[{"data":1,"prerenderedAt":41},["ShallowReactive",2],{"chapter:kernels\u002Fin-a-model\u002Fbackward.json":3},{"project":4,"route":5,"title":6,"titleHtml":6,"navTitle":6,"part":7,"sourcePath":8,"editUrl":9,"html":10,"toc":11,"hasMermaid":34,"prev":35,"next":38},"kernels","\u002Fkernels\u002Fin-a-model\u002Fbackward","Training: The Backward Kernel","Kernels in a Real Model","in-a-model\u002Fbackward.md","https:\u002F\u002Fgithub.com\u002Fteenygrad\u002Fteenygrad\u002Fedit\u002Fmain\u002Fbooks\u002Fkernels\u002Fsrc\u002Fin-a-model\u002Fbackward.md","\u003Cp>Everything so far runs a model forwards. Training needs it to run backwards\ntoo — and a custom op that cannot produce gradients is a wall across the middle\nof your network.\u003C\u002Fp>\n\u003Ch2 id=\"what-a-backward-pass-needs\">What a backward pass needs\u003C\u002Fh2>\n\u003Cp>Training works by chain rule. Each operation is asked: given how much the loss\nchanges with respect to \u003Cem>your output\u003C\u002Fem>, how much does it change with respect to\nyour \u003Cem>inputs\u003C\u002Fem>?\u003C\u002Fp>\n\u003Cp>So a backward kernel takes the upstream gradient \u003Ccode>dy\u003C\u002Fcode> and produces \u003Ccode>dx\u003C\u002Fcode> — one\nper input the forward pass consumed, plus one per parameter that gets trained.\u003C\u002Fp>\n\u003Cp>Two consequences shape everything below.\u003C\u002Fp>\n\u003Cp>\u003Cstrong>It needs values from the forward pass.\u003C\u002Fstrong> Most gradients depend on what went in\nor came out. So the executor keeps the forward activations and hands them back.\u003C\u002Fp>\n\u003Cp>\u003Cstrong>The shapes are mirrored.\u003C\u002Fstrong> Where forward went inputs → output, backward goes\noutput-gradient → input-gradients. A forward with two inputs has a backward that\nwrites two buffers.\u003C\u002Fp>\n\u003Ch2 id=\"declaring-the-pair\">Declaring the pair\u003C\u002Fh2>\n\u003Cpre data-lang=\"rust\" class=\"shiki teeny-datasheet\" style=\"background-color:#16181a;color:#e6e8e3\" tabindex=\"0\">\u003Ccode>\u003Cspan class=\"line\">\u003Cspan style=\"color:#8A9088\">#[\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\">kernel\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">(\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\">backward \u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">=\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\"> GeluBackward\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">)]\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#FF5F9E\">pub\u003C\u002Fspan>\u003Cspan style=\"color:#FF5F9E\"> fn\u003C\u002Fspan>\u003Cspan style=\"color:#7FB6D9\"> gelu_forward\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\"><\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\">T\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">:\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\"> Triton\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">,\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\"> D\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">:\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\"> Float\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">,\u003C\u002Fspan>\u003Cspan style=\"color:#FF5F9E\"> const\u003C\u002Fspan>\u003Cspan style=\"color:#B79AD4\"> BLOCK_SIZE\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">:\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\"> i32\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">>(\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#E6E8E3\">    x_ptr\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">:\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\"> T\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">::\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\">Pointer\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\"><\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\">D\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">>,\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#E6E8E3\">    y_ptr\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">:\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\"> T\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">::\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\">Pointer\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\"><\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\">D\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">>,\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#E6E8E3\">    n_elements\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">:\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\"> i32\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">,\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#8A9088\">)\u003C\u002Fspan>\u003Cspan style=\"color:#FF5F9E\"> where\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#6FBF98\">    T\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">::\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\">I32Tensor\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">:\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\"> types\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">::\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\">Tensor\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\"><\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\">i32\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">,\u003C\u002Fspan>\u003Cspan style=\"color:#E6E8E3\"> 1\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">>,\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#6FBF98\">    T\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">::\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\">I32Tensor\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">:\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\"> Comparison\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\"><\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\">i32\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">,\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\"> BoolTensor\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\"> =\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\"> T\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">::\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\">BoolTensor\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">>,\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#6FBF98\">    T\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">::\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\">Pointer\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\"><\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\">D\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">>:\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\"> AddOffsets\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\"><\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\">i32\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">,\u003C\u002Fspan>\u003Cspan style=\"color:#E6E8E3\"> 1\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">,\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\"> T\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">::\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\">I32Tensor\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">,\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\"> Output\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\"> =\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\"> T\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">::\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\">Tensor\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\"><\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\">T\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">::\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\">Pointer\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\"><\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\">D\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">>>>,\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#8A9088\">{\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#FF5F9E\">    let\u003C\u002Fspan>\u003Cspan style=\"color:#E6E8E3\"> pid \u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">=\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\"> T\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">::\u003C\u002Fspan>\u003Cspan style=\"color:#7FB6D9\">program_id\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">(\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\">Axis\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">::\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\">X\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">);\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#FF5F9E\">    let\u003C\u002Fspan>\u003Cspan style=\"color:#E6E8E3\"> block_start \u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">=\u003C\u002Fspan>\u003Cspan style=\"color:#E6E8E3\"> pid \u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">*\u003C\u002Fspan>\u003Cspan style=\"color:#B79AD4\"> BLOCK_SIZE\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">;\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#FF5F9E\">    let\u003C\u002Fspan>\u003Cspan style=\"color:#E6E8E3\"> offsets \u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">=\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\"> T\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">::\u003C\u002Fspan>\u003Cspan style=\"color:#7FB6D9\">arange\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">(\u003C\u002Fspan>\u003Cspan style=\"color:#B79AD4\">0\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">,\u003C\u002Fspan>\u003Cspan style=\"color:#B79AD4\"> BLOCK_SIZE\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">)\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\"> +\u003C\u002Fspan>\u003Cspan style=\"color:#E6E8E3\"> block_start\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">;\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#FF5F9E\">    let\u003C\u002Fspan>\u003Cspan style=\"color:#E6E8E3\"> in_bounds \u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">=\u003C\u002Fspan>\u003Cspan style=\"color:#E6E8E3\"> offsets\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">.\u003C\u002Fspan>\u003Cspan style=\"color:#7FB6D9\">lt\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">(\u003C\u002Fspan>\u003Cspan style=\"color:#E6E8E3\">n_elements\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">);\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#FF5F9E\">    let\u003C\u002Fspan>\u003Cspan style=\"color:#E6E8E3\"> x \u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">=\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\"> T\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">::\u003C\u002Fspan>\u003Cspan style=\"color:#7FB6D9\">load\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">(\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#E6E8E3\">        x_ptr\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">.\u003C\u002Fspan>\u003Cspan style=\"color:#7FB6D9\">add_offsets\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">(\u003C\u002Fspan>\u003Cspan style=\"color:#E6E8E3\">offsets\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">),\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#6FBF98\">        Some\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">(\u003C\u002Fspan>\u003Cspan style=\"color:#E6E8E3\">in_bounds\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">),\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#6FBF98\">        None\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">,\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#8A9088\">        &[],\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#6FBF98\">        None\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">,\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#6FBF98\">        None\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">,\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#6FBF98\">        None\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">,\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#B79AD4\">        false\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">,\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#8A9088\">    );\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#FF5F9E\">    let\u003C\u002Fspan>\u003Cspan style=\"color:#E6E8E3\"> one \u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">=\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\"> T\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">::\u003C\u002Fspan>\u003Cspan style=\"color:#7FB6D9\">full\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">(&[\u003C\u002Fspan>\u003Cspan style=\"color:#B79AD4\">BLOCK_SIZE\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">],\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\"> D\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">::\u003C\u002Fspan>\u003Cspan style=\"color:#7FB6D9\">from_f64\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">(\u003C\u002Fspan>\u003Cspan style=\"color:#B79AD4\">1\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">.\u003C\u002Fspan>\u003Cspan style=\"color:#B79AD4\">0\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">));\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#FF5F9E\">    let\u003C\u002Fspan>\u003Cspan style=\"color:#E6E8E3\"> neg2c \u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">=\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\"> T\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">::\u003C\u002Fspan>\u003Cspan style=\"color:#7FB6D9\">full\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">(&[\u003C\u002Fspan>\u003Cspan style=\"color:#B79AD4\">BLOCK_SIZE\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">],\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\"> D\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">::\u003C\u002Fspan>\u003Cspan style=\"color:#7FB6D9\">from_f64\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">(-\u003C\u002Fspan>\u003Cspan style=\"color:#B79AD4\">2\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">.\u003C\u002Fspan>\u003Cspan style=\"color:#B79AD4\">0\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\"> *\u003C\u002Fspan>\u003Cspan style=\"color:#B79AD4\"> 0\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">.\u003C\u002Fspan>\u003Cspan style=\"color:#B79AD4\">7978845608028654\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">));\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#FF5F9E\">    let\u003C\u002Fspan>\u003Cspan style=\"color:#E6E8E3\"> coeff \u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">=\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\"> T\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">::\u003C\u002Fspan>\u003Cspan style=\"color:#7FB6D9\">full\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">(&[\u003C\u002Fspan>\u003Cspan style=\"color:#B79AD4\">BLOCK_SIZE\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">],\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\"> D\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">::\u003C\u002Fspan>\u003Cspan style=\"color:#7FB6D9\">from_f64\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">(\u003C\u002Fspan>\u003Cspan style=\"color:#B79AD4\">0\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">.\u003C\u002Fspan>\u003Cspan style=\"color:#B79AD4\">044715\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">));\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#7F877D;font-style:italic\">    \u002F\u002F tanh-GELU: y = x * 0.5 * (1 + tanh(c*(x + a*x³)))\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#7F877D;font-style:italic\">    \u002F\u002F              = x \u002F (1 + exp(-2c*(x + a*x³)))\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#FF5F9E\">    let\u003C\u002Fspan>\u003Cspan style=\"color:#E6E8E3\"> inner \u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">=\u003C\u002Fspan>\u003Cspan style=\"color:#E6E8E3\"> x \u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">+\u003C\u002Fspan>\u003Cspan style=\"color:#E6E8E3\"> coeff \u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">*\u003C\u002Fspan>\u003Cspan style=\"color:#E6E8E3\"> x \u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">*\u003C\u002Fspan>\u003Cspan style=\"color:#E6E8E3\"> x \u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">*\u003C\u002Fspan>\u003Cspan style=\"color:#E6E8E3\"> x\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">;\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#FF5F9E\">    let\u003C\u002Fspan>\u003Cspan style=\"color:#E6E8E3\"> y \u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">=\u003C\u002Fspan>\u003Cspan style=\"color:#E6E8E3\"> x \u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">\u002F\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\"> (\u003C\u002Fspan>\u003Cspan style=\"color:#E6E8E3\">one \u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">+\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\"> T\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">::\u003C\u002Fspan>\u003Cspan style=\"color:#7FB6D9\">exp\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">(\u003C\u002Fspan>\u003Cspan style=\"color:#E6E8E3\">neg2c \u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">*\u003C\u002Fspan>\u003Cspan style=\"color:#E6E8E3\"> inner\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">));\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#6FBF98\">    T\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">::\u003C\u002Fspan>\u003Cspan style=\"color:#7FB6D9\">store\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">(\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#E6E8E3\">        y_ptr\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">.\u003C\u002Fspan>\u003Cspan style=\"color:#7FB6D9\">add_offsets\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">(\u003C\u002Fspan>\u003Cspan style=\"color:#E6E8E3\">offsets\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">),\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#E6E8E3\">        y\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">,\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#6FBF98\">        Some\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">(\u003C\u002Fspan>\u003Cspan style=\"color:#E6E8E3\">in_bounds\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">),\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#8A9088\">        &[],\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#6FBF98\">        None\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">,\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#6FBF98\">        None\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">,\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#8A9088\">    );\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#8A9088\">}\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003C\u002Fspan>\u003C\u002Fcode>\u003C\u002Fpre>\n\u003Cp>\u003Ccode>#[kernel(backward = GeluBackward)]\u003C\u002Fcode> names the struct that computes this\nkernel’s gradient. \u003Ccode>GeluBackward\u003C\u002Fcode> is generated from another \u003Ccode>#[kernel]\u003C\u002Fcode> function\nin the same file, exactly like the forward one — there is no separate macro for\nbackward kernels.\u003C\u002Fp>\n\u003Cp>This attribute is used throughout the tree: every activation with a derivative,\nthe elementwise ops, the losses. It also opts the kernel into dtype dispatch,\nwhich is where the implicit \u003Ccode>f32\u003C\u002Fcode>\u002F\u003Ccode>f64\u003C\u002Fcode> set from Chapter 15 comes from.\u003C\u002Fp>\n\u003Ch2 id=\"writing-the-gradient\">Writing the gradient\u003C\u002Fh2>\n\u003Cp>The simplest case is addition, where the gradient flows through unchanged to\nboth inputs:\u003C\u002Fp>\n\u003Cpre data-lang=\"rust\" class=\"shiki teeny-datasheet\" style=\"background-color:#16181a;color:#e6e8e3\" tabindex=\"0\">\u003Ccode>\u003Cspan class=\"line\">\u003Cspan style=\"color:#7F877D;font-style:italic\">\u002F\u002F grad_a[i] = dy[i],  grad_b[i] = dy[i]\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#FF5F9E\">let\u003C\u002Fspan>\u003Cspan style=\"color:#E6E8E3\"> dy \u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">=\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\"> T\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">::\u003C\u002Fspan>\u003Cspan style=\"color:#7FB6D9\">load\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">(\u003C\u002Fspan>\u003Cspan style=\"color:#E6E8E3\">dy_ptr\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">.\u003C\u002Fspan>\u003Cspan style=\"color:#7FB6D9\">add_offsets\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">(\u003C\u002Fspan>\u003Cspan style=\"color:#E6E8E3\">offsets\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">),\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\"> Some\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">(\u003C\u002Fspan>\u003Cspan style=\"color:#E6E8E3\">in_bounds\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">),\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\"> ...);\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#6FBF98\">T\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">::\u003C\u002Fspan>\u003Cspan style=\"color:#7FB6D9\">store\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">(\u003C\u002Fspan>\u003Cspan style=\"color:#E6E8E3\">grad_a_ptr\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">.\u003C\u002Fspan>\u003Cspan style=\"color:#7FB6D9\">add_offsets\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">(\u003C\u002Fspan>\u003Cspan style=\"color:#E6E8E3\">offsets\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">),\u003C\u002Fspan>\u003Cspan style=\"color:#E6E8E3\"> dy\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">,\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\"> Some\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">(\u003C\u002Fspan>\u003Cspan style=\"color:#E6E8E3\">in_bounds\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">),\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\"> &[],\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\"> None\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">,\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\"> None\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">);\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#6FBF98\">T\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">::\u003C\u002Fspan>\u003Cspan style=\"color:#7FB6D9\">store\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">(\u003C\u002Fspan>\u003Cspan style=\"color:#E6E8E3\">grad_b_ptr\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">.\u003C\u002Fspan>\u003Cspan style=\"color:#7FB6D9\">add_offsets\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">(\u003C\u002Fspan>\u003Cspan style=\"color:#E6E8E3\">offsets\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">),\u003C\u002Fspan>\u003Cspan style=\"color:#E6E8E3\"> dy\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">,\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\"> Some\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">(\u003C\u002Fspan>\u003Cspan style=\"color:#E6E8E3\">in_bounds\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">),\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\"> &[],\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\"> None\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">,\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\"> None\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">);\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003C\u002Fspan>\u003C\u002Fcode>\u003C\u002Fpre>\n\u003Cp>\u003Cem>From \u003Ccode>kernels\u002Fteeny-kernels\u002Fsrc\u002Fnn\u002Ftensor\u002Felemwise_add.rs\u003C\u002Fcode>.\u003C\u002Fem>\u003C\u002Fp>\n\u003Cp>Softmax is the more representative shape. Its gradient needs the forward\n\u003Cem>output\u003C\u002Fem>, not the input:\u003C\u002Fp>\n\u003Cpre class=\"code-panel\" data-lang=\"text\">\u003Ccode>dx_i = y_i * (dy_i - sum_j(y_j * dy_j))\n\u003C\u002Fcode>\u003C\u002Fpre>\n\u003Cp>so \u003Ccode>softmax_backward\u003C\u002Fcode> takes \u003Ccode>dy_ptr\u003C\u002Fcode>, \u003Ccode>y_ptr\u003C\u002Fcode> and \u003Ccode>dx_ptr\u003C\u002Fcode>. The saved \u003Ccode>y\u003C\u002Fcode> is\nwhat the executor hands back, and the inner sum is a row-wide reduction —\nChapter 10’s machinery, in a backward kernel.\u003C\u002Fp>\n\u003Cp>That is the usual pattern: a backward kernel is a normal kernel with more\npointer arguments.\u003C\u002Fp>\n\u003Ch2 id=\"wiring-it-to-the-runtime\">Wiring it to the runtime\u003C\u002Fh2>\n\u003Cp>\u003Ccode>RuntimeOp\u003C\u002Fcode> has a parallel set of methods for the backward pass, all behind the\n\u003Ccode>training\u003C\u002Fcode> cargo feature:\u003C\u002Fp>\n\u003Cpre data-lang=\"rust\" class=\"shiki teeny-datasheet\" style=\"background-color:#16181a;color:#e6e8e3\" tabindex=\"0\">\u003Ccode>\u003Cspan class=\"line\">\u003Cspan style=\"color:#8A9088\">#[\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\">cfg\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">(\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\">feature \u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">=\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\"> \"\u003C\u002Fspan>\u003Cspan style=\"color:#D8A76B\">training\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">\"\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">)]\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#FF5F9E\">fn\u003C\u002Fspan>\u003Cspan style=\"color:#7FB6D9\"> has_backward\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">(&\u003C\u002Fspan>\u003Cspan style=\"color:#FF5F9E\">self\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">)\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\"> ->\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\"> bool\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\"> {\u003C\u002Fspan>\u003Cspan style=\"color:#B79AD4\"> true\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\"> }\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#8A9088\">#[\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\">cfg\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">(\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\">feature \u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">=\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\"> \"\u003C\u002Fspan>\u003Cspan style=\"color:#D8A76B\">training\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">\"\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">)]\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#FF5F9E\">fn\u003C\u002Fspan>\u003Cspan style=\"color:#7FB6D9\"> pack_backward_args\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">(\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#8A9088\">    &\u003C\u002Fspan>\u003Cspan style=\"color:#FF5F9E\">self\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">,\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#E6E8E3\">    inputs\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">:\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\"> &[(\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\">RawPtr\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">,\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\"> &[\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\">usize\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">])],\u003C\u002Fspan>\u003Cspan style=\"color:#7F877D;font-style:italic\">   \u002F\u002F forward inputs, from the activation cache\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#E6E8E3\">    params\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">:\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\"> &[\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\">RawPtr\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">],\u003C\u002Fspan>\u003Cspan style=\"color:#7F877D;font-style:italic\">               \u002F\u002F forward parameters\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#E6E8E3\">    output\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">:\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\"> RawPtr\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">,\u003C\u002Fspan>\u003Cspan style=\"color:#7F877D;font-style:italic\">                  \u002F\u002F forward output — the saved `y`\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#E6E8E3\">    output_shape\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">:\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\"> &[\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\">usize\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">],\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#E6E8E3\">    grad_output\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">:\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\"> RawPtr\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">,\u003C\u002Fspan>\u003Cspan style=\"color:#7F877D;font-style:italic\">             \u002F\u002F dy, from the consumer\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#E6E8E3\">    grad_output_row_stride\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">:\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\"> i32\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">,\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#E6E8E3\">    grad_inputs\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">:\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\"> &[\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\">RawPtr\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">],\u003C\u002Fspan>\u003Cspan style=\"color:#7F877D;font-style:italic\">          \u002F\u002F dx, one per activation input — you write these\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#E6E8E3\">    grad_params\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">:\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\"> &[\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\">RawPtr\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">],\u003C\u002Fspan>\u003Cspan style=\"color:#7F877D;font-style:italic\">          \u002F\u002F dW, one per parameter\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#E6E8E3\">    visitor\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">:\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\"> &\u003C\u002Fspan>\u003Cspan style=\"color:#FF5F9E\">mut\u003C\u002Fspan>\u003Cspan style=\"color:#FF5F9E\"> dyn\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\"> ArgVisitor\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">,\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#8A9088\">);\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#8A9088\">#[\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\">cfg\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">(\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\">feature \u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">=\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\"> \"\u003C\u002Fspan>\u003Cspan style=\"color:#D8A76B\">training\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">\"\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">)]\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#FF5F9E\">fn\u003C\u002Fspan>\u003Cspan style=\"color:#7FB6D9\"> backward_block\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">(&\u003C\u002Fspan>\u003Cspan style=\"color:#FF5F9E\">self\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">)\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\"> ->\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\"> [\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\">u32\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">;\u003C\u002Fspan>\u003Cspan style=\"color:#B79AD4\"> 3\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">];\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#8A9088\">#[\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\">cfg\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">(\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\">feature \u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">=\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\"> \"\u003C\u002Fspan>\u003Cspan style=\"color:#D8A76B\">training\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">\"\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">)]\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#FF5F9E\">fn\u003C\u002Fspan>\u003Cspan style=\"color:#7FB6D9\"> backward_grid\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">(&\u003C\u002Fspan>\u003Cspan style=\"color:#FF5F9E\">self\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">,\u003C\u002Fspan>\u003Cspan style=\"color:#E6E8E3\"> input_shapes\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">:\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\"> &[&[\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\">usize\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">]],\u003C\u002Fspan>\u003Cspan style=\"color:#E6E8E3\"> output_shape\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">:\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\"> &[\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\">usize\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">])\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\"> ->\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\"> [\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\">u32\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">;\u003C\u002Fspan>\u003Cspan style=\"color:#B79AD4\"> 3\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">];\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003C\u002Fspan>\u003C\u002Fcode>\u003C\u002Fpre>\n\u003Cp>Same contract as Chapter 21, same absence of checking, with more buffers to get\nin the right order.\u003C\u002Fp>\n\u003Cp>Two defaults are worth knowing because they fail quietly:\u003C\u002Fp>\n\u003Cul>\n\u003Cli>\u003Cstrong>\u003Ccode>has_backward\u003C\u002Fcode> defaults to \u003Ccode>false\u003C\u002Fcode>.\u003C\u002Fstrong> Forget it and your op contributes no\ngradient. Nothing errors; the parameters upstream of it simply never learn.\u003C\u002Fli>\n\u003Cli>\u003Cstrong>\u003Ccode>backward_grid\u003C\u002Fcode> defaults to \u003Ccode>[0, 0, 0]\u003C\u002Fcode>.\u003C\u002Fstrong> A grid of zero launches no\nprograms. Also silent, also produces zeros.\u003C\u002Fli>\n\u003C\u002Ful>\n\u003Cp>If gradients are zero everywhere behind your op, check those two first.\u003C\u002Fp>\n\u003Cp>The elementwise-add op shows the whole shape:\u003C\u002Fp>\n\u003Cpre data-lang=\"rust\" class=\"shiki teeny-datasheet\" style=\"background-color:#16181a;color:#e6e8e3\" tabindex=\"0\">\u003Ccode>\u003Cspan class=\"line\">\u003Cspan style=\"color:#8A9088\">#[\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\">cfg\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">(\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\">feature \u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">=\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\"> \"\u003C\u002Fspan>\u003Cspan style=\"color:#D8A76B\">training\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">\"\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">)]\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#FF5F9E\">fn\u003C\u002Fspan>\u003Cspan style=\"color:#7FB6D9\"> pack_backward_args\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">(&\u003C\u002Fspan>\u003Cspan style=\"color:#FF5F9E\">self\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">,\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\"> ...,\u003C\u002Fspan>\u003Cspan style=\"color:#E6E8E3\"> grad_output\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">,\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\"> ...,\u003C\u002Fspan>\u003Cspan style=\"color:#E6E8E3\"> grad_inputs\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">,\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\"> ...,\u003C\u002Fspan>\u003Cspan style=\"color:#E6E8E3\"> visitor\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">)\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\"> {\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#FF5F9E\">    let\u003C\u002Fspan>\u003Cspan style=\"color:#E6E8E3\"> n\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">:\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\"> usize\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\"> =\u003C\u002Fspan>\u003Cspan style=\"color:#E6E8E3\"> output_shape\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">.\u003C\u002Fspan>\u003Cspan style=\"color:#7FB6D9\">iter\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">().\u003C\u002Fspan>\u003Cspan style=\"color:#7FB6D9\">product\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">();\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#E6E8E3\">    visitor\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">.\u003C\u002Fspan>\u003Cspan style=\"color:#7FB6D9\">visit_ptr\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">(\u003C\u002Fspan>\u003Cspan style=\"color:#E6E8E3\">grad_output\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">);\u003C\u002Fspan>\u003Cspan style=\"color:#7F877D;font-style:italic\">    \u002F\u002F dy_ptr\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#E6E8E3\">    visitor\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">.\u003C\u002Fspan>\u003Cspan style=\"color:#7FB6D9\">visit_ptr\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">(\u003C\u002Fspan>\u003Cspan style=\"color:#E6E8E3\">grad_inputs\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">[\u003C\u002Fspan>\u003Cspan style=\"color:#B79AD4\">0\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">]);\u003C\u002Fspan>\u003Cspan style=\"color:#7F877D;font-style:italic\"> \u002F\u002F grad_a_ptr\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#E6E8E3\">    visitor\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">.\u003C\u002Fspan>\u003Cspan style=\"color:#7FB6D9\">visit_ptr\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">(\u003C\u002Fspan>\u003Cspan style=\"color:#E6E8E3\">grad_inputs\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">[\u003C\u002Fspan>\u003Cspan style=\"color:#B79AD4\">1\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">]);\u003C\u002Fspan>\u003Cspan style=\"color:#7F877D;font-style:italic\"> \u002F\u002F grad_b_ptr\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#E6E8E3\">    visitor\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">.\u003C\u002Fspan>\u003Cspan style=\"color:#7FB6D9\">visit_i32\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">(\u003C\u002Fspan>\u003Cspan style=\"color:#E6E8E3\">n \u003C\u002Fspan>\u003Cspan style=\"color:#FF5F9E\">as\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\"> i32\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">);\u003C\u002Fspan>\u003Cspan style=\"color:#7F877D;font-style:italic\">       \u002F\u002F n_elements\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#8A9088\">}\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003C\u002Fspan>\u003C\u002Fcode>\u003C\u002Fpre>\n\u003Ch2 id=\"for-a-custom-op\">For a custom op\u003C\u002Fh2>\n\u003Cp>A \u003Ccode>CustomOp\u003C\u002Fcode> supplies its backward source through a separate method:\u003C\u002Fp>\n\u003Cpre data-lang=\"rust\" class=\"shiki teeny-datasheet\" style=\"background-color:#16181a;color:#e6e8e3\" tabindex=\"0\">\u003Ccode>\u003Cspan class=\"line\">\u003Cspan style=\"color:#FF5F9E\">fn\u003C\u002Fspan>\u003Cspan style=\"color:#7FB6D9\"> lower_backward_source\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">(&\u003C\u002Fspan>\u003Cspan style=\"color:#FF5F9E\">self\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">)\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\"> ->\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\"> String\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\"> {\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#6FBF98\">    MyOpBackward\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">::<\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\">D\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">>::\u003C\u002Fspan>\u003Cspan style=\"color:#7FB6D9\">new\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">(\u003C\u002Fspan>\u003Cspan style=\"color:#FF5F9E\">self\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">.\u003C\u002Fspan>\u003Cspan style=\"color:#E6E8E3\">block_size\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">).\u003C\u002Fspan>\u003Cspan style=\"color:#E6E8E3\">source\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">.\u003C\u002Fspan>\u003Cspan style=\"color:#7FB6D9\">clone\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">()\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#8A9088\">}\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003C\u002Fspan>\u003C\u002Fcode>\u003C\u002Fpre>\n\u003Cp>The default returns an empty string, which the lowering reads as “no backward”,\nso an op without it is inference-only.\u003C\u002Fp>\n\u003Ch2 id=\"inference-and-training-modes\">Inference and training modes\u003C\u002Fh2>\n\u003Cp>The lowering is told which it is building for:\u003C\u002Fp>\n\u003Cpre data-lang=\"rust\" class=\"shiki teeny-datasheet\" style=\"background-color:#16181a;color:#e6e8e3\" tabindex=\"0\">\u003Ccode>\u003Cspan class=\"line\">\u003Cspan style=\"color:#FF5F9E\">pub\u003C\u002Fspan>\u003Cspan style=\"color:#FF5F9E\"> enum\u003C\u002Fspan>\u003Cspan style=\"color:#6FBF98\"> LoweringMode\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\"> {\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#6FBF98\">    Inference\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">,\u003C\u002Fspan>\u003Cspan style=\"color:#7F877D;font-style:italic\">  \u002F\u002F default — no backward kernels\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#6FBF98\">    Training\u003C\u002Fspan>\u003Cspan style=\"color:#8A9088\">,\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003Cspan style=\"color:#8A9088\">}\u003C\u002Fspan>\u003C\u002Fspan>\n\u003Cspan class=\"line\">\u003C\u002Fspan>\u003C\u002Fcode>\u003C\u002Fpre>\n\u003Cp>In \u003Ccode>Inference\u003C\u002Fcode> the backward kernels are never compiled, which is the right\ndefault: a deployed model should not pay for machinery it will not use.\u003C\u002Fp>\n\u003Cp>The \u003Ccode>training\u003C\u002Fcode> cargo feature is the compile-time half of the same distinction.\nIt gates the backward methods on \u003Ccode>RuntimeOp\u003C\u002Fcode> entirely, so the trait’s shape\nchanges with your feature flags. Build without it and \u003Ccode>has_backward\u003C\u002Fcode> does not\nexist to be overridden.\u003C\u002Fp>\n\u003Ch2 id=\"checking-a-gradient\">Checking a gradient\u003C\u002Fh2>\n\u003Cp>An analytically-derived gradient with a sign error still produces plausible\nnumbers, and a model that trains slightly worse than it should is an\nextraordinarily hard bug to find later.\u003C\u002Fp>\n\u003Cp>Check it numerically instead. The derivative is a limit, so approximate it:\u003C\u002Fp>\n\u003Cpre class=\"code-panel\" data-lang=\"text\">\u003Ccode>dx_i  ≈  (f(x + h·e_i) - f(x - h·e_i)) \u002F 2h\n\u003C\u002Fcode>\u003C\u002Fpre>\n\u003Cp>Perturb one input by a small \u003Ccode>h\u003C\u002Fcode>, run the forward pass twice, and compare\nagainst what your backward kernel produced. With \u003Ccode>h\u003C\u002Fcode> around \u003Ccode>1e-3\u003C\u002Fcode> in \u003Ccode>f32\u003C\u002Fcode> this\nagrees to a few decimal places. Too small and rounding dominates; too large and\nthe approximation does.\u003C\u002Fp>\n\u003Cp>Do it once per backward kernel, on a small input, and keep it as a test. Chapter\n14’s warning applies: if your backward uses atomics it is not bit-reproducible,\nso compare with a tolerance.\u003C\u002Fp>\n\u003Cp>Next: building all of this for a machine you do not have in front of you.\u003C\u002Fp>\n",[12,16,19,22,25,28,31],{"id":13,"text":14,"level":15},"what-a-backward-pass-needs","What a backward pass needs",2,{"id":17,"text":18,"level":15},"declaring-the-pair","Declaring the pair",{"id":20,"text":21,"level":15},"writing-the-gradient","Writing the gradient",{"id":23,"text":24,"level":15},"wiring-it-to-the-runtime","Wiring it to the runtime",{"id":26,"text":27,"level":15},"for-a-custom-op","For a custom op",{"id":29,"text":30,"level":15},"inference-and-training-modes","Inference and training modes",{"id":32,"text":33,"level":15},"checking-a-gradient","Checking a gradient",false,{"title":36,"titleHtml":36,"route":37},"Wiring the Runtime","\u002Fkernels\u002Fin-a-model\u002Fruntime-op",{"title":39,"titleHtml":39,"route":40},"Building for Another Target","\u002Fkernels\u002Fin-a-model\u002Fcross-building",1786271830108]