="o">=</span> <span class="n">apex</span><span class="o">.</span><span class="n">normalization</span><span class="o">.</span><span class="n">FusedLayerNorm</span><span class="p">(</span><span class="n">w_shape</span><span class="p">)</span><span class="o">.</span><span class="n">to</span><span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">device</span><span class="p">)</span><span class="o">.</span><span class="n">to</span><span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">dtype</span><span class="p">)</span>
<span class="n">y_fwd</span> <span class="o">=</span> <span class="k">lambda</span><span class="p">:</span> <span class="n">apex_layer_norm</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="c1"># forward pass</span>
<span class="k">if</span> <span class="n">mode</span> <span class="o">==</span> <span class="s1">&#39;forward&#39;</span><span class="p">:</span>
<span class="n">gbps</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">ms</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">numel</span><span class="p">()</span> <span class="o">*</span> <span class="n">x</span><span class="o">.</span><span class="n">element_size</span><span class="p">()</span> <span class="o">/</span> <span class="n">ms</span> <span class="o">*</span> <span class="mf">1e-6</span>
<span class="n">ms</span><span class="p">,</span> <span class="n">min_ms</span><span class="p">,</span> <span class="n">max_ms</span> <span class="o">=</span> <span class="n">triton</span><span class="o">.</span><span class="n">testing</span><span class="o">.</span><span class="n">do_bench</span><span class="p">(</span><span class="n">y_fwd</span><span class="p">,</span> <span class="n">rep</span><span class="o">=</span><span class="mi">500</span><span class="p">)</span>
<span class="c1"># backward pass</span>
<span class="k">if</span> <span class="n">mode</span> <span class="o">==</span> <span class="s1">&#39;backward&#39;</span><span class="p">:</span>
<span class="n">gbps</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">ms</span><span class="p">:</span> <span class="mi">3</span> <span class="o">*</span> <span class="n">x</span><span class="o">.</span><span class="n">numel</span><span class="p">()</span> <span class="o">*</span> <span class="n">x</span><span class="o">.</span><span class="n">element_size</span><span class="p">()</span> <span class="o">/</span> <span class="n">ms</span> <span class="o">*</span> <span class="mf">1e-6</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">y_fwd</span><span class="p">()</span>
<span class="n">ms</span><span class="p">,</span> <span class="n">min_ms</span><span class="p">,</span> <span class="n">max_ms</span> <span class="o">=</span> <span class="n">triton</span><span class="o">.</span><span class="n">testing</span><span class="o">.</span><span class="n">do_bench</span><span class="p">(</span><span class="k">lambda</span><span class="p">:</span> <span class="n">y</span><span class="o">.</span><span class="n">backward</span><span class="p">(</span><span class="n">dy</span><span class="p">,</span> <span class="n">retain_graph</span><span class="o">=</span><span class="kc">True</span><span class="p">),</span>
<span class="n">grad_to_none</span><span class="o">=</span><span class="p">[</span><span class="n">x</span><span class="p">],</span> <span class="n">rep</span><span class="o">=</span><span class="mi">500</span><span class="p">)</span>
<span class="k">return</span> <span class="n">gbps</span><span class="p">(</span><span class="n">ms</span><span class="p">),</span> <span class="n">gbps</span><span class="p">(</span><span class="n">max_ms</span><span class="p">),</span> <span class="n">gbps</span><span class="p">(</span><span class="n">min_ms</span><span class="p">)</span>
<span class="c1"># test_layer_norm(1151, 8192, torch.float16)</span>
<span class="n">bench_layer_norm</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">save_path</span><span class="o">=</span><span class="s1">&#39;.&#39;</span><span class="p">,</span> <span class="n">print_data</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
</pre></div>
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