{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "hide_input": false
   },
   "outputs": [],
   "source": [
    "#hide\n",
    "from fastai.gen_doc.nbdoc import *"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# A neural net from the foundations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## A neural net layer from scratch"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Modeling a neuron"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Matrix multiplication from scratch"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import torch\n",
    "from torch import tensor"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def matmul(a,b):\n",
    "    ar,ac = a.shape # n_rows * n_cols\n",
    "    br,bc = b.shape\n",
    "    assert ac==br\n",
    "    c = torch.zeros(ar, bc)\n",
    "    for i in range(ar):\n",
    "        for j in range(bc):\n",
    "            for k in range(ac): c[i,j] += a[i,k] * b[k,j]\n",
    "    return c"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "m1 = torch.randn(5,28*28)\n",
    "m2 = torch.randn(784,10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "CPU times: user 1.15 s, sys: 4.09 ms, total: 1.15 s\n",
      "Wall time: 1.15 s\n"
     ]
    }
   ],
   "source": [
    "%time t1=matmul(m1, m2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "14 µs ± 8.95 µs per loop (mean ± std. dev. of 7 runs, 20 loops each)\n"
     ]
    }
   ],
   "source": [
    "%timeit -n 20 t2=m1@m2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([12., 14.,  3.])"
      ]
     },
     "execution_count": null,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = tensor([10., 6, -4])\n",
    "b = tensor([2., 8, 7])\n",
    "a + b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([False,  True,  True])"
      ]
     },
     "execution_count": null,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a < b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(tensor(False), tensor(False))"
      ]
     },
     "execution_count": null,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(a < b).all(), (a==b).all()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "9.666666984558105"
      ]
     },
     "execution_count": null,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(a + b).mean().item()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([[ 1.,  4.,  9.],\n",
       "        [16., 25., 36.],\n",
       "        [49., 64., 81.]])"
      ]
     },
     "execution_count": null,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m = tensor([[1., 2, 3], [4,5,6], [7,8,9]])\n",
    "m*m"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "ename": "RuntimeError",
     "evalue": "The size of tensor a (3) must match the size of tensor b (2) at non-singleton dimension 0",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mRuntimeError\u001b[0m                           Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-12-add73c4f74e0>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m\u001b[0m\n\u001b[1;32m      1\u001b[0m \u001b[0mn\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mtensor\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m1.\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;36m2\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;36m3\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;36m4\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m5\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m6\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 2\u001b[0;31m \u001b[0mm\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mn\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;31mRuntimeError\u001b[0m: The size of tensor a (3) must match the size of tensor b (2) at non-singleton dimension 0"
     ]
    }
   ],
   "source": [
    "n = tensor([[1., 2, 3], [4,5,6]])\n",
    "m*n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def matmul(a,b):\n",
    "    ar,ac = a.shape\n",
    "    br,bc = b.shape\n",
    "    assert ac==br\n",
    "    c = torch.zeros(ar, bc)\n",
    "    for i in range(ar):\n",
    "        for j in range(bc): c[i,j] = (a[i] * b[:,j]).sum()\n",
    "    return c"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.7 ms ± 88.1 µs per loop (mean ± std. dev. of 7 runs, 20 loops each)\n"
     ]
    }
   ],
   "source": [
    "%timeit -n 20 t3 = matmul(m1,m2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Broadcasting"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Broadcasting with a scalar"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([ True,  True, False])"
      ]
     },
     "execution_count": null,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = tensor([10., 6, -4])\n",
    "a > 0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([[-1.4652, -1.0989, -0.7326],\n",
       "        [-0.3663,  0.0000,  0.3663],\n",
       "        [ 0.7326,  1.0989,  1.4652]])"
      ]
     },
     "execution_count": null,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m = tensor([[1., 2, 3], [4,5,6], [7,8,9]])\n",
    "(m - 5) / 2.73"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Broadcasting a vector to a matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(torch.Size([3, 3]), torch.Size([3]))"
      ]
     },
     "execution_count": null,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c = tensor([10.,20,30])\n",
    "m = tensor([[1., 2, 3], [4,5,6], [7,8,9]])\n",
    "m.shape,c.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([[11., 22., 33.],\n",
       "        [14., 25., 36.],\n",
       "        [17., 28., 39.]])"
      ]
     },
     "execution_count": null,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m + c"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([[10., 20., 30.],\n",
       "        [10., 20., 30.],\n",
       "        [10., 20., 30.]])"
      ]
     },
     "execution_count": null,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c.expand_as(m)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       " 10.0\n",
       " 20.0\n",
       " 30.0\n",
       "[torch.FloatStorage of size 3]"
      ]
     },
     "execution_count": null,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t = c.expand_as(m)\n",
    "t.storage()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "((0, 1), torch.Size([3, 3]))"
      ]
     },
     "execution_count": null,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t.stride(), t.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([[11., 22., 33.],\n",
       "        [14., 25., 36.],\n",
       "        [17., 28., 39.]])"
      ]
     },
     "execution_count": null,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c + m"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([[11., 22., 33.],\n",
       "        [14., 25., 36.]])"
      ]
     },
     "execution_count": null,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c = tensor([10.,20,30])\n",
    "m = tensor([[1., 2, 3], [4,5,6]])\n",
    "c+m"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "ename": "RuntimeError",
     "evalue": "The size of tensor a (2) must match the size of tensor b (3) at non-singleton dimension 1",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mRuntimeError\u001b[0m                           Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-25-64bbbad4d99c>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m\u001b[0m\n\u001b[1;32m      1\u001b[0m \u001b[0mc\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mtensor\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m10.\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m20\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m      2\u001b[0m \u001b[0mm\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mtensor\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m1.\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;36m2\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;36m3\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;36m4\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m5\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m6\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 3\u001b[0;31m \u001b[0mc\u001b[0m\u001b[0;34m+\u001b[0m\u001b[0mm\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;31mRuntimeError\u001b[0m: The size of tensor a (2) must match the size of tensor b (3) at non-singleton dimension 1"
     ]
    }
   ],
   "source": [
    "c = tensor([10.,20])\n",
    "m = tensor([[1., 2, 3], [4,5,6]])\n",
    "c+m"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(torch.Size([3, 3]), torch.Size([3, 1]))"
      ]
     },
     "execution_count": null,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c = tensor([10.,20,30])\n",
    "m = tensor([[1., 2, 3], [4,5,6], [7,8,9]])\n",
    "c = c.unsqueeze(1)\n",
    "m.shape,c.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([[11., 12., 13.],\n",
       "        [24., 25., 26.],\n",
       "        [37., 38., 39.]])"
      ]
     },
     "execution_count": null,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c+m"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       " 10.0\n",
       " 20.0\n",
       " 30.0\n",
       "[torch.FloatStorage of size 3]"
      ]
     },
     "execution_count": null,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t = c.expand_as(m)\n",
    "t.storage()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "((1, 0), torch.Size([3, 3]))"
      ]
     },
     "execution_count": null,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t.stride(), t.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(torch.Size([3]), torch.Size([1, 3]), torch.Size([3, 1]))"
      ]
     },
     "execution_count": null,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c = tensor([10.,20,30])\n",
    "c.shape, c.unsqueeze(0).shape,c.unsqueeze(1).shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(torch.Size([3]), torch.Size([1, 3]), torch.Size([3, 1]))"
      ]
     },
     "execution_count": null,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c.shape, c[None,:].shape,c[:,None].shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(torch.Size([1, 3]), torch.Size([3, 1]))"
      ]
     },
     "execution_count": null,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c[None].shape,c[...,None].shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def matmul(a,b):\n",
    "    ar,ac = a.shape\n",
    "    br,bc = b.shape\n",
    "    assert ac==br\n",
    "    c = torch.zeros(ar, bc)\n",
    "    for i in range(ar):\n",
    "#       c[i,j] = (a[i,:]          * b[:,j]).sum() # previous\n",
    "        c[i]   = (a[i  ].unsqueeze(-1) * b).sum(dim=0)\n",
    "    return c"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "357 µs ± 7.2 µs per loop (mean ± std. dev. of 7 runs, 20 loops each)\n"
     ]
    }
   ],
   "source": [
    "%timeit -n 20 t4 = matmul(m1,m2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Broadcasting Rules"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Einstein summation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def matmul(a,b): return torch.einsum('ik,kj->ij', a, b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "68.7 µs ± 4.06 µs per loop (mean ± std. dev. of 7 runs, 20 loops each)\n"
     ]
    }
   ],
   "source": [
    "%timeit -n 20 t5 = matmul(m1,m2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The forward and backward passes"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Defining and initializing a layer"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def lin(x, w, b): return x @ w + b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = torch.randn(200, 100)\n",
    "y = torch.randn(200)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "w1 = torch.randn(100,50)\n",
    "b1 = torch.zeros(50)\n",
    "w2 = torch.randn(50,1)\n",
    "b2 = torch.zeros(1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "torch.Size([200, 50])"
      ]
     },
     "execution_count": null,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "l1 = lin(x, w1, b1)\n",
    "l1.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(tensor(0.0019), tensor(10.1058))"
      ]
     },
     "execution_count": null,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "l1.mean(), l1.std()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([[nan, nan, nan, nan, nan],\n",
       "        [nan, nan, nan, nan, nan],\n",
       "        [nan, nan, nan, nan, nan],\n",
       "        [nan, nan, nan, nan, nan],\n",
       "        [nan, nan, nan, nan, nan]])"
      ]
     },
     "execution_count": null,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = torch.randn(200, 100)\n",
    "for i in range(50): x = x @ torch.randn(100,100)\n",
    "x[0:5,0:5]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([[0., 0., 0., 0., 0.],\n",
       "        [0., 0., 0., 0., 0.],\n",
       "        [0., 0., 0., 0., 0.],\n",
       "        [0., 0., 0., 0., 0.],\n",
       "        [0., 0., 0., 0., 0.]])"
      ]
     },
     "execution_count": null,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = torch.randn(200, 100)\n",
    "for i in range(50): x = x @ (torch.randn(100,100) * 0.01)\n",
    "x[0:5,0:5]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([[ 0.7554,  0.6167, -0.1757, -1.5662,  0.5644],\n",
       "        [-0.1987,  0.6292,  0.3283, -1.1538,  0.5416],\n",
       "        [ 0.6106,  0.2556, -0.0618, -0.9463,  0.4445],\n",
       "        [ 0.4484,  0.7144,  0.1164, -0.8626,  0.4413],\n",
       "        [ 0.3463,  0.5930,  0.3375, -0.9486,  0.5643]])"
      ]
     },
     "execution_count": null,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = torch.randn(200, 100)\n",
    "for i in range(50): x = x @ (torch.randn(100,100) * 0.1)\n",
    "x[0:5,0:5]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor(0.7042)"
      ]
     },
     "execution_count": null,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x.std()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = torch.randn(200, 100)\n",
    "y = torch.randn(200)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from math import sqrt\n",
    "w1 = torch.randn(100,50) / sqrt(100)\n",
    "b1 = torch.zeros(50)\n",
    "w2 = torch.randn(50,1) / sqrt(50)\n",
    "b2 = torch.zeros(1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(tensor(-0.0050), tensor(1.0000))"
      ]
     },
     "execution_count": null,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "l1 = lin(x, w1, b1)\n",
    "l1.mean(),l1.std()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def relu(x): return x.clamp_min(0.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(tensor(0.3961), tensor(0.5783))"
      ]
     },
     "execution_count": null,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "l2 = relu(l1)\n",
    "l2.mean(),l2.std()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([[0.0000e+00, 1.9689e-08, 4.2820e-08, 0.0000e+00, 0.0000e+00],\n",
       "        [0.0000e+00, 1.6701e-08, 4.3501e-08, 0.0000e+00, 0.0000e+00],\n",
       "        [0.0000e+00, 1.0976e-08, 3.0411e-08, 0.0000e+00, 0.0000e+00],\n",
       "        [0.0000e+00, 1.8457e-08, 4.9469e-08, 0.0000e+00, 0.0000e+00],\n",
       "        [0.0000e+00, 1.9949e-08, 4.1643e-08, 0.0000e+00, 0.0000e+00]])"
      ]
     },
     "execution_count": null,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = torch.randn(200, 100)\n",
    "for i in range(50): x = relu(x @ (torch.randn(100,100) * 0.1))\n",
    "x[0:5,0:5]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([[0.2871, 0.0000, 0.0000, 0.0000, 0.0026],\n",
       "        [0.4546, 0.0000, 0.0000, 0.0000, 0.0015],\n",
       "        [0.6178, 0.0000, 0.0000, 0.0180, 0.0079],\n",
       "        [0.3333, 0.0000, 0.0000, 0.0545, 0.0000],\n",
       "        [0.1940, 0.0000, 0.0000, 0.0000, 0.0096]])"
      ]
     },
     "execution_count": null,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = torch.randn(200, 100)\n",
    "for i in range(50): x = relu(x @ (torch.randn(100,100) * sqrt(2/100)))\n",
    "x[0:5,0:5]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = torch.randn(200, 100)\n",
    "y = torch.randn(200)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "w1 = torch.randn(100,50) * sqrt(2 / 100)\n",
    "b1 = torch.zeros(50)\n",
    "w2 = torch.randn(50,1) * sqrt(2 / 50)\n",
    "b2 = torch.zeros(1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(tensor(0.5661), tensor(0.8339))"
      ]
     },
     "execution_count": null,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "l1 = lin(x, w1, b1)\n",
    "l2 = relu(l1)\n",
    "l2.mean(), l2.std()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def model(x):\n",
    "    l1 = lin(x, w1, b1)\n",
    "    l2 = relu(l1)\n",
    "    l3 = lin(l2, w2, b2)\n",
    "    return l3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "torch.Size([200, 1])"
      ]
     },
     "execution_count": null,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "out = model(x)\n",
    "out.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def mse(output, targ): return (output.squeeze(-1) - targ).pow(2).mean()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "loss = mse(out, y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Gradients and backward pass"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def mse_grad(inp, targ): \n",
    "    # grad of loss with respect to output of previous layer\n",
    "    inp.g = 2. * (inp.squeeze() - targ).unsqueeze(-1) / inp.shape[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def relu_grad(inp, out):\n",
    "    # grad of relu with respect to input activations\n",
    "    inp.g = (inp>0).float() * out.g"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def lin_grad(inp, out, w, b):\n",
    "    # grad of matmul with respect to input\n",
    "    inp.g = out.g @ w.t()\n",
    "    w.g = inp.t() @ out.g\n",
    "    b.g = out.g.sum(0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Sidebar: SymPy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 2 sx$"
      ],
      "text/plain": [
       "2*sx"
      ]
     },
     "execution_count": null,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy import symbols,diff\n",
    "sx,sy = symbols('sx sy')\n",
    "diff(sx**2, sx)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### End sidebar"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def forward_and_backward(inp, targ):\n",
    "    # forward pass:\n",
    "    l1 = inp @ w1 + b1\n",
    "    l2 = relu(l1)\n",
    "    out = l2 @ w2 + b2\n",
    "    # we don't actually need the loss in backward!\n",
    "    loss = mse(out, targ)\n",
    "    \n",
    "    # backward pass:\n",
    "    mse_grad(out, targ)\n",
    "    lin_grad(l2, out, w2, b2)\n",
    "    relu_grad(l1, l2)\n",
    "    lin_grad(inp, l1, w1, b1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Refactor the model"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "class Relu():\n",
    "    def __call__(self, inp):\n",
    "        self.inp = inp\n",
    "        self.out = inp.clamp_min(0.)\n",
    "        return self.out\n",
    "    \n",
    "    def backward(self): self.inp.g = (self.inp>0).float() * self.out.g"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "class Lin():\n",
    "    def __init__(self, w, b): self.w,self.b = w,b\n",
    "        \n",
    "    def __call__(self, inp):\n",
    "        self.inp = inp\n",
    "        self.out = inp@self.w + self.b\n",
    "        return self.out\n",
    "    \n",
    "    def backward(self):\n",
    "        self.inp.g = self.out.g @ self.w.t()\n",
    "        self.w.g = self.inp.t() @ self.out.g\n",
    "        self.b.g = self.out.g.sum(0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "class Mse():\n",
    "    def __call__(self, inp, targ):\n",
    "        self.inp = inp\n",
    "        self.targ = targ\n",
    "        self.out = (inp.squeeze() - targ).pow(2).mean()\n",
    "        return self.out\n",
    "    \n",
    "    def backward(self):\n",
    "        x = (self.inp.squeeze()-self.targ).unsqueeze(-1)\n",
    "        self.inp.g = 2.*x/self.targ.shape[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "class Model():\n",
    "    def __init__(self, w1, b1, w2, b2):\n",
    "        self.layers = [Lin(w1,b1), Relu(), Lin(w2,b2)]\n",
    "        self.loss = Mse()\n",
    "        \n",
    "    def __call__(self, x, targ):\n",
    "        for l in self.layers: x = l(x)\n",
    "        return self.loss(x, targ)\n",
    "    \n",
    "    def backward(self):\n",
    "        self.loss.backward()\n",
    "        for l in reversed(self.layers): l.backward()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "model = Model(w1, b1, w2, b2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "loss = model(x, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "model.backward()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Going to PyTorch"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "class LayerFunction():\n",
    "    def __call__(self, *args):\n",
    "        self.args = args\n",
    "        self.out = self.forward(*args)\n",
    "        return self.out\n",
    "    \n",
    "    def forward(self):  raise Exception('not implemented')\n",
    "    def bwd(self):      raise Exception('not implemented')\n",
    "    def backward(self): self.bwd(self.out, *self.args)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "class Relu(LayerFunction):\n",
    "    def forward(self, inp): return inp.clamp_min(0.)\n",
    "    def bwd(self, out, inp): inp.g = (inp>0).float() * out.g"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "class Lin(LayerFunction):\n",
    "    def __init__(self, w, b): self.w,self.b = w,b\n",
    "        \n",
    "    def forward(self, inp): return inp@self.w + self.b\n",
    "    \n",
    "    def bwd(self, out, inp):\n",
    "        inp.g = out.g @ self.w.t()\n",
    "        self.w.g = self.inp.t() @ self.out.g\n",
    "        self.b.g = out.g.sum(0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "class Mse(LayerFunction):\n",
    "    def forward (self, inp, targ): return (inp.squeeze() - targ).pow(2).mean()\n",
    "    def bwd(self, out, inp, targ): \n",
    "        inp.g = 2*(inp.squeeze()-targ).unsqueeze(-1) / targ.shape[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from torch.autograd import Function\n",
    "\n",
    "class MyRelu(Function):\n",
    "    @staticmethod\n",
    "    def forward(ctx, i):\n",
    "        result = i.clamp_min(0.)\n",
    "        ctx.save_for_backward(i)\n",
    "        return result\n",
    "    \n",
    "    @staticmethod\n",
    "    def backward(ctx, grad_output):\n",
    "        i, = ctx.saved_tensors\n",
    "        return grad_output * (i>0).float()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import torch.nn as nn\n",
    "\n",
    "class LinearLayer(nn.Module):\n",
    "    def __init__(self, n_in, n_out):\n",
    "        super().__init__()\n",
    "        self.weight = nn.Parameter(torch.randn(n_out, n_in) * sqrt(2/n_in))\n",
    "        self.bias = nn.Parameter(torch.zeros(n_out))\n",
    "    \n",
    "    def forward(self, x): return x @ self.weight.t() + self.bias"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(torch.Size([2, 10]), torch.Size([2]))"
      ]
     },
     "execution_count": null,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lin = LinearLayer(10,2)\n",
    "p1,p2 = lin.parameters()\n",
    "p1.shape,p2.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "class Model(nn.Module):\n",
    "    def __init__(self, n_in, nh, n_out):\n",
    "        super().__init__()\n",
    "        self.layers = nn.Sequential(\n",
    "            nn.Linear(n_in,nh), nn.ReLU(), nn.Linear(nh,n_out))\n",
    "        self.loss = mse\n",
    "        \n",
    "    def forward(self, x, targ): return self.loss(self.layers(x).squeeze(), targ)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "class Model(Module):\n",
    "    def __init__(self, n_in, nh, n_out):\n",
    "        self.layers = nn.Sequential(\n",
    "            nn.Linear(n_in,nh), nn.ReLU(), nn.Linear(nh,n_out))\n",
    "        self.loss = mse\n",
    "        \n",
    "    def forward(self, x, targ): return self.loss(self.layers(x).squeeze(), targ)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Things to remember"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Questionnaire"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Further research"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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