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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>
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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
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0 1024.0 585.142849 277.694907 468.114273
[GH-PAGES] Updated website
2022-06-24 00:46:49 +00:00
1 1536.0 630.153868 323.368435 511.999982
2 2048.0 682.666643 334.367358 520.126988
[GH-PAGES] Updated website
2022-06-27 00:48:22 +00:00
3 2560.0 694.237267 365.714281 518.481028
[GH-PAGES] Updated website
2022-06-26 00:50:12 +00:00
4 3072.0 712.347810 378.092307 501.551037
[GH-PAGES] Updated website
2022-06-27 00:48:22 +00:00
5 3584.0 725.873439 384.859062 458.751978
[GH-PAGES] Updated website
2022-06-26 00:50:12 +00:00
6 4096.0 728.177767 381.023256 458.293714
[GH-PAGES] Updated website
2022-06-27 00:48:22 +00:00
7 4608.0 670.254540 396.387087 426.173427
8 5120.0 694.237267 397.669909 426.666652
9 5632.0 704.000002 396.969169 413.357796
[GH-PAGES] Updated website
2022-06-24 00:46:49 +00:00
10 6144.0 702.171410 402.885254 411.313806
[GH-PAGES] Updated website
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11 6656.0 700.631610 400.360920 400.360920
12 7168.0 695.078767 396.844306 388.772874
[GH-PAGES] Updated website
2022-06-26 00:50:12 +00:00
13 7680.0 682.666656 393.846167 387.634072
[GH-PAGES] Updated website
2022-06-27 00:48:22 +00:00
14 8192.0 642.509816 393.609605 372.363633
15 8704.0 627.315309 389.005597 380.502740
16 9216.0 606.814809 407.337026 383.999986
[GH-PAGES] Updated website
2022-06-25 00:46:57 +00:00
17 9728.0 589.575753 409.599987 383.369452
[GH-PAGES] Updated website
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18 10240.0 566.920437 408.578556 382.803739
[GH-PAGES] Updated website
2022-06-25 00:46:57 +00:00
19 10752.0 549.623009 411.559798 381.445676
[GH-PAGES] Updated website
2022-06-24 00:46:49 +00:00
20 11264.0 536.380957 406.826188 373.134567
[GH-PAGES] Updated website
2022-06-27 00:48:22 +00:00
21 11776.0 523.377770 410.492372 377.587162
22 12288.0 517.389457 414.784810 383.251457
23 12800.0 505.679014 410.420828 376.470582
[GH-PAGES] Updated website
2022-06-26 00:50:12 +00:00
24 13312.0 494.180982 405.699062 376.976995
[GH-PAGES] Updated website
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25 13824.0 482.934503 411.888257 379.389355
26 14336.0 471.967074 406.695045 374.185964
27 14848.0 461.297068 408.192434 375.304904
[GH-PAGES] Updated website
2022-06-23 00:46:26 +00:00
28 15360.0 454.269882 406.214870 378.092307
[GH-PAGES] Updated website
2022-06-26 00:50:12 +00:00
29 15872.0 447.887117 407.627589 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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