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Backpropagation by hand ✍️ ~ 11 steps walkthrough below Backpropagation is the algorithm that actually trains a neural network, and it is where most people stop following along. It is not calculus you cannot do. It is matrix multiplication, working backward, one layer at a time. So I drew... show more
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![[RLHF] by Hand ✍️ Yesterday, Jan Leike (Jan Leike) announced he is joining #Anthropic to lead their "super-alignment" mission. He is the co-inventor of Reinforcement Learning with Human Feedback (#RLHF). How does RLHF work? [1] Given ↳ Reward Model (RM) ↳ Large Language Model (LLM) ↳ Two (Prompt, Next) Pairs 🟪 TRAIN RM Goal: Learn to give higher rewards to winners [2] Preferences ↳ A human reviews the two pairs and picks a "winner" ↳ (doc is, him) Embeddings ↳ This prompt has never received human feedback directly ↳ [S] is the special start symbol [11] Transformer ↳ Attention (yellow) ↳ Feed Forward (4x2 weight and bias matrix) ↳ Output: 3 "transformed" feature vector, one per position ↳ More details in my previous post 8. Transformer [] [12] Output Probabilities ↳ Apply a linear layer to map each transformed feature vector to a probability distribution over the vocabulary. [13] Sample ↳ Apply the greedy method, which is to pick the word with the highest score ↳ For output 1 and 2, the model accurately predicts the next word ↳ For 3rd output position, the model's predicts "him" [14] Reward Model ↳ The new pair (CEO is, him) is fed to the reward model ↳ The process is same as [3]-[6] ↳ Output: Reward = 3 [15] Loss Gradient ↳ We set the loss as the negative of the reward. ↳ The loss gradient is simply a constant -1. ↳ Run backpropagation and gradient descent to update LLM's weights and biases (red border)](https://image.24vids.com/tw-1795803665845899511/ext_tw_video_thumb/1795803020149534720/pu/img/jb8SBLT_1drHtYd_.jpg)
![SORA by Hand ✍️ OpenAI’s #SORA took over the Internet when it was announced earlier this year. The technology behind Sora is the Diffusion Transformer (DiT) developed by William Peebles and Shining Xie. How does DiT work? 𝗚𝗼𝗮𝗹: Generate a video conditioned by a text prompt and a series of diffusion steps [1] Given ↳ Video ↳ Prompt: "sora is sky" ↳ Diffusion step: t = 3 [2] Video → Patches ↳ Divide all pixels in all frames into 4 spacetime patches [3] Visual Encoder: Pixels 🟨 → Latent 🟩 ↳ Multiply the patches with weights and biases, followed by ReLU ↳ The result is a latent feature vector per patch ↳ The purpose is dimension reduction from 4 (2x2x1) to 2 (2x1). ↳ In the paper, the reduction is 196,608 (256x256x3)→ 4096 (32x32x4) [4] ⬛ Add Noise ↳ Sample a noise according to the diffusion time step t. Typically, the larger the t, the smaller the noise. ↳ Add the Sampled Noise to latent features to obtain Noised Latent. ↳ The goal is to purposely add noise to a video and ask the model to guess what that noise is. ↳ This is analogous to training a language model by purposely deleting a word in a sentence and ask the model to guess what the deleted word was. [5-7] 🟪 Conditioning by Adaptive Layer Norm [5] Encode Conditions ↳ Encode "sora is sky" into a text embedding vector [0,1,-1]. ↳ Encode t = 3 to as a binary vector [1,1]. ↳ Concatenate the two vectors in to a 5D column vector. [6] Estimate Scale/Shift ↳ Multiply the combined vector with weights and biases ↳ The goal is to estimate the scale [2,-1] and shift [-1,5]. ↳ Copy the result to (X) and (+) [7] Apply Scale/Sift ↳ Scale the noised latent by [2,-1] ↳ Shifted the scaled noised latent by [-1, 5] ↳ The result is "conditioned" noise latent. [8-10] Transformer [8] Self-Attention ↳ Feed the conditioned noised latent to Query-Key function to obtain a self-attention matrix ↳ Value is omitted for simplicity [9] Attention Pooling ↳ Multiply the conditioned noised latent with the self-attention matrix ↳ The result are attention weighted features [10] Pointwise Feed Forward Network ↳ Multiply the attention weighted features with weights and biases ↳ The result is the Predicted Noise 🏋️♂️ 𝗧𝗿𝗮𝗶𝗻 [11] ↳ Calculate MSE loss gradients by taking the different between the Predicted Noise and the Sampled Noise (ground truth). ↳ Use the loss gradients to kick off backpropagation to update all learnable parameters (red borders) ↳ Note the visual encoder and decoder's parameters are frozen (blue borders) 🎨 𝗚𝗲𝗻𝗲𝗿𝗮𝘁𝗲 (𝗦𝗮𝗺𝗽𝗹𝗲) [12] Denoise ↳ Subtract the predicted noise from the noised latent to obtain the noise-free latent [13] Visual Decoder: Latent 🟩 → Pixels 🟨 ↳ Multiply the patches with weights and biases, followed by ReLU [14] Patches → Video ↳ Rearrange patches into a sequence of video frames.](https://image.24vids.com/tw-1795449683285848509/ext_tw_video_thumb/1795449360802525185/pu/img/j-hppkMq667X9mjK.jpg)
![[CLIP] by Hand ✍️ The CLIP (Contrastive Language–Image Pre-training) model, a groundbreaking work by OpenAI, redefines the intersection of computer vision and natural language processing. It is the basis of all the multi-modal foundation models we see today. How does CLIP work? Goal: 🟨 Learn a shared embedding space for text and image [1] Given ↳ A mini batch of 3 text-image pairs ↳ OpenAI used 400 million text-image pairs to train its original CLIP model. Process 1st pair: "big table" [2] 🟪 Text → 2 Vectors (3D) ↳ Look up word embedding vectors using word2vec. [3] 🟩 Image → 2 Vectors (4D) ↳ Divide the image into two patches. ↳ Flatten each patch [4] Process other pairs ↳ Repeat [2]-[3] [5] 🟪 Text Encoder & 🟩 Image Encoder ↳ Encode input vectors into feature vectors ↳ Here, both encoders are simple one layer perceptron (linear + ReLU) ↳ In practice, the encoders are usually transformer models. [6] 🟪 🟩 Mean Pooling: 2 → 1 vector ↳ Average 2 feature vectors into a single vector by averaging across the columns ↳ The goal is to have one vector to represent each image or text [7] 🟪 🟩 -> 🟨 Projection ↳ Note that the text and image feature vectors from the encoders have different dimensions (3D vs. 4D). ↳ Use a linear layer to project image and text vectors to a 2D shared embedding space. 🏋️ Contrastive Pre-training 🏋️ [8] Prepare for MatMul ↳ Copy text vectors (T1,T2,T3) ↳ Copy the transpose of image vectors (I1,I2,I3) ↳ They are all in the 2D shared embedding space. [9] 🟦 MatMul ↳ Multiply T and I matrices. ↳ This is equivalent to taking dot product between every pair of image and text vectors. ↳ The purpose is to use dot product to estimate the similarity between a pair of image-text. [10] 🟦 Softmax: e^x ↳ Raise e to the power of the number in each cell ↳ To simplify hand calculation, we approximate e^□ with 3^□. [11] 🟦 Softmax: ∑ ↳ Sum each row for 🟩 image→🟪 text ↳ Sum each column for 🟪 text→ 🟩 image [12] 🟦 Softmax: 1 / sum ↳ Divide each element by the column sum to obtain a similarity matrix for 🟪 text→🟩 image ↳ Divide each element by the row sum to obtain a similarity matrix for 🟩 image→🟪 text [13] 🟥 Loss Gradients ↳ The "Targets" for the similarity matrices are Identity Matrices. ↳ Why? If I and T come from the same pair (i=j), we want the highest value, which is 1, and 0 otherwise. ↳ Apply the simple equation of [Similarity - Target] to compute gradients of for both directions. ↳ Why so simple? Because when Softmax and Cross-Entropy Loss are used together, the math magically works out that way. ↳ These gradients kick off the backpropagation process to update weights and biases of the encoders and projection layers (red borders).](https://image.24vids.com/tw-1801370086152016034/ext_tw_video_thumb/1801368762937131008/pu/img/GuRh5lcJ6M7HsrkY.jpg)
![[Self-Attention] by Hand ✍️ Self-attention is what enables LLMs to understand context. How does it work? This exercise demonstrates how to calculate a 6-3 attention head by hand. Note that if we have two instances of this, we get 6-6 attention (i.e., multi-head attention, n=2). -- 𝗚𝗼𝗮𝗹 -- Transform [6D Features 🟧] to [3D Attention Weighted Features 🟦] -- 𝗪𝗮𝗹𝗸𝘁𝗵𝗿𝗼𝘂𝗴𝗵 -- [1] Given ↳ A set of 4 feature vectors (6-D): x1,x2,x3,x4 [2] Query, Key, Value ↳ Multiply features x's with linear transformation matrices WQ, WK, and WV, to obtain query vectors (q1,q2,q3,q4), key vectors (k1,k2,k3,k4), and value vectors (v1,v2,v3,v4). ↳ "Self" refers to the fact that both queries and keys are derived from the same set of features. [3] 🟪 Prepare for MatMul ↳ Copy query vectors ↳ Copy the transpose of key vectors [4] 🟪 MatMul ↳ Multiply K^T and Q ↳ This is equivalent to taking dot product between every pair of query and key vectors. ↳ The purpose is to use dot product as an estimate of the "matching score" between every key-value pair. ↳ This estimate makes sense because dot product is the numerator of Cosine Similarity between two vectors. [5] 🟨 Scale ↳ Scale each element by the square root of dk, which is the dimension of key vectors (dk=3). ↳ The purpose is to normalize the impact of the dk on matching scores, even if we scale dk to 32, 64, or 128. ↳ To simplify hand calculation, we approximate [ □/sqrt(3) ] with [ floor(□/2) ]. [6] 🟩 Softmax: e^x ↳ Raise e to the power of the number in each cell ↳ To simplify hand calculation, we approximate e^□ with 3^□. [7] 🟩 Softmax: ∑ ↳ Sum across each column [8] 🟩 Softmax: 1 / sum ↳ For each column, divide each element by the column sum ↳ The purpose is normalize each column so that the numbers sum to 1. In other words, each column is a probability distribution of attention, and we have four of them. ↳ The result is the Attention Weight Matrix (A) (yellow) [9] 🟦 MatMul ↳ Multiply the value vectors (Vs) with the Attention Weight Matrix (A) ↳ The results are the attention weighted features Zs. ↳ They are fed to the position-wise feed forward network in the next layer.](https://image.24vids.com/tw-1797249951325434315/ext_tw_video_thumb/1797249707795697665/pu/img/bRQFS-RcgUFbmOnk.jpg)
