{ "nbformat": 4, "nbformat_minor": 5, "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "name": "python" }, "lecture_migration": { "project_id": "c0b185e7-d7e3-48b6-a605-8f1acdfe243c", "notebook_id": "0b7ff2d0e61347cd917939e62deb701c", "destination": "L2", "cloud_run": false }, "title": "LAB 01 · 学会一个卷积核" }, "cells": [ { "cell_type": "markdown", "metadata": { "deepnote": { "block_id": "cf05560d5dad47c792e26261a9010ad3", "type": "text-cell-h1", "source_block_id": null, "logical_key": null }, "html_anchor": "cell-1" }, "source": "LAB 01 · 学会一个卷积核", "id": "cf05560d5dad47c792e26261a9010ad3" }, { "cell_type": "markdown", "metadata": { "deepnote": { "block_id": "2d1acab2ddb54f0f935594a4bbe401a1", "type": "markdown", "source_block_id": null, "logical_key": null }, "html_anchor": "cell-2" }, "source": [ "[返回第二讲 · 局部乘加与共享反馈](https://codingai-lec02.pages.dev/#/convolution-four-products/0) · [第二讲导读与学习材料](https://codingai-lec02.pages.dev/course/guide.html) · [问卷作业 ↗](https://tally.so/r/VLroVv)（右键在新标签页打开）。\n", "\n", "**一个核在多个位置使用，更新时听谁的意见？** 这本材料把局部乘加、共享梯度和框架实现接起来；不需要先训练图像分类器。\n", "\n", "| 想解决的问题 | 去哪里、怎样做 | 先看哪个输出 |\n", "| --- | --- | --- |\n", "| 四个输出为什么共用一组权重 | 第1节 → 第3节；先运行“准备与使用”的导入代码，再运行各节代码 | `response` 与各位置梯度贡献的和 |\n", "| 输入通道求和、输出通道堆叠有什么区别 | 文末“从循环到 Parameter”；这个代码块自带导入，可单独运行 | `Input / Weight / Output`、注册参数和三项误差 |\n", "| 拟合对响应是否等于找到唯一核 | 文末“输出都拟合对了……”；独立运行 | 常数输入与新探针的响应差、设计矩阵的秩 |\n", "\n", "**课堂先选中间一项。** 写出默认输出形状，再只把 `C_IN=2` 改为 `3`，其余三个量保持 `C_OUT=3, KH=2, KW=2`。重新运行同一个独立代码块：输出通道数和空间形状应保持，核参数与每个输出的求和项会变化。用输出和参数表检查预测。完成后再决定是否读第3节的完整梯度推导。\n", "\n", "本册参考图是已保存的本地结果；你在副本中运行的输出单独记录，不覆盖或混称为课件原结果。" ], "id": "2d1acab2ddb54f0f935594a4bbe401a1" }, { "cell_type": "markdown", "metadata": { "deepnote": { "block_id": "9bcfa38df923443ea1949e4521099d7c", "type": "markdown", "source_block_id": null, "logical_key": null }, "html_anchor": "cell-3" }, "source": "**主路径：1 → 2 → 3 → 6。** 把前向局部乘加、参数共享和反向梯度连成一条链；4、5、7 为深入阅读。\n\n我们要回答：卷积相比普通全连接层究竟排除了哪些函数？同一个 kernel 在许多位置被调用，更新时听谁的意见？框架源码为什么只剩一次函数调用？\n\n给二维隐藏层保留空间索引，线性映射可写作\n\\[\nY_{i,j}=b_{i,j}+\\sum_{a,b}V_{i,j,a,b}X_{i+a,j+b}.\n\\]\n重索引还没有减少参数。要求各位置采用同一规则，令 $V_{i,j,a,b}=K_{a,b}$；要求局部连接，再将窗口外权重设为零。这是对函数类的约束：如果绝对位置携带任务信息，此先验可能有代价。\n\n来源：[D2L 7.1](https://d2l.ai/chapter_convolutional-neural-networks/why-conv.html)、[7.2](https://d2l.ai/chapter_convolutional-neural-networks/conv-layer.html)、[7.4](https://d2l.ai/chapter_convolutional-neural-networks/channels.html)。", "id": "9bcfa38df923443ea1949e4521099d7c" }, { "cell_type": "markdown", "metadata": { "deepnote": { "block_id": "4b79aa8e3c714604823b96cc69be853a", "type": "text-cell-h2", "source_block_id": null, "logical_key": null }, "html_anchor": "cell-4" }, "source": "## 目标", "id": "4b79aa8e3c714604823b96cc69be853a" }, { "cell_type": "markdown", "metadata": { "deepnote": { "block_id": "d11c4ca51f474281aae65dad9478668d", "type": "markdown", "source_block_id": null, "logical_key": null }, "html_anchor": "cell-5" }, "source": "从手算输出进入多通道，再把共享梯度写成公式和代码；改变stride与边界，检验等变条件。", "id": "d11c4ca51f474281aae65dad9478668d" }, { "cell_type": "markdown", "metadata": { "deepnote": { "block_id": "8dcc3d14da674744b7db1831a80ed208", "type": "text-cell-h2", "source_block_id": null, "logical_key": null }, "html_anchor": "cell-6" }, "source": "## 准备与使用", "id": "8dcc3d14da674744b7db1831a80ed208" }, { "cell_type": "markdown", "metadata": { "deepnote": { "block_id": "310b91618cef467e84651df157799b35", "type": "markdown", "source_block_id": null, "logical_key": null }, "html_anchor": "cell-7" }, "source": [ "本册可独立从头运行；跨册文件依赖在相应准备节单独说明。需要 Python 3.10+ 与 `numpy matplotlib torch ipython`；保存/检查 notebook 另需 `nbformat matplotlib-inline`。安装命令：\n", "\n", "```bash\n", "python -m pip install numpy matplotlib torch ipython nbformat matplotlib-inline\n", "```\n", "\n", "保存图与文末记录来自此前 Python 3.12 / PyTorch 2.14.0+cpu 的本地执行。你自己的运行环境版本由下一代码块打印；本次整理未重跑这些训练。图轴使用英文，正文提供中文读图说明。缺依赖时再安装，已有环境不必重复安装；课堂按开头任务入口运行，不必 Run all。\n", "\n", "本材料改编自 Aston Zhang、Zachary C. Lipton、Mu Li、Alexander J. Smola 的 [Dive into Deep Learning](https://d2l.ai/)。D2L 正文 CC BY-SA 4.0，示例代码 modified MIT，许可随完整包 `licenses/` 保存。我们补充多通道梯度、采样反例、受控实验、真实 PyTorch 状态语义；不要把教学小实现视为框架底层实现。\n", "\n", "[下载完整已执行 notebooks、MNIST 与许可](https://codingai-lec02.pages.dev/downloads/Lecture02-CNN-Labs.zip)。" ], "id": "310b91618cef467e84651df157799b35" }, { "cell_type": "code", "metadata": { "deepnote": { "block_id": "985103a972ba47dd90c55d758a1df719", "type": "code", "source_block_id": null, "logical_key": null }, "html_anchor": "cell-8" }, "source": "from pathlib import Path\nimport copy, hashlib, inspect, json, math, platform, time\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport torch\nfrom torch import nn\nfrom torch.nn import functional as F\nfrom IPython.display import display, Markdown\n\ntorch.manual_seed(20260921)\ntorch.set_num_threads(2)\ntorch.use_deterministic_algorithms(True)\nOUT=Path('results'); OUT.mkdir(exist_ok=True)\nFIG=Path('figures'); FIG.mkdir(exist_ok=True)\nBLUE,GOLD,TEAL,RED='#002676','#FDB515','#00897B','#B83A3A'\nplt.rcParams.update({'font.size':11,'axes.spines.top':False,'axes.spines.right':False,\n 'figure.dpi':110,'savefig.facecolor':'white'})\ndef savefig(fig,name):\n fig.savefig(FIG/f'{name}.png',dpi=160,bbox_inches='tight')\n fig.savefig(FIG/f'{name}.svg',bbox_inches='tight')\ndef table(headers,rows):\n display(Markdown('| '+' | '.join(headers)+' |\\n| '+ ' | '.join(['---']*len(headers))+' |\\n'+\n '\\n'.join('| '+' | '.join(map(str,row))+' |' for row in rows)))\nprint('PyTorch',torch.__version__,'/ Python',platform.python_version(),'/ device CPU')", "id": "985103a972ba47dd90c55d758a1df719", "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "metadata": { "deepnote": { "block_id": "760796b1ad6746249188b24284ce53f0", "type": "text-cell-h2", "source_block_id": null, "logical_key": null }, "html_anchor": "cell-9" }, "source": "## 1. 一个输出：先算 19，再追踪共享", "id": "760796b1ad6746249188b24284ce53f0" }, { "cell_type": "markdown", "metadata": { "deepnote": { "block_id": "76dda77d304b47a7bddaaa76e1eb666e", "type": "markdown", "source_block_id": null, "logical_key": null }, "html_anchor": "cell-10" }, "source": "深度学习的 `Conv2d` 默认实现互相关，不翻转 kernel。以下四个输出用了同一组参数；这不是四个独立的神经元权重副本。\n\n左上角是 $0\\times0+1\\times1+3\\times2+4\\times3=19$。核翻转会改变固定权重的输出；对可学习模型，卷积与互相关可通过权重重参数化对应，不能把固定核的差异也说成不存在。", "id": "76dda77d304b47a7bddaaa76e1eb666e" }, { "cell_type": "code", "metadata": { "deepnote": { "block_id": "4ed7ceee625c48e4b679ce70fb794a20", "type": "code", "source_block_id": null, "logical_key": null }, "html_anchor": "cell-11" }, "source": "from numpy.lib.stride_tricks import sliding_window_view\nimage=np.arange(9,dtype=np.float64).reshape(3,3)\nkernel=np.array([[0.,1.],[2.,3.]])\npatches=sliding_window_view(image,kernel.shape)\nresponse=np.einsum('ijab,ab->ij',patches,kernel)\nassert np.array_equal(response,[[19.,25.],[37.,43.]])\nfig,axes=plt.subplots(1,3,figsize=(9,2.7))\nfor ax,a,title in zip(axes,[image,kernel,response],['Input','One shared kernel','Response']):\n ax.imshow(a,cmap='Blues')\n for (i,j),v in np.ndenumerate(a):\n ax.text(j,i,f'{v:g}',ha='center',va='center',color='white' if v>a.max()*.6 else BLUE)\n ax.set(title=title,xticks=[],yticks=[])\nsavefig(fig,'lab01-shared-correlation');plt.show()\nprint(response)", "id": "4ed7ceee625c48e4b679ce70fb794a20", "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "metadata": { "deepnote": { "block_id": "a5d9bc3e48034df49642f1a81501b491", "type": "markdown", "source_block_id": null, "logical_key": null }, "html_anchor": "cell-12" }, "source": [ "**保存结果｜四个窗口，同一个核**\n", "\n", "![输入、共享卷积核与四个输出](https://codingai-lec02.pages.dev/assets/lab01-shared-correlation.png)\n", "\n", "先用左上窗口算出19，再在图上找到产生25的窗口：变的是输入片段，核没有变。要检验这件事，改一个输入元素前先圈出所有覆盖它的窗口，预测哪些输出会改变。" ], "id": "a5d9bc3e48034df49642f1a81501b491" }, { "cell_type": "markdown", "metadata": { "deepnote": { "block_id": "f194a189bed74b53aadbbcfa2170b933", "type": "text-cell-h2", "source_block_id": null, "logical_key": null }, "html_anchor": "cell-13" }, "source": "## 2. 多通道：对输入通道求和，对输出通道堆叠", "id": "f194a189bed74b53aadbbcfa2170b933" }, { "cell_type": "markdown", "metadata": { "deepnote": { "block_id": "5d4c649ec16a458692a2c03c1ffdbf37", "type": "markdown", "source_block_id": null, "logical_key": null }, "html_anchor": "cell-14" }, "source": "\\[\nY_{n,o,i,j}=b_o+\\sum_{c,a,b}K_{o,c,a,b}X_{n,c,i+a,j+b}.\n\\]\n\n每个输出通道有一组覆盖所有输入通道的核。输入 RGB 三通道不意味着输出也必须为三通道。以下 NumPy 实现保留所有轴，与 PyTorch `F.conv2d` 对照。\n\n`Conv2d(3,16,3)` 的权重 shape 是 `(16,3,3,3)`；加 16 个 bias 共 448 参数。改变输入空间分辨率不会改变这 448 个参数，却会改变计算和中间激活内存。", "id": "5d4c649ec16a458692a2c03c1ffdbf37" }, { "cell_type": "code", "metadata": { "deepnote": { "block_id": "68287d25f0df4264b788f7aaf7dfb682", "type": "code", "source_block_id": null, "logical_key": null }, "html_anchor": "cell-15" }, "source": "rng=np.random.default_rng(20260921)\nX=rng.normal(size=(2,3,6,7));K=rng.normal(size=(4,3,3,2));b=rng.normal(size=4)\nP=sliding_window_view(X,(3,2),axis=(-2,-1))\nY_numpy=np.einsum('ncijab,ocab->noij',P,K)+b[None,:,None,None]\nY_torch=F.conv2d(torch.from_numpy(X),torch.from_numpy(K),torch.from_numpy(b))\nnp.testing.assert_allclose(Y_numpy,Y_torch.numpy(),rtol=1e-12,atol=1e-12)\nconv=nn.Conv2d(3,16,3,padding=1)\nassert sum(p.numel() for p in conv.parameters())==448\nprint('NumPy vs PyTorch max error:',np.abs(Y_numpy-Y_torch.numpy()).max())\nprint('weight:',tuple(conv.weight.shape),'bias:',tuple(conv.bias.shape),'total parameters:',448)\nprint('32x32 output:',tuple(conv(torch.zeros(1,3,32,32)).shape))\nprint('64x64 output:',tuple(conv(torch.zeros(1,3,64,64)).shape))", "id": "68287d25f0df4264b788f7aaf7dfb682", "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "metadata": { "deepnote": { "block_id": "456af611195b4380870b4a32245ddc14", "type": "text-cell-h2", "source_block_id": null, "logical_key": null }, "html_anchor": "cell-16" }, "source": "## 3. 共享权重的梯度：每个位置贡献，再加到一起", "id": "456af611195b4380870b4a32245ddc14" }, { "cell_type": "markdown", "metadata": { "deepnote": { "block_id": "8252cf5436a84368870bdd66bea6a9fd", "type": "markdown", "source_block_id": null, "logical_key": null }, "html_anchor": "cell-17" }, "source": [ "设 $\\delta=\\partial L/\\partial Y$，对 valid、stride 1 的卷积有\n", "\\[\n", "\\frac{\\partial L}{\\partial K_{o,c,a,b}}\n", "=\\sum_{n,i,j}\\delta_{n,o,i,j}X_{n,c,i+a,j+b}.\n", "\\]\n", "\n", "我们用 $L=\\sum Y\\delta$ 固定上游梯度，隔离本层链式法则。先构造每个位置对同一个权重的贡献热图，再验证完整多通道梯度。loss 采用 mean 或 sum 时，上游 $\\delta$ 的尺度也会变；这不是梯度公式矛盾。\n", "\n", "共享使不同位置对同一探测规则积累证据，但相邻 patches 高度相关，不能直接称为 $H\\times W$ 个独立训练样本。\n", "\n", "![同一个核元素接收多个窗口的梯度贡献，位置贡献求和后才更新一次](https://codingai-lec02.pages.dev/course/visuals/deepnote-final/lab01-shared-gradient.png)\n", "\n", "[打开矢量图](https://codingai-lec02.pages.dev/course/visuals/deepnote-final/lab01-shared-gradient.svg)\n", "\n", "**先沿一条支路读。** 固定同一个 `K[o,c,a,b]`，每个窗口贡献的是 `δ[n,o,i,j] × X[n,c,i+a,j+b]`，最后对样本和位置求和。这里没有对固定的输出通道 `o` 再求和。\n", "\n", "**预测再核对：** 若把代码中的 `delta` 整体乘2，核梯度应怎样变？保留输入与权重、只改这一项并重跑第3节；图上的每个贡献与总和应同比变化。第4节依赖这里的 `x,w,y,delta`，不能直接跳跑。" ], "id": "8252cf5436a84368870bdd66bea6a9fd" }, { "cell_type": "code", "metadata": { "deepnote": { "block_id": "d3bcfb189160472aaa642b8c714129d6", "type": "code", "source_block_id": null, "logical_key": null }, "html_anchor": "cell-18" }, "source": "torch.manual_seed(20260921)\nx=torch.randn(2,2,5,6,dtype=torch.float64)\nw=torch.randn(3,2,3,2,dtype=torch.float64,requires_grad=True)\ny=F.conv2d(x,w);delta=torch.randn_like(y)\n(y*delta).sum().backward()\nmanual=torch.zeros_like(w)\nfor n in range(x.shape[0]):\n for o in range(w.shape[0]):\n for c in range(w.shape[1]):\n for a in range(w.shape[2]):\n for b_ in range(w.shape[3]):\n manual[o,c,a,b_] += (delta[n,o]*x[n,c,a:a+y.shape[2],b_:b_+y.shape[3]]).sum()\ntorch.testing.assert_close(manual,w.grad)\ncontributions=delta[:,0]*x[:,0,:y.shape[2],:y.shape[3]]\nfig,axes=plt.subplots(1,2,figsize=(9,3.4),layout='constrained')\nlimit=float(contributions.abs().max())\nfor n,ax in enumerate(axes):\n im=ax.imshow(contributions[n],cmap='RdBu_r',vmin=-limit,vmax=limit)\n ax.set(title=f'Sample {n}',xlabel='Output column',ylabel='Output row')\nfig.suptitle('Per-position contributions to the shared weight K[0,0,0,0]')\nfig.colorbar(im,ax=axes,shrink=.8);savefig(fig,'lab01-gradient-contributions');plt.show()\nprint('Sum over samples and positions:',float(contributions.sum()))\nprint('Autograd weight gradient:',float(w.grad[0,0,0,0]))\ngradient_error=float((manual-w.grad).abs().max())\nprint('Full gradient maximum difference:',gradient_error)", "id": "d3bcfb189160472aaa642b8c714129d6", "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "metadata": { "deepnote": { "block_id": "792cbd3b1953499eb3a7d6c8a79d3020", "type": "markdown", "source_block_id": null, "logical_key": null }, "html_anchor": "cell-19" }, "source": [ "**保存结果｜先看局部贡献，再看总和**\n", "\n", "![两个样本各位置对同一个共享权重的有符号梯度贡献](https://codingai-lec02.pages.dev/assets/lab01-gradient-contributions.png)\n", "\n", "两图使用同一对称色标；正负贡献可能抵消。先判断哪一块贡献的符号与相邻位置不同，再核对代码打印的 `Sum over samples and positions` 与 `Autograd weight gradient`。一张热图的峰值不是最终参数更新量。" ], "id": "792cbd3b1953499eb3a7d6c8a79d3020" }, { "cell_type": "markdown", "metadata": { "deepnote": { "block_id": "79714d915d504dc588a07bf02c4d9b93", "type": "text-cell-h2", "source_block_id": null, "logical_key": null }, "html_anchor": "cell-20" }, "source": "## 4. 深入：滑窗也可以变成矩阵乘法", "id": "79714d915d504dc588a07bf02c4d9b93" }, { "cell_type": "markdown", "metadata": { "deepnote": { "block_id": "373e467480cf4fa8adf097e5f29ae4c8", "type": "markdown", "source_block_id": null, "logical_key": null }, "html_anchor": "cell-21" }, "source": "`unfold` 把每个局部窗口摊成一个列向量。空间扫描转换为许多列、所有 filter 转换为矩阵的行，前向就是一次矩阵乘法。这是「卷积与线性代数」的桥，不是声明底层 backend 一定显式构造这个大矩阵。显式 unfold 会复制重叠窗口，可能增加内存。", "id": "373e467480cf4fa8adf097e5f29ae4c8" }, { "cell_type": "code", "metadata": { "deepnote": { "block_id": "19c460b2b8ca457fbcafca08a9ffc268", "type": "code", "source_block_id": null, "logical_key": null }, "html_anchor": "cell-22" }, "source": "columns=F.unfold(x,kernel_size=(3,2))\nY_matrix=(w.detach().reshape(3,-1)@columns).reshape_as(y)\ntorch.testing.assert_close(Y_matrix,y)\nmatrix_gradient=torch.einsum('nol,nkl->ok',delta.flatten(2),columns).reshape_as(w)\ntorch.testing.assert_close(matrix_gradient,w.grad)\nprint('Input:',tuple(x.shape),'unfold:',tuple(columns.shape),'kernel matrix:',tuple(w.reshape(3,-1).shape))\nprint('Input entries:',x.numel(),'unfold entries:',columns.numel())", "id": "19c460b2b8ca457fbcafca08a9ffc268", "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "metadata": { "deepnote": { "block_id": "315a2aa60c1a44929ec74ade47d35c53", "type": "text-cell-h2", "source_block_id": null, "logical_key": null }, "html_anchor": "cell-23" }, "source": "## 5. 深入：从例子学习核，更新的仍是同一个参数", "id": "315a2aa60c1a44929ec74ade47d35c53" }, { "cell_type": "markdown", "metadata": { "deepnote": { "block_id": "49605103e1b14310a31b5a229940879a", "type": "markdown", "source_block_id": null, "logical_key": null }, "html_anchor": "cell-24" }, "source": "给随机输入配上 `[1,-1]` 的目标响应，用均方误差学习两个参数。已知算子恢复让我们能知道正确答案；真实分类只提供最终类别，不规定某一层必须成为边缘核。\n\n更新放在 `torch.no_grad()` 中避免记录参数更新的计算图。它与 `model.eval()` 是不同操作；LAB 03 将用 BatchNorm 验证。", "id": "49605103e1b14310a31b5a229940879a" }, { "cell_type": "code", "metadata": { "deepnote": { "block_id": "37050c4a02d84964924f4612d3b95dd1", "type": "code", "source_block_id": null, "logical_key": null }, "html_anchor": "cell-25" }, "source": "torch.manual_seed(20260921)\ninputs=torch.randn(32,1,8,8)\ntargets=F.conv2d(inputs,torch.tensor([[[[1.,-1.]]]]))\nlearner=nn.Conv2d(1,1,(1,2),bias=False)\nlosses=[]\nfor step in range(60):\n learner.zero_grad(set_to_none=True)\n loss=F.mse_loss(learner(inputs),targets);loss.backward()\n with torch.no_grad(): learner.weight-=.15*learner.weight.grad\n losses.append(float(loss.detach()))\nprint('Learned kernel:',learner.weight.detach().flatten().tolist())\nassert torch.max(torch.abs(learner.weight.detach().flatten()-torch.tensor([1.,-1.])))<1e-5\nfig,ax=plt.subplots(figsize=(6,3));ax.semilogy(losses,color=BLUE)\nax.set(xlabel='Update',ylabel='MSE',title='Recover a known operator');savefig(fig,'lab01-learned-kernel');plt.show()", "id": "37050c4a02d84964924f4612d3b95dd1", "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "metadata": { "deepnote": { "block_id": "41b6bf8bdc264124ba64dec0c096bac7", "type": "markdown", "source_block_id": null, "logical_key": null }, "html_anchor": "cell-26" }, "source": [ "**保存结果｜学回一个已知算子**\n", "\n", "![恢复目标卷积核时的均方误差轨迹](https://codingai-lec02.pages.dev/assets/lab01-learned-kernel.png)\n", "\n", "纵轴是对数 MSE，曲线记录60次参数更新。先看误差的量级变化，再核对最终两个核值是否接近 `[1,-1]`。目标响应是我们事先用已知核构造的；这个实验不表示真实分类网络必须学成某个指定边缘核。课堂无需为读此图重跑这段训练。" ], "id": "41b6bf8bdc264124ba64dec0c096bac7" }, { "cell_type": "markdown", "metadata": { "deepnote": { "block_id": "bc44ebdfed46488c898bfd0887479f06", "type": "text-cell-h2", "source_block_id": null, "logical_key": null }, "html_anchor": "cell-27" }, "source": "## 6. 平移、边界与 stride：检验先验的准确边界", "id": "bc44ebdfed46488c898bfd0887479f06" }, { "cell_type": "markdown", "metadata": { "deepnote": { "block_id": "38fa34d9c9a048af8dd56c3d4dbe977f", "type": "markdown", "source_block_id": null, "logical_key": null }, "html_anchor": "cell-28" }, "source": "尺寸公式中的 $p$ 是每侧 padding：\n\\[\nH_{out}=\\left\\lfloor\\frac{H+2p-d(k-1)-1}{s}\\right\\rfloor+1.\n\\]\n\n不能整除也可以运行，框架向下取整。用 circular padding 把边界也变成周期结构，stride 1 应满足 `f(roll(x,1)) = roll(f(x),1)`。stride 2 后，输入移动 2 像素对应输出移动 1 格；输入移动 1 像素改变采样相位，没有整数输出平移与之对应。\n\n下面零填充反例使用非对称核，让边界差异可被观察。平移等变是表示跟着移动，分类不变是输出保持相同，不能互换。", "id": "38fa34d9c9a048af8dd56c3d4dbe977f" }, { "cell_type": "code", "metadata": { "deepnote": { "block_id": "1ce9baae05944175953dec2c72df9c14", "type": "code", "source_block_id": null, "logical_key": null }, "html_anchor": "cell-29" }, "source": "H,p,d,k,s=8,1,1,3,3\nexpected=(H+2*p-d*(k-1)-1)//s+1\nactual=F.conv2d(torch.randn(1,1,H,H),torch.ones(1,1,k,k),stride=s,padding=p).shape[-1]\nassert expected==actual==3\ntorch.manual_seed(20260921)\nz=torch.randn(1,1,8,8);weight=torch.randn(1,1,3,3)\ndef circular(z,s=1): return F.conv2d(F.pad(z,(1,1,1,1),mode='circular'),weight,stride=s)\nerrors={\n 'periodic stride1 / input shift1 / output shift1':float((circular(z.roll(1,-1))-circular(z).roll(1,-1)).abs().max()),\n 'periodic stride2 / input shift2 / output shift1':float((circular(z.roll(2,-1),2)-circular(z,2).roll(1,-1)).abs().max()),\n 'periodic stride2 / input shift1 / output shift1':float((circular(z.roll(1,-1),2)-circular(z,2).roll(1,-1)).abs().max()),\n 'zero pad stride1 / circular input shift1':float((F.conv2d(z.roll(1,-1),weight,padding=1)-F.conv2d(z,weight,padding=1).roll(1,-1)).abs().max()),\n}\nassert list(errors.values())[0]<1e-10 and list(errors.values())[1]<1e-10\ntable(['Condition','Maximum absolute difference'],[[k,f'{v:.6g}'] for k,v in errors.items()])", "id": "1ce9baae05944175953dec2c72df9c14", "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "metadata": { "deepnote": { "block_id": "2f6ab2ede0024a688f3a43614577b800", "type": "text-cell-h2", "source_block_id": null, "logical_key": null }, "html_anchor": "cell-30" }, "source": "## 7. 读真实 PyTorch 源码：Module → functional → backend", "id": "2f6ab2ede0024a688f3a43614577b800" }, { "cell_type": "markdown", "metadata": { "deepnote": { "block_id": "d0e39008a01e4a07b11b53f520b5c3b7", "type": "markdown", "source_block_id": null, "logical_key": null }, "html_anchor": "cell-31" }, "source": "下面的文本来自**当前运行环境**的 `inspect`，不是我们伪写的框架源码。`Conv2d` 保存参数、stride、padding、groups 等配置；`forward` 转交 `_conv_forward`。非零 padding 模式先 `F.pad`；数值运算转交 `F.conv2d`，其后的 CPU/CUDA kernel 不在这段 Python wrapper 中。\n\n读源码时应把两种问题分开：数学如何定义，模块如何管理状态并调用 backend。", "id": "d0e39008a01e4a07b11b53f520b5c3b7" }, { "cell_type": "code", "metadata": { "deepnote": { "block_id": "ef8c65db107a4dbf88c949699126d17a", "type": "code", "source_block_id": null, "logical_key": null }, "html_anchor": "cell-32" }, "source": "source_record={'torch':torch.__version__,'git_version':torch.version.git_version,\n 'file':inspect.getsourcefile(nn.Conv2d)}\nprint(json.dumps(source_record,indent=2))\nprint(inspect.getsource(nn.Conv2d.forward))\nprint(inspect.getsource(nn.Conv2d._conv_forward))\nsummary={'hand_output':response.tolist(),'rgb_conv_parameters':448,'shared_gradient_max_error':gradient_error,\n 'equivariance_errors':errors,'learned_kernel':learner.weight.detach().flatten().tolist(),\n 'source':source_record,'execution':'local CPU; complete notebook'}\n(OUT/'lab01-results.json').write_text(json.dumps(summary,indent=2))", "id": "ef8c65db107a4dbf88c949699126d17a", "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "metadata": { "deepnote": { "block_id": "b77baeb540e147dcabbe4c782a1a9e2b", "type": "text-cell-h2", "source_block_id": null, "logical_key": null }, "html_anchor": "cell-33" }, "source": "## 带走的问题", "id": "b77baeb540e147dcabbe4c782a1a9e2b" }, { "cell_type": "markdown", "metadata": { "deepnote": { "block_id": "d24d8e54104a420faff67b6a5cab6c7f", "type": "markdown", "source_block_id": null, "logical_key": null }, "html_anchor": "cell-34" }, "source": "1. 如果每个位置拥有独立 kernel，哪些梯度还会相加，哪些不再相加？\n2. 为什么减小参数量不等价于保证更好的泛化？为自己的研究数据举一个位置相关的反例。\n3. 输入分辨率加倍时，参数、MAC 和激活内存各如何变化？哪些结论依赖 stride 和通道数不变？\n\n可以让 AI 修改实现，但先写下你预计变的量与应保持不变的量，再用数值对照检验。", "id": "d24d8e54104a420faff67b6a5cab6c7f" }, { "cell_type": "markdown", "metadata": { "deepnote": { "block_id": "a29de41a2b384b0cb81c281d9dc28bc5", "type": "text-cell-h2", "source_block_id": null, "logical_key": null }, "html_anchor": "cell-35" }, "source": "## 既有实验的执行记录（沿用）", "id": "a29de41a2b384b0cb81c281d9dc28bc5" }, { "cell_type": "markdown", "metadata": { "deepnote": { "block_id": "c0bc13ec86c947719a3de848c2ef8799", "type": "markdown", "source_block_id": null, "logical_key": null }, "html_anchor": "cell-36" }, "source": "以下为既有本地执行记录，重组时沿用来源与原数值；本轮未重跑这一源实验。\n\n本地 CPU / PyTorch 2.14.0+cpu，所有单元按顺序执行。\n\n| 检查 | 实测 |\n| --- | --- |\n| 手算输出 | `[[19,25],[37,43]]` |\n| 3→16、3×3、带bias参数 | 448 |\n| 手算共享梯度与autograd最大差 | 1.78e-15 |\n| 恢复目标 `[1,-1]` 的核 | `[0.9999999403953552, -0.9999999403953552]` |\n\n周期边界下，stride=1配输入/输出各移1格误差为0；stride=2配输入移2格/输出移1格也为0。stride=2配输入移1格时误差为7.4331，给出不兼容采样相位的具体反例；随机输入下的误差数值不是普遍常数。", "id": "c0bc13ec86c947719a3de848c2ef8799" }, { "cell_type": "markdown", "metadata": { "deepnote": { "block_id": "7e27c6ba030b460385a4645fe17d43cc", "type": "text-cell-h2", "source_block_id": null, "logical_key": null }, "html_anchor": "cell-37" }, "source": "## 协议与来源沿革", "id": "7e27c6ba030b460385a4645fe17d43cc" }, { "cell_type": "markdown", "metadata": { "deepnote": { "block_id": "147369055e9a49da8586fcf386b954f4", "type": "markdown", "source_block_id": null, "logical_key": null }, "html_anchor": "cell-38" }, "source": "历次备课记录与原数值表保留在[教师记录](https://codingai-lec02.pages.dev/course/teacher.html)。本实验使用前文明确的当前协议与模型身份；历史数值不与新运行合并。", "id": "147369055e9a49da8586fcf386b954f4" }, { "cell_type": "markdown", "metadata": { "deepnote": { "block_id": "91a62481dd1644f9b3134df08b9029ba", "type": "markdown", "source_block_id": null, "logical_key": null }, "html_anchor": "cell-39" }, "source": "**返回 CLS 00 · 第二讲导读**", "id": "91a62481dd1644f9b3134df08b9029ba" }, { "cell_type": "markdown", "metadata": { "deepnote": { "block_id": "40698be812d747d29d7c6e48b3e3a561", "type": "text-cell-h2", "source_block_id": null, "logical_key": null }, "html_anchor": "cell-40" }, "source": "## 历史实验再读 1 · 从循环到 Parameter：同一个核怎样进入自动微分？", "id": "40698be812d747d29d7c6e48b3e3a561" }, { "cell_type": "markdown", "metadata": { "deepnote": { "block_id": "24f3a7ed18394385ba47760a020c6853", "type": "markdown", "source_block_id": null, "logical_key": null }, "html_anchor": "cell-41" }, "source": "**先预测。** 前文已用 NumPy 向量化核对多通道卷积，也算过共享梯度。本节补上中间一层：把旧课的显式双循环真正包进 `nn.Module`，让 `nn.Parameter` 管理同一份权重。输入通道应该 `sum`，输出通道应该 `stack`；如果把二者交换，会错在数值还是形状？先写下 `(C_in,H,W)=(2,4,5)`、`(C_out,C_in,k_h,k_w)=(3,2,2,2)` 的输出形状。\n\n**修改参数再运行。** 先执行默认值，再把开头四项改为 `C_IN=3, C_OUT=1, KH=1, KW=3`。观察输入通道贡献、完整输出、注册的参数名，以及显式循环与 `F.conv2d` 的前向/梯度误差。代码只做一次求导，不做参数更新，不依赖前文变量。\n\n**历史来源与改写。** Spring26 `Lecture3-1_convnet`，project `03ca31da-8b5e-4562-82d0-0898f6eb7b7a`，notebook `3560ca0d49804f409f2db00795b76ae5`；Fall25 同源 `Lecture4-5_convnet`，project `f60a1775-066c-48b1-a0c1-181c392044b6`，notebook `b8161140028a4cf1aa9ec9f78da119d0`。保留旧课/D2L 的 `corr2d` 双循环、`Parameter`、通道 `sum/stack` 思路；本次补 dtype/device 一致的分配、输入通道检查，以及共享同值参数的框架和梯度核对。范围限于单样本、valid、stride 1、groups 1。", "id": "24f3a7ed18394385ba47760a020c6853" }, { "cell_type": "code", "metadata": { "deepnote": { "block_id": "5bc3ca70b7b74f22852821cbced13b6b", "type": "code", "source_block_id": null, "logical_key": null }, "html_anchor": "cell-42" }, "source": "import torch\nfrom torch import nn\nfrom torch.nn import functional as F\n\ntorch.set_num_threads(2)\nC_IN, C_OUT, KH, KW = 2, 3, 2, 2 # 再试：3, 1, 1, 3\nassert 1 <= C_IN <= 4 and 1 <= C_OUT <= 4\nassert 1 <= KH <= 4 and 1 <= KW <= 5\n\ndef corr2d_loop(x, kernel):\n kh, kw = kernel.shape\n h, w = x.shape[0] - kh + 1, x.shape[1] - kw + 1\n result = x.new_zeros((h, w))\n for i in range(h):\n for j in range(w):\n result[i, j] = (x[i:i+kh, j:j+kw] * kernel).sum()\n return result\n\ndef multi_in(x, kernels):\n assert x.shape[0] == kernels.shape[0] # 防止 zip 静默截断通道\n return sum(corr2d_loop(xc, kc) for xc, kc in zip(x, kernels))\n\nclass LoopConv2d(nn.Module):\n def __init__(self, cin, cout, kh, kw):\n super().__init__()\n values = torch.arange(cout*cin*kh*kw, dtype=torch.float64)\n self.weight = nn.Parameter(values.reshape(cout, cin, kh, kw) / 20)\n self.bias = nn.Parameter(torch.linspace(-0.2, 0.2, cout, dtype=torch.float64))\n def forward(self, x):\n return torch.stack([multi_in(x, k) for k in self.weight]) + self.bias[:, None, None]\n\nx = torch.arange(C_IN*4*5, dtype=torch.float64).reshape(C_IN, 4, 5) / 10\nlayer = LoopConv2d(C_IN, C_OUT, KH, KW)\ny = layer(x)\nref_weight = layer.weight.detach().clone().requires_grad_()\nref_bias = layer.bias.detach().clone().requires_grad_()\ny_ref = F.conv2d(x[None], ref_weight, ref_bias)[0]\ntorch.testing.assert_close(y, y_ref)\nupstream = torch.linspace(-1, 1, y.numel(), dtype=x.dtype).reshape_as(y)\n(y * upstream).sum().backward()\n(y_ref * upstream).sum().backward()\ntorch.testing.assert_close(layer.weight.grad, ref_weight.grad)\ntorch.testing.assert_close(layer.bias.grad, ref_bias.grad)\nparts = torch.stack([corr2d_loop(xc, kc.detach()) for xc, kc in zip(x, layer.weight[0])])\nprint('Input:', tuple(x.shape), 'Weight:', tuple(layer.weight.shape), 'Output:', tuple(y.shape))\nprint('Registered parameters:', [(name, tuple(p.shape)) for name, p in layer.named_parameters()])\nprint('For output[0,0,0], one contribution per input channel:', parts[:, 0, 0].tolist())\nprint('Their sum + bias:', float(parts[:, 0, 0].sum() + layer.bias[0].detach()))\nprint('Full output:\\n', y.detach())\nHISTORY_RESULT = {\n 'output_shape': list(y.shape),\n 'parameters': sum(p.numel() for p in layer.parameters()),\n 'forward_max_error': float((y.detach()-y_ref.detach()).abs().max()),\n 'weight_grad_max_error': float((layer.weight.grad-ref_weight.grad).abs().max()),\n 'bias_grad_max_error': float((layer.bias.grad-ref_bias.grad).abs().max()),\n}\nprint('Checks:', HISTORY_RESULT)", "id": "5bc3ca70b7b74f22852821cbced13b6b", "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "metadata": { "deepnote": { "block_id": "337c156a1bd046b58e4c61fa2d6cae7e", "type": "markdown", "source_block_id": null, "logical_key": null }, "html_anchor": "cell-43" }, "source": "**核对与解释。** 默认输出是 `(3,3,4)`，参数是 24 个核权重加 3 个偏置。`sum` 消去输入通道轴，`stack` 建立输出通道轴；它们承担不同操作。默认和建议的替代设置下，三项误差都应只有浮点舍入量。`nn.Parameter` 的作用是注册参数；能求导来自前向操作仍保留计算图。参数登记、梯度计算、优化器更新是三件事。\n\n**边界与追问。** 这不是 PyTorch 的底层高性能实现，也没有证明一次正确梯度会降低任意损失。把 `zip` 的通道检查删掉再给错通道数，为什么可能得到“能运行但少算”的结果？把前向中的权重 `.detach()` 后，哪一项核对会失败？请先预测，再在副本中试验。", "id": "337c156a1bd046b58e4c61fa2d6cae7e" }, { "cell_type": "markdown", "metadata": { "deepnote": { "block_id": "323212a027b249ff98f83321f5b61a1f", "type": "text-cell-h2", "source_block_id": null, "logical_key": null }, "html_anchor": "cell-44" }, "source": "## 历史实验再读 2 · 输出都拟合对了，卷积核就一定被唯一找到了吗？", "id": "323212a027b249ff98f83321f5b61a1f" }, { "cell_type": "markdown", "metadata": { "deepnote": { "block_id": "3c0ecec1a8284a6d90fdb23f8ce5e440", "type": "markdown", "source_block_id": null, "logical_key": null }, "html_anchor": "cell-45" }, "source": "**先预测。** 前文用随机输入恢复 `[1,-1]`；旧课也有从目标响应学习边缘核。现在去掉训练，仅检查信息是否足够：如果观察到的输入全是常数图，`[1,-1]` 与 `[2,-2]` 能否被输出区分？在没有独立输入的情况下，“输出误差为零”能否证明找回了唯一真实核？\n\n**修改参数再运行。** `DELTA` 改变候选核的差异，`BRIGHTNESS` 改变常数图的亮度。默认后再试 `DELTA=2.5, BRIGHTNESS=3.0`。代码显示两个输入、两组响应、窗口设计矩阵的秩。所有数据是明确构造的小张量，无训练、无数据下载。\n\n**历史来源与改写。** 承接 Spring26 project `03ca31da-8b5e-4562-82d0-0898f6eb7b7a` / notebook `3560ca0d49804f409f2db00795b76ae5`（`Lecture3-1_convnet`）以及 Fall25 project `f60a1775-066c-48b1-a0c1-181c392044b6` / notebook `b8161140028a4cf1aa9ec9f78da119d0` 的“学习卷积核”。本段是针对核恢复结论新写的不可辨识反例；不复用旧课训练日志，不把构造输出称为历史训练结果。", "id": "3c0ecec1a8284a6d90fdb23f8ce5e440" }, { "cell_type": "code", "metadata": { "deepnote": { "block_id": "ac6f2703870a4986b4f79744ffc25c46", "type": "code", "source_block_id": null, "logical_key": null }, "html_anchor": "cell-46" }, "source": "import torch\nfrom torch.nn import functional as F\n\ntorch.set_num_threads(2)\nDELTA, BRIGHTNESS = 1.0, 1.0 # 再试：2.5, 3.0\nassert BRIGHTNESS != 0 and abs(DELTA) <= 100\nconstant = torch.full((1, 1, 2, 5), BRIGHTNESS, dtype=torch.float64)\nprobe = torch.tensor([[[[0., 1., 0., 1., 0.], [1., 0., 1., 0., 1.]]]], dtype=torch.float64)\nk1 = torch.tensor([[[[1., -1.]]]], dtype=torch.float64)\nk2 = k1 + DELTA * torch.tensor([[[[1., -1.]]]], dtype=torch.float64)\ny1, y2 = F.conv2d(constant, k1), F.conv2d(constant, k2)\np1, p2 = F.conv2d(probe, k1), F.conv2d(probe, k2)\nA_constant = F.unfold(constant, (1, 2))[0].T\nA_probe = F.unfold(probe, (1, 2))[0].T\nnull_direction = torch.tensor([1., -1.], dtype=torch.float64)\ntorch.testing.assert_close(y1, y2)\ntorch.testing.assert_close(A_constant @ null_direction, torch.zeros(A_constant.shape[0], dtype=torch.float64))\nrank_constant = int(torch.linalg.matrix_rank(A_constant))\nrank_combined = int(torch.linalg.matrix_rank(torch.cat([A_constant, A_probe])))\nassert rank_constant == 1 and rank_combined == 2\nif DELTA != 0:\n assert not torch.allclose(p1, p2)\nprint('K1:', k1.flatten().tolist(), 'K2:', k2.flatten().tolist())\nfor name, inp, a, b in [('Constant observations', constant, y1, y2), ('New alternating probe', probe, p1, p2)]:\n print('\\n' + name, '\\ninput:\\n', inp[0, 0])\n print('K1 response:\\n', a[0, 0], '\\nK2 response:\\n', b[0, 0])\nHISTORY_RESULT = {\n 'constant_response_max_gap': float((y1-y2).abs().max()),\n 'probe_response_max_gap': float((p1-p2).abs().max()),\n 'constant_design_rank': rank_constant,\n 'combined_design_rank': rank_combined,\n 'unknown_kernel_values': k1.numel(),\n}\nprint('\\nChecks:', HISTORY_RESULT)", "id": "ac6f2703870a4986b4f79744ffc25c46", "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "metadata": { "deepnote": { "block_id": "003c69f06d904ba79ecd225a417391b1", "type": "markdown", "source_block_id": null, "logical_key": null }, "html_anchor": "cell-47" }, "source": "**核对与解释。** 默认常数图的两组输出都为 0；新探针的响应最大差为 1。原窗口只有一种方向 `[b,b]`，只能约束两个核值的和；沿 `[1,-1]` 改变核，原观察完全不变。加入非恒定探针后，此例的设计矩阵才达到秩 2。\n\n**边界与讨论。** 这是无偏置、线性、无噪声的两参数问题。满列秩能保证这个线性参数问题的唯一性；它不保证有噪声时估计稳定，更不保证深层分类网络的参数唯一。把 `DELTA=0` 设为对照：两组核相同，新探针也不应制造差异。解释“恢复目标算子”时，应说明输入提供了哪些约束，而非只报告训练误差。", "id": "003c69f06d904ba79ecd225a417391b1" } ] }