├── NoProp.pdf
├── README.md
└── NoProp.ipynb


/NoProp.pdf:
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https://raw.githubusercontent.com/Sid3503/NoProp/HEAD/NoProp.pdf


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/README.md:
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  1 | # NoProp: Training Neural Networks Without Forward/Backward Propagation 🚀
  2 | 
  3 | [![Paper](https://img.shields.io/badge/arXiv-Paper-<COLOR>.svg)](https://arxiv.org/abs/2503.24322)
  4 | [![PyTorch](https://img.shields.io/badge/PyTorch-2.0+-red.svg)](https://pytorch.org)
  5 | 
  6 | Official implementation of **NoProp**, a novel neural network training method that eliminates both forward and backward propagation through denoising diffusion. Achieves competitive performance on MNIST/CIFAR while enabling parallel layer training.
  7 | 
  8 | 
  9 | ## 🔍 Table of Contents
 10 | - [Key Innovations](#-key-innovations)
 11 | - [Mathematical Foundations](#-mathematical-foundations)
 12 | - [Implementation](#-implementation)
 13 | - [Results](#-results)
 14 |   
 15 | ## 🚀 Key Innovations
 16 | NoProp introduces three paradigm shifts:
 17 | 
 18 | 1. **Propagation-Free Training**  
 19 |    - No sequential forward/backward passes
 20 |    - Layers train **independently** via denoising
 21 | 
 22 | 2. **Diffusion-Based Learning**  
 23 |    - Corrupts labels progressively (forward diffusion)
 24 |    - Each layer learns to denoise a specific noise level
 25 | 
 26 | 3. **Biological Plausibility**  
 27 |    - Avoids weight transport problem
 28 |    - Local learning only (no global gradients)
 29 | 
 30 | ---
 31 | 
 32 | ## 📜 Mathematical Foundations
 33 | 
 34 | ### A. Forward Diffusion Process (Noising)
 35 | 
 36 | NoProp gradually corrupts clean labels through a Markov chain of noise additions:
 37 | 
 38 | #### 1. Noise Corruption Equation
 39 | ```math
 40 | z_t = \sqrt{\alpha_t} z_{t-1} + \sqrt{1-\alpha_t} \epsilon_t
 41 | ```
 42 | Where:
 43 | - $z_t$: Noisy label at step $t$
 44 | - $\alpha_t$: Noise schedule ($\alpha_0=1 \rightarrow \alpha_T\approx0$)
 45 | - $\epsilon_t \sim \mathcal{N}(0,I)$: Gaussian noise
 46 | - $z_0 = u_y$: Ground truth one-hot label
 47 | 
 48 | #### 2. Noise Schedule Properties
 49 | | Parameter | Role | Typical Value |
 50 | |-----------|------|---------------|
 51 | | `αₜ` | Controls noise level | Linear: `1.0 → 0.1` | - [1]
 52 | | `T` | Total steps | 10-1000 |
 53 | | `ᾱₜ = ∏ αₛ` (s=1 to t) | Cumulative product | (auto-computed) | - [2]
 54 | 
 55 | [1]: `αₜ` decreases linearly  
 56 | [2]: `ᾱₜ = α₁ × α₂ × ... × αₜ`
 57 | 
 58 | #### 3. Step-by-Step Example (MNIST)
 59 | Given label "2" ($u_y = [0,0,1,0,...]$):
 60 | 
 61 | | Step $t$ | $\alpha_t$ | $z_t$ (Visualized) | Noise Level |
 62 | |---------|-----------|--------------------|------------|
 63 | | 0 | 1.0 | [0, 0, 1.0, 0] | 0% |
 64 | | 1 | 0.9 | [0, 0.1, 0.85, 0.05] | 10% |
 65 | | 2 | 0.7 | [0.05, 0.15, 0.7, 0.1] | 30% |
 66 | | ... | ... | ... | ... |
 67 | | T | 0.1 | [0.25, 0.25, 0.3, 0.2] | 90% |
 68 | 
 69 | #### 4. Key Properties
 70 | 1. **Gradual Corruption**:
 71 |    ```math
 72 |    \text{SNR}(t) = \frac{\alpha_t}{1-\alpha_t} \quad \text{(Monotonically decreases)}
 73 |    ```
 74 | 2. **Variance-Preserving**:
 75 |    ```math
 76 |    \text{Var}(z_t) = \text{Var}(z_{t-1}) = 1
 77 |    ```
 78 | 3. **Closed-Form Sampling**:
 79 |    ```math
 80 |    q(z_t|u_y) = \mathcal{N}(z_t; \sqrt{\bar{\alpha}_t}u_y, (1-\bar{\alpha}_t)I)
 81 |    ```
 82 | 
 83 | 
 84 | ### B. Reverse Process (Training)
 85 | Each MLP layer $t$ predicts clean labels from noisy inputs:
 86 | 
 87 | ```math
 88 | \mathcal{L}_t = \mathbb{E} \| \hat{u}_\theta(z_t,x) - u_y \|^2
 89 | ```
 90 | 
 91 | Where:
 92 | - $\hat{u}_\theta$: MLP prediction  
 93 | - $x$: Input image features  
 94 | - $u_y$: Ground truth one-hot label
 95 | 
 96 | 
 97 | During inference, NoProp iteratively refines noisy labels through learned denoising steps:
 98 | 
 99 | #### 1. Denoising Update Rule
100 | ```math
101 | z_{t-1} = \sqrt{\alpha_{l-1}} \underbrace{\hat{u}_\theta(z_t,x)}_{\text{Predicted clean label}} + \sqrt{1-\alpha_{t-1}} \epsilon_t
102 | ```
103 | 
104 | Where:
105 | - $z_t$: Noisy label at step $t$
106 | - $\hat{u}_\theta(z_t,x)$: MLP's prediction of clean label
107 | - $\alpha_{t-1}$: Noise schedule value
108 | - $\epsilon_t \sim \mathcal{N}(0,I)$: Fresh Gaussian noise
109 | 
110 | #### 2. Step-by-Step Process
111 | 1. **Start from noise**: $z_T \sim \mathcal{N}(0,I)$
112 | 2. **Iterate for** $t=T$ to $1$:
113 |    - Predict $\hat{u}_\theta(z_t,x)$ using MLP
114 |    - Compute $z_{t-1}$ via denoising update
115 | 3. **Final prediction**: $\arg\max(z_0)$
116 | 
117 | #### 3. Example (MNIST)
118 | | Step | $z_t$ (Noisy)          | $\hat{u}_\theta(z_t,x)$ (Predicted) | $z_{t-1}$ (Refined)       |
119 | |------|-------------------------|-------------------------------------|---------------------------|
120 | | t=3  | [0.4, 0.3, 0.3]         | [0.1, 0.0, 0.9]                     | [0.25, 0.05, 0.7]         |
121 | | t=2  | [0.25, 0.05, 0.7]       | [0.0, 0.0, 1.0]                     | [0.1, 0.0, 0.9]           |
122 | | t=1  | [0.1, 0.0, 0.9]         | [0.0, 0.0, 1.0]                     | [0.0, 0.0, 1.0] (Final)   |
123 | 
124 | #### 4. Key Properties
125 | 1. **Noise Injection**: 
126 |    - The $\sqrt{1-\alpha_{t-1}} \epsilon_t$ term prevents deterministic collapse
127 | 2. **Geometric Interpolation**:
128 |    - Balances between prediction ($\sqrt{\alpha_{t-1}}$) and noise ($\sqrt{1-\alpha_{t-1}}$)
129 | 3. **Stochasticity**:
130 |    - Different noise samples $\epsilon_t$ yield varied trajectories
131 | 
132 | 
133 | ## ⚙️ Implementation
134 | 
135 | ### Core Components
136 | ```python
137 | # 1. Diffusion Noise Scheduler
138 | alpha = torch.linspace(1.0, 0.1, T)  # T diffusion steps
139 | 
140 | # 2. Denoising MLP (per layer)
141 | class DenoisingMLP(nn.Module):
142 |     def forward(self, x_features, z_t):
143 |         return self.mlp(torch.cat([x_features, z_t], dim=1))
144 | 
145 | # 3. Training Loop
146 | for t in range(T):
147 |     u_hat = mlps[t](x_features, z[t+1].detach())
148 |     loss = F.mse_loss(u_hat, u_y)
149 | ```
150 | 
151 | ### Key Features
152 | - **Parallel Training**: All `T` MLPs update simultaneously
153 | - **Memory Efficient**: No activation storage
154 | - **Flexible**: Works with CNNs/Transformers
155 |   
156 | 
157 | ## 👤 Author
158 | 
159 | For any questions or issues, please open an issue on GitHub: [@Siddharth Mishra](https://github.com/Sid3503)
160 | 
161 | ---
162 | 
163 | <p align="center">
164 |   Made with ❤️ and lots of ☕
165 | </p>
166 | 


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/NoProp.ipynb:
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  7 |       "gpuType": "T4",
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 10 |     },
 11 |     "kernelspec": {
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 15 |     "language_info": {
 16 |       "name": "python"
 17 |     },
 18 |     "accelerator": "GPU"
 19 |   },
 20 |   "cells": [
 21 |     {
 22 |       "cell_type": "markdown",
 23 |       "metadata": {
 24 |         "id": "view-in-github",
 25 |         "colab_type": "text"
 26 |       },
 27 |       "source": [
 28 |         "<a href=\"https://colab.research.google.com/github/Sid3503/NoProp/blob/main/NoProp.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
 29 |       ]
 30 |     },
 31 |     {
 32 |       "cell_type": "markdown",
 33 |       "source": [
 34 |         "# Setup Libraries"
 35 |       ],
 36 |       "metadata": {
 37 |         "id": "zCu1mXACaKdx"
 38 |       }
 39 |     },
 40 |     {
 41 |       "cell_type": "code",
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 43 |       "metadata": {
 44 |         "colab": {
 45 |           "base_uri": "https://localhost:8080/"
 46 |         },
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 48 |         "outputId": "2503536e-6b14-403c-c38e-b64dd36983bf"
 49 |       },
 50 |       "outputs": [
 51 |         {
 52 |           "output_type": "stream",
 53 |           "name": "stdout",
 54 |           "text": [
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142 |             "    Uninstalling nvidia-cusparse-cu12-12.5.1.3:\n",
143 |             "      Successfully uninstalled nvidia-cusparse-cu12-12.5.1.3\n",
144 |             "  Attempting uninstall: nvidia-cudnn-cu12\n",
145 |             "    Found existing installation: nvidia-cudnn-cu12 9.3.0.75\n",
146 |             "    Uninstalling nvidia-cudnn-cu12-9.3.0.75:\n",
147 |             "      Successfully uninstalled nvidia-cudnn-cu12-9.3.0.75\n",
148 |             "  Attempting uninstall: nvidia-cusolver-cu12\n",
149 |             "    Found existing installation: nvidia-cusolver-cu12 11.6.3.83\n",
150 |             "    Uninstalling nvidia-cusolver-cu12-11.6.3.83:\n",
151 |             "      Successfully uninstalled nvidia-cusolver-cu12-11.6.3.83\n",
152 |             "Successfully installed nvidia-cublas-cu12-12.4.5.8 nvidia-cuda-cupti-cu12-12.4.127 nvidia-cuda-nvrtc-cu12-12.4.127 nvidia-cuda-runtime-cu12-12.4.127 nvidia-cudnn-cu12-9.1.0.70 nvidia-cufft-cu12-11.2.1.3 nvidia-curand-cu12-10.3.5.147 nvidia-cusolver-cu12-11.6.1.9 nvidia-cusparse-cu12-12.3.1.170 nvidia-nvjitlink-cu12-12.4.127\n"
153 |           ]
154 |         }
155 |       ],
156 |       "source": [
157 |         "!pip install torch torchvision"
158 |       ]
159 |     },
160 |     {
161 |       "cell_type": "code",
162 |       "source": [
163 |         "import torch\n",
164 |         "import torch.nn as nn\n",
165 |         "import torch.optim as optim\n",
166 |         "from torchvision import datasets, transforms\n",
167 |         "from torch.utils.data import DataLoader"
168 |       ],
169 |       "metadata": {
170 |         "id": "ZoxseEX7_Fy1"
171 |       },
172 |       "execution_count": 2,
173 |       "outputs": []
174 |     },
175 |     {
176 |       "cell_type": "markdown",
177 |       "source": [
178 |         "# Hyperparameters"
179 |       ],
180 |       "metadata": {
181 |         "id": "nxES-OQfaZaF"
182 |       }
183 |     },
184 |     {
185 |       "cell_type": "code",
186 |       "source": [
187 |         "# Hyperparameters\n",
188 |         "T = 10  # Diffusion steps\n",
189 |         "embed_dim = 10  # Label embedding dimension(No. of Classes)\n",
190 |         "batch_size = 128\n",
191 |         "lr = 0.001\n",
192 |         "epochs = 50"
193 |       ],
194 |       "metadata": {
195 |         "id": "GApOWb8K_QbC"
196 |       },
197 |       "execution_count": 22,
198 |       "outputs": []
199 |     },
200 |     {
201 |       "cell_type": "code",
202 |       "source": [
203 |         "# Noise schedule (linear)\n",
204 |         "alpha = torch.linspace(1.0, 0.1, T)  # α_t from 1.0 → 0.1"
205 |       ],
206 |       "metadata": {
207 |         "id": "vzSRvuN-_QUv"
208 |       },
209 |       "execution_count": 4,
210 |       "outputs": []
211 |     },
212 |     {
213 |       "cell_type": "code",
214 |       "source": [
215 |         "print(alpha)"
216 |       ],
217 |       "metadata": {
218 |         "colab": {
219 |           "base_uri": "https://localhost:8080/"
220 |         },
221 |         "id": "Sqfg91HP_QSx",
222 |         "outputId": "10d03ab1-b7ad-4d32-92fc-47d80b4aae24"
223 |       },
224 |       "execution_count": 6,
225 |       "outputs": [
226 |         {
227 |           "output_type": "stream",
228 |           "name": "stdout",
229 |           "text": [
230 |             "tensor([1.0000, 0.9000, 0.8000, 0.7000, 0.6000, 0.5000, 0.4000, 0.3000, 0.2000,\n",
231 |             "        0.1000])\n"
232 |           ]
233 |         }
234 |       ]
235 |     },
236 |     {
237 |       "cell_type": "markdown",
238 |       "source": [
239 |         "# Setting Up Models"
240 |       ],
241 |       "metadata": {
242 |         "id": "eeZ3MSd3adhn"
243 |       }
244 |     },
245 |     {
246 |       "cell_type": "code",
247 |       "source": [
248 |         "# MLP for denoising\n",
249 |         "class DenoisingMLP(nn.Module):\n",
250 |         "    def __init__(self):\n",
251 |         "        super().__init__()\n",
252 |         "        self.mlp = nn.Sequential(\n",
253 |         "            nn.Linear(128 + embed_dim, 256),  # Input: image features + noisy label\n",
254 |         "            nn.ReLU(),\n",
255 |         "            nn.Linear(256, embed_dim)        # Output: denoised label\n",
256 |         "        )\n",
257 |         "\n",
258 |         "    def forward(self, x_features, z_t):\n",
259 |         "        combined = torch.cat([x_features, z_t], dim=1)\n",
260 |         "        return self.mlp(combined)"
261 |       ],
262 |       "metadata": {
263 |         "id": "uJ26Tb7Z_QQR"
264 |       },
265 |       "execution_count": 7,
266 |       "outputs": []
267 |     },
268 |     {
269 |       "cell_type": "code",
270 |       "source": [
271 |         "# CNN for image features\n",
272 |         "class CNN(nn.Module):\n",
273 |         "    def __init__(self):\n",
274 |         "        super().__init__()\n",
275 |         "        self.features = nn.Sequential(\n",
276 |         "            nn.Conv2d(1, 32, 3, 1),\n",
277 |         "            nn.ReLU(),\n",
278 |         "            nn.MaxPool2d(2),\n",
279 |         "            nn.Conv2d(32, 64, 3, 1),\n",
280 |         "            nn.ReLU(),\n",
281 |         "            nn.MaxPool2d(2),\n",
282 |         "            nn.Flatten(),\n",
283 |         "            nn.Linear(1600, 128)  # MNIST: 28x28 → 1600-dim\n",
284 |         "        )\n",
285 |         "\n",
286 |         "    def forward(self, x):\n",
287 |         "        return self.features(x)"
288 |       ],
289 |       "metadata": {
290 |         "id": "JDYgzrdr_tsx"
291 |       },
292 |       "execution_count": 8,
293 |       "outputs": []
294 |     },
295 |     {
296 |       "cell_type": "code",
297 |       "source": [
298 |         "# Initialize models\n",
299 |         "cnn = CNN()\n",
300 |         "mlps = nn.ModuleList([DenoisingMLP() for _ in range(T)])  # One MLP per layer\n",
301 |         "optimizers = [optim.Adam(mlp.parameters(), lr=lr) for mlp in mlps]"
302 |       ],
303 |       "metadata": {
304 |         "id": "bTXUdHIm_wEw"
305 |       },
306 |       "execution_count": 9,
307 |       "outputs": []
308 |     },
309 |     {
310 |       "cell_type": "code",
311 |       "source": [
312 |         "cnn"
313 |       ],
314 |       "metadata": {
315 |         "colab": {
316 |           "base_uri": "https://localhost:8080/"
317 |         },
318 |         "id": "3c_c5vIBaE1o",
319 |         "outputId": "348103a5-19c8-4a99-db33-d5997ab6a314"
320 |       },
321 |       "execution_count": 55,
322 |       "outputs": [
323 |         {
324 |           "output_type": "execute_result",
325 |           "data": {
326 |             "text/plain": [
327 |               "CNN(\n",
328 |               "  (features): Sequential(\n",
329 |               "    (0): Conv2d(1, 32, kernel_size=(3, 3), stride=(1, 1))\n",
330 |               "    (1): ReLU()\n",
331 |               "    (2): MaxPool2d(kernel_size=2, stride=2, padding=0, dilation=1, ceil_mode=False)\n",
332 |               "    (3): Conv2d(32, 64, kernel_size=(3, 3), stride=(1, 1))\n",
333 |               "    (4): ReLU()\n",
334 |               "    (5): MaxPool2d(kernel_size=2, stride=2, padding=0, dilation=1, ceil_mode=False)\n",
335 |               "    (6): Flatten(start_dim=1, end_dim=-1)\n",
336 |               "    (7): Linear(in_features=1600, out_features=128, bias=True)\n",
337 |               "  )\n",
338 |               ")"
339 |             ]
340 |           },
341 |           "metadata": {},
342 |           "execution_count": 55
343 |         }
344 |       ]
345 |     },
346 |     {
347 |       "cell_type": "code",
348 |       "source": [
349 |         "mlps"
350 |       ],
351 |       "metadata": {
352 |         "colab": {
353 |           "base_uri": "https://localhost:8080/"
354 |         },
355 |         "id": "uwA8rRCqaF4C",
356 |         "outputId": "47eed38e-48ce-4186-a945-d0f0795f1f95"
357 |       },
358 |       "execution_count": 56,
359 |       "outputs": [
360 |         {
361 |           "output_type": "execute_result",
362 |           "data": {
363 |             "text/plain": [
364 |               "ModuleList(\n",
365 |               "  (0-9): 10 x DenoisingMLP(\n",
366 |               "    (mlp): Sequential(\n",
367 |               "      (0): Linear(in_features=138, out_features=256, bias=True)\n",
368 |               "      (1): ReLU()\n",
369 |               "      (2): Linear(in_features=256, out_features=10, bias=True)\n",
370 |               "    )\n",
371 |               "  )\n",
372 |               ")"
373 |             ]
374 |           },
375 |           "metadata": {},
376 |           "execution_count": 56
377 |         }
378 |       ]
379 |     },
380 |     {
381 |       "cell_type": "markdown",
382 |       "source": [
383 |         "# Preparing Dataset"
384 |       ],
385 |       "metadata": {
386 |         "id": "TY3BUc9Dah44"
387 |       }
388 |     },
389 |     {
390 |       "cell_type": "code",
391 |       "source": [
392 |         "# Load MNIST\n",
393 |         "transform = transforms.ToTensor()\n",
394 |         "train_data = datasets.MNIST(root='./data', train=True, download=True, transform=transform)\n",
395 |         "train_loader = DataLoader(train_data, batch_size=batch_size, shuffle=True)"
396 |       ],
397 |       "metadata": {
398 |         "colab": {
399 |           "base_uri": "https://localhost:8080/"
400 |         },
401 |         "id": "zZAWOkNB_xuF",
402 |         "outputId": "3acd5dba-09e9-40a9-cd8c-16e8a87d6ffc"
403 |       },
404 |       "execution_count": 10,
405 |       "outputs": [
406 |         {
407 |           "output_type": "stream",
408 |           "name": "stderr",
409 |           "text": [
410 |             "100%|██████████| 9.91M/9.91M [00:02<00:00, 4.56MB/s]\n",
411 |             "100%|██████████| 28.9k/28.9k [00:00<00:00, 65.1kB/s]\n",
412 |             "100%|██████████| 1.65M/1.65M [00:01<00:00, 1.26MB/s]\n",
413 |             "100%|██████████| 4.54k/4.54k [00:00<00:00, 6.77MB/s]\n"
414 |           ]
415 |         }
416 |       ]
417 |     },
418 |     {
419 |       "cell_type": "markdown",
420 |       "source": [
421 |         "# Training"
422 |       ],
423 |       "metadata": {
424 |         "id": "0lgU9EC0amwQ"
425 |       }
426 |     },
427 |     {
428 |       "cell_type": "markdown",
429 |       "source": [
430 |         "---\n",
431 |         "\n",
432 |         "### **NoProp Training Dry Run (1 Epoch Example)**\n",
433 |         "\n",
434 |         "\n",
435 |         "# 🧪 NoProp Training Dry Run (1 Epoch)\n",
436 |         "\n",
437 |         "Let's simulate **one epoch** of training with:\n",
438 |         "- Batch size = 3\n",
439 |         "- Classes = 3 (\"cat\", \"dog\", \"bird\")\n",
440 |         "- Diffusion steps (T) = 2\n",
441 |         "\n",
442 |         "## 📥 Batch Data\n",
443 |         "**Input (x):** 3 images  \n",
444 |         "**Labels (y):** [\"cat\"=0, \"dog\"=1, \"bird\"=2]  \n",
445 |         "→ One-hot encoded `u_y`:\n",
446 |         "```\n",
447 |         "tensor([\n",
448 |         " [1., 0., 0.],  # cat\n",
449 |         " [0., 1., 0.],  # dog\n",
450 |         " [0., 0., 1.]   # bird\n",
451 |         "])\n",
452 |         "```\n",
453 |         "\n",
454 |         "---\n",
455 |         "\n",
456 |         "## 🌪️ Forward Diffusion (Corrupt Labels)\n",
457 |         "**Noise schedule (α):** [α₀=1.0, α₁=0.6, α₂=0.3]  \n",
458 |         "1. **t=0:** `z₀ = u_y` (clean)\n",
459 |         "\n",
460 |         "2. **t=1:**  \n",
461 |         "   `z₁ = √0.6*z₀ + √0.4*noise` ≈\n",
462 |         "```\n",
463 |         "tensor([\n",
464 |         " [0.77, 0.20, 0.03],  # noisy cat\n",
465 |         " [0.10, 0.85, 0.05],  # noisy dog\n",
466 |         " [0.05, 0.10, 0.85]   # noisy bird\n",
467 |         "])\n",
468 |         "```\n",
469 |         "\n",
470 |         "\n",
471 |         "3. **t=2:**  \n",
472 |         "   `z₂ = √0.3*z₁ + √0.7*noise` ≈\n",
473 |         "\n",
474 |         "```\n",
475 |         "tensor([\n",
476 |         " [0.40, 0.35, 0.25],  # very noisy cat\n",
477 |         " [0.25, 0.45, 0.30],  # very noisy dog\n",
478 |         " [0.20, 0.25, 0.55]   # very noisy bird\n",
479 |         "])\n",
480 |         "```\n",
481 |         "\n",
482 |         "---\n",
483 |         "\n",
484 |         "**Example for MLP1 (t=1):**\n",
485 |         "- **Input:** Image features + `z₁ = [0.77,0.20,0.03]` (noisy cat)\n",
486 |         "- **Prediction:** `[0.9, 0.1, 0.0]` (should approach `[1,0,0]`)\n",
487 |         "- **Loss:** MSE([0.9,0.1,0.0], [1,0,0]) = 0.01\n",
488 |         "\n",
489 |         "## 🔄 Weight Updates\n",
490 |         "1. Sum losses from all MLPs → `total_loss`\n",
491 |         "2. Backpropagate → Update all MLPs independently\n",
492 |         "\n",
493 |         "## 📊 Epoch Output\n",
494 |         "`Epoch 1/10 | Avg Loss: 0.85`  \n",
495 |         "*(Loss decreases as MLPs learn to denoise better)*\n",
496 |         "\n",
497 |         "---\n",
498 |         "\n",
499 |         "### **Key Takeaways**\n",
500 |         "1. **Diffusion**: Labels are progressively noised from clean → random.\n",
501 |         "2. **Specialization**: Each MLP handles a specific noise level.\n",
502 |         "3. **Independence**: No backprop between layers → parallel training."
503 |       ],
504 |       "metadata": {
505 |         "id": "HjthYG3JddhO"
506 |       }
507 |     },
508 |     {
509 |       "cell_type": "code",
510 |       "source": [
511 |         "# Training loop\n",
512 |         "for epoch in range(epochs):\n",
513 |         "    epoch_loss = 0.0\n",
514 |         "    batch_count = 0\n",
515 |         "\n",
516 |         "    for x, y in train_loader:\n",
517 |         "        current_batch_size = x.shape[0]\n",
518 |         "        u_y = torch.zeros(current_batch_size, embed_dim).scatter_(1, y.unsqueeze(1), 1)\n",
519 |         "\n",
520 |         "        # Forward diffusion(Adding Noise for each 'T')\n",
521 |         "        z = [u_y]\n",
522 |         "        for t in range(1, T + 1):\n",
523 |         "            eps = torch.randn_like(u_y)\n",
524 |         "            z_t = torch.sqrt(alpha[t-1]) * z[-1] + torch.sqrt(1 - alpha[t-1]) * eps\n",
525 |         "            z.append(z_t)\n",
526 |         "\n",
527 |         "        # Train MLPs\n",
528 |         "        x_features = cnn(x)\n",
529 |         "        losses = []\n",
530 |         "        for t in range(T):\n",
531 |         "            u_hat = mlps[t](x_features, z[t+1].detach())\n",
532 |         "            losses.append(torch.mean((u_hat - u_y) ** 2))\n",
533 |         "\n",
534 |         "        total_loss = sum(losses)\n",
535 |         "        for opt in optimizers:\n",
536 |         "            opt.zero_grad()\n",
537 |         "        total_loss.backward()\n",
538 |         "        for opt in optimizers:\n",
539 |         "            opt.step()\n",
540 |         "\n",
541 |         "        epoch_loss += total_loss.item()\n",
542 |         "        batch_count += 1\n",
543 |         "\n",
544 |         "    # Epoch summary print\n",
545 |         "    avg_loss = epoch_loss / batch_count\n",
546 |         "    print(f\"Epoch {epoch+1}/{epochs} | Avg Loss: {avg_loss:.4f}\")\n",
547 |         "\n",
548 |         "# Final message\n",
549 |         "print(\"Training complete!\")"
550 |       ],
551 |       "metadata": {
552 |         "colab": {
553 |           "base_uri": "https://localhost:8080/"
554 |         },
555 |         "id": "KtGT_c1H_y9r",
556 |         "outputId": "2ae2430a-4e96-4e08-9ce6-1e7bc7b719ed"
557 |       },
558 |       "execution_count": 23,
559 |       "outputs": [
560 |         {
561 |           "output_type": "stream",
562 |           "name": "stdout",
563 |           "text": [
564 |             "Epoch 1/50 | Avg Loss: 0.0835\n",
565 |             "Epoch 2/50 | Avg Loss: 0.0819\n",
566 |             "Epoch 3/50 | Avg Loss: 0.0806\n",
567 |             "Epoch 4/50 | Avg Loss: 0.0791\n",
568 |             "Epoch 5/50 | Avg Loss: 0.0778\n",
569 |             "Epoch 6/50 | Avg Loss: 0.0766\n",
570 |             "Epoch 7/50 | Avg Loss: 0.0753\n",
571 |             "Epoch 8/50 | Avg Loss: 0.0744\n",
572 |             "Epoch 9/50 | Avg Loss: 0.0734\n",
573 |             "Epoch 10/50 | Avg Loss: 0.0725\n",
574 |             "Epoch 11/50 | Avg Loss: 0.0714\n",
575 |             "Epoch 12/50 | Avg Loss: 0.0706\n",
576 |             "Epoch 13/50 | Avg Loss: 0.0700\n",
577 |             "Epoch 14/50 | Avg Loss: 0.0692\n",
578 |             "Epoch 15/50 | Avg Loss: 0.0685\n",
579 |             "Epoch 16/50 | Avg Loss: 0.0677\n",
580 |             "Epoch 17/50 | Avg Loss: 0.0671\n",
581 |             "Epoch 18/50 | Avg Loss: 0.0664\n",
582 |             "Epoch 19/50 | Avg Loss: 0.0660\n",
583 |             "Epoch 20/50 | Avg Loss: 0.0654\n",
584 |             "Epoch 21/50 | Avg Loss: 0.0648\n",
585 |             "Epoch 22/50 | Avg Loss: 0.0643\n",
586 |             "Epoch 23/50 | Avg Loss: 0.0637\n",
587 |             "Epoch 24/50 | Avg Loss: 0.0631\n",
588 |             "Epoch 25/50 | Avg Loss: 0.0627\n",
589 |             "Epoch 26/50 | Avg Loss: 0.0621\n",
590 |             "Epoch 27/50 | Avg Loss: 0.0617\n",
591 |             "Epoch 28/50 | Avg Loss: 0.0615\n",
592 |             "Epoch 29/50 | Avg Loss: 0.0614\n",
593 |             "Epoch 30/50 | Avg Loss: 0.0604\n",
594 |             "Epoch 31/50 | Avg Loss: 0.0600\n",
595 |             "Epoch 32/50 | Avg Loss: 0.0596\n",
596 |             "Epoch 33/50 | Avg Loss: 0.0594\n",
597 |             "Epoch 34/50 | Avg Loss: 0.0589\n",
598 |             "Epoch 35/50 | Avg Loss: 0.0588\n",
599 |             "Epoch 36/50 | Avg Loss: 0.0583\n",
600 |             "Epoch 37/50 | Avg Loss: 0.0581\n",
601 |             "Epoch 38/50 | Avg Loss: 0.0577\n",
602 |             "Epoch 39/50 | Avg Loss: 0.0575\n",
603 |             "Epoch 40/50 | Avg Loss: 0.0570\n",
604 |             "Epoch 41/50 | Avg Loss: 0.0568\n",
605 |             "Epoch 42/50 | Avg Loss: 0.0564\n",
606 |             "Epoch 43/50 | Avg Loss: 0.0562\n",
607 |             "Epoch 44/50 | Avg Loss: 0.0560\n",
608 |             "Epoch 45/50 | Avg Loss: 0.0558\n",
609 |             "Epoch 46/50 | Avg Loss: 0.0555\n",
610 |             "Epoch 47/50 | Avg Loss: 0.0552\n",
611 |             "Epoch 48/50 | Avg Loss: 0.0551\n",
612 |             "Epoch 49/50 | Avg Loss: 0.0548\n",
613 |             "Epoch 50/50 | Avg Loss: 0.0545\n",
614 |             "Training complete!\n"
615 |           ]
616 |         }
617 |       ]
618 |     },
619 |     {
620 |       "cell_type": "markdown",
621 |       "source": [
622 |         "# Inferencing"
623 |       ],
624 |       "metadata": {
625 |         "id": "kGrsvukHaqGX"
626 |       }
627 |     },
628 |     {
629 |       "cell_type": "code",
630 |       "source": [
631 |         "# Inference (denoising)\n",
632 |         "def predict(x):\n",
633 |         "    z_t = torch.randn(1, embed_dim)  # Start from noise\n",
634 |         "    x_features = cnn(x.unsqueeze(0))\n",
635 |         "    for t in reversed(range(T)):\n",
636 |         "        u_hat = mlps[t](x_features, z_t)\n",
637 |         "        z_t = torch.sqrt(alpha[t]) * u_hat + torch.sqrt(1 - alpha[t]) * torch.randn_like(u_hat)\n",
638 |         "    return torch.argmax(z_t)  # Final prediction"
639 |       ],
640 |       "metadata": {
641 |         "id": "LINA1yBV_0XH"
642 |       },
643 |       "execution_count": 24,
644 |       "outputs": []
645 |     },
646 |     {
647 |       "cell_type": "code",
648 |       "source": [
649 |         "# Test on an example\n",
650 |         "x_test, y_test = next(iter(train_loader))\n",
651 |         "pred = predict(x_test[0])\n",
652 |         "print(f\"Predicted: {pred}, True: {y_test[0]}\")"
653 |       ],
654 |       "metadata": {
655 |         "colab": {
656 |           "base_uri": "https://localhost:8080/"
657 |         },
658 |         "id": "Qs8kIZcb_0Tp",
659 |         "outputId": "8823f70e-d3f6-4eb8-c2a6-bd771136c828"
660 |       },
661 |       "execution_count": 48,
662 |       "outputs": [
663 |         {
664 |           "output_type": "stream",
665 |           "name": "stdout",
666 |           "text": [
667 |             "Predicted: 1, True: 1\n"
668 |           ]
669 |         }
670 |       ]
671 |     },
672 |     {
673 |       "cell_type": "code",
674 |       "source": [
675 |         "import matplotlib.pyplot as plt\n",
676 |         "import numpy as np\n",
677 |         "\n",
678 |         "def plot_prediction(x, true_label, pred_label, class_names=None):\n",
679 |         "    \"\"\"\n",
680 |         "    Plot image with true and predicted labels.\n",
681 |         "\n",
682 |         "    Args:\n",
683 |         "        x (torch.Tensor): Input image tensor (1, C, H, W)\n",
684 |         "        true_label (int): Ground truth class index\n",
685 |         "        pred_label (int): Predicted class index\n",
686 |         "        class_names (list): Optional list of class names\n",
687 |         "    \"\"\"\n",
688 |         "    # Convert tensor to numpy and denormalize if needed\n",
689 |         "    img = x.squeeze().cpu().numpy()\n",
690 |         "    if img.min() < 0 or img.max() > 1:  # Assuming [0,1] or [-1,1] range\n",
691 |         "        img = (img - img.min()) / (img.max() - img.min())\n",
692 |         "\n",
693 |         "    # Create figure\n",
694 |         "    plt.figure(figsize=(6, 3))\n",
695 |         "\n",
696 |         "    # Plot image\n",
697 |         "    plt.subplot(1, 2, 1)\n",
698 |         "    plt.imshow(img, cmap='gray' if img.ndim == 2 else None)\n",
699 |         "    plt.axis('off')\n",
700 |         "    plt.title('Input Image', pad=10)\n",
701 |         "\n",
702 |         "    # Plot labels\n",
703 |         "    plt.subplot(1, 2, 2)\n",
704 |         "    plt.axis('off')\n",
705 |         "\n",
706 |         "    if class_names:\n",
707 |         "        true_str = f\"True: {class_names[true_label]} ({true_label})\"\n",
708 |         "        pred_str = f\"Predicted: {class_names[pred_label]} ({pred_label})\"\n",
709 |         "    else:\n",
710 |         "        true_str = f\"True label: {true_label}\"\n",
711 |         "        pred_str = f\"Predicted: {pred_label}\"\n",
712 |         "\n",
713 |         "    plt.text(0.1, 0.7, true_str, fontsize=12, color='green')\n",
714 |         "    plt.text(0.1, 0.5, pred_str,\n",
715 |         "             fontsize=12,\n",
716 |         "             color='red' if true_label != pred_label else 'green')\n",
717 |         "\n",
718 |         "    # Highlight incorrect predictions\n",
719 |         "    if true_label != pred_label:\n",
720 |         "        plt.text(0.1, 0.3, \"INCORRECT\", fontsize=14, color='red', weight='bold')\n",
721 |         "\n",
722 |         "    plt.tight_layout()\n",
723 |         "    plt.show()"
724 |       ],
725 |       "metadata": {
726 |         "id": "XTHv45S5_0RO"
727 |       },
728 |       "execution_count": 29,
729 |       "outputs": []
730 |     },
731 |     {
732 |       "cell_type": "code",
733 |       "source": [
734 |         "# Example usage with MNIST\n",
735 |         "class_names = [str(i) for i in range(10)]  # ['0', '1', ..., '9']\n",
736 |         "x_test, y_test = next(iter(train_loader))\n",
737 |         "pred = predict(x_test[0])\n",
738 |         "\n",
739 |         "plot_prediction(x_test[0],\n",
740 |         "               y_test[0].item(),\n",
741 |         "               pred.item(),\n",
742 |         "               class_names)"
743 |       ],
744 |       "metadata": {
745 |         "colab": {
746 |           "base_uri": "https://localhost:8080/",
747 |           "height": 316
748 |         },
749 |         "id": "WK1nzjCt_0O-",
750 |         "outputId": "d3afcc0a-d137-4ad0-dec9-e1a7fe0faa07"
751 |       },
752 |       "execution_count": 57,
753 |       "outputs": [
754 |         {
755 |           "output_type": "display_data",
756 |           "data": {
757 |             "text/plain": [
758 |               "<Figure size 600x300 with 2 Axes>"
759 |             ],
760 |             "image/png": 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\n"
761 |           },
762 |           "metadata": {}
763 |         }
764 |       ]
765 |     },
766 |     {
767 |       "cell_type": "code",
768 |       "source": [],
769 |       "metadata": {
770 |         "id": "nGEr22BecAXl"
771 |       },
772 |       "execution_count": null,
773 |       "outputs": []
774 |     }
775 |   ]
776 | }
777 | 


--------------------------------------------------------------------------------