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![Dropout by hand ✍️ ~ 10 steps walkthrough below Dropout is the simplest trick in deep learning that actually works: during training you randomly switch neurons off, so the network cannot lean on any one of them. It is two lines of code and almost nobody has worked through what those lines do to the numbers. So I drew and calculated one entirely by hand. Goal: train one pass through a small network with two dropout layers, then run inference with dropout switched off. The network: Linear(2,4), ReLU, Dropout(0.5), Linear(4,3), ReLU, Dropout(0.33), Linear(3,2). = 1. Given = A training set of two examples, X1 and X2, and the weight matrices for all three linear layers. = 2. Draw the first random numbers = Let us draw 4 random numbers, one per neuron in the first hidden layer. Above 0.5 we keep (◯), below we drop (╳). Here that gives [◯, ╳, ◯, ╳]. = 3. Build the first dropout matrix = We turn that pattern into a diagonal matrix. The scaling factor is 1/(1-p) = 2, so a kept neuron gets 2 and a dropped one gets 0. Multiplying by it does both jobs at once: it deletes the 2nd and 4th neurons and doubles the two that survive. = 4. Draw the second random numbers = Let us do it again for the 3 neurons in the next layer, this time against p = 0.33. The result is [◯, ◯, ╳]. = 5. Build the second dropout matrix = We set the diagonal to 1.5 where kept and 0 where dropped. Only the 3rd neuron goes. = 6. Feed forward = Let us run the whole thing top to bottom: one matrix multiplication per layer, ReLU setting the negatives to zero, and the two dropout matrices doing their work in between. The outputs Y come out at the bottom. = 7. MSE loss gradients = We compare Y against the targets Y', subtract, and multiply each element by 2. That is the whole gradient of the mean squared error. = 8. Update the weights = Let us push those gradients back through the network and update the weights (marked in light red). = 9. Deactivate dropout = Training is over, so we set both dropout matrices to the identity. Every neuron is back, and nothing is scaled. = 10. Feed forward again = One more pass, this time on unseen data, to make the prediction. You have just trained and run a network with dropout by hand. ✍️ The outputs: Training outputs Y = [-6, 9; 13, 4] Loss gradients = [-4, 4; 6, -2] Inference outputs = [13, 13; 4, 3] 💾 Save this post! #AIbyHand #Dropout #DeepLearning #NeuralNetworks](https://image.24vids.com/tw-2078846104460783996/ext_tw_video_thumb/2078846092997763074/pu/img/pljAsdaJjvGEEjPE.jpg)
!["Warren Buffett has this old metaphor: the yes basket, the no basket, and the too-hard basket. For me, software is generally in the too-hard basket." — Brad Gerstner Brad shows how the recent rerating of software stocks simply brought them down to the average market P/E ratio. "So when I hear everybody crying, 'Hey, these multiples aren't fair,' it looks to me like the multiples reset from an above-market multiple — where everybody thought the revenue and earnings were impenetrable — to now, they're saying, 'Maybe 3-5 years out, they'll be replaced. So we're going to raise the discount rate and lower the multiple.' They've only lowered it to the market multiple." "Let me just suggest there's a possibility [software] will trade well below the market multiple. I'm not wishing for that, I'm just saying there's a distribution of potential outcomes here." "If you get on the AI train, if you get in the token flow, you're gonna get an above-market multiple. If you don't, if you slow down and it looks like every time that computational intelligence improves, your business gets worse, then I promise you, [you] will trade below the market multiple."](https://image.24vids.com/tw-2060483256202801272/amplify_video_thumb/2060483071066136576/img/mU1afiKq4bV-wTdb.jpg)
