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%load_ext autoreload
%autoreload 2
%matplotlib inline
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#export
from exp.nb_01 import *
def get_data():
path = datasets.download_data(MNIST_URL, ext='.gz')
with gzip.open(path, 'rb') as f:
((x_train, y_train), (x_valid, y_valid), _) = pickle.load(f, encoding='latin-1')
return map(tensor, (x_train,y_train,x_valid,y_valid))
def normalize(x, m, s): return (x-m)/s
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x_train,y_train,x_valid,y_valid = get_data()
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train_mean,train_std = x_train.mean(),x_train.std()
train_mean,train_std
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x_train = normalize(x_train, train_mean, train_std)
# NB: Use training, not validation mean for validation set
x_valid = normalize(x_valid, train_mean, train_std)
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train_mean,train_std = x_train.mean(),x_train.std()
train_mean,train_std
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#export
def test_near_zero(a,tol=1e-3): assert a.abs()<tol, f"Near zero: {a}"
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test_near_zero(x_train.mean())
test_near_zero(1-x_train.std())
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n,m = x_train.shape
c = y_train.max()+1
n,m,c
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# num hidden
nh = 50
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# simplified kaiming init / he init
w1 = torch.randn(m,nh)/math.sqrt(m)
b1 = torch.zeros(nh)
w2 = torch.randn(nh,1)/math.sqrt(nh)
b2 = torch.zeros(1)
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test_near_zero(w1.mean())
test_near_zero(w1.std()-1/math.sqrt(m))
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# This should be ~ (0,1) (mean,std)...
x_valid.mean(),x_valid.std()
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def lin(x, w, b): return x@w + b
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t = lin(x_valid, w1, b1)
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#...so should this, because we used kaiming init, which is designed to do this
t.mean(),t.std()
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def relu(x): return x.clamp_min(0.)
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t = relu(lin(x_valid, w1, b1))
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#...actually it really should be this!
t.mean(),t.std()
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From pytorch docs: a: the negative slope of the rectifier used after this layer (0 for ReLU by default)
This was introduced in the paper that described the Imagenet-winning approach from He et al: Delving Deep into Rectifiers, which was also the first paper that claimed "super-human performance" on Imagenet (and, most importantly, it introduced resnets!)
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# kaiming init / he init for relu
w1 = torch.randn(m,nh)*math.sqrt(2/m)
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w1.mean(),w1.std()
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t = relu(lin(x_valid, w1, b1))
t.mean(),t.std()
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#export
from torch.nn import init
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w1 = torch.zeros(m,nh)
init.kaiming_normal_(w1, mode='fan_out')
t = relu(lin(x_valid, w1, b1))
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init.kaiming_normal_??
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w1.mean(),w1.std()
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t.mean(),t.std()
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w1.shape
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import torch.nn
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torch.nn.Linear(m,nh).weight.shape
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torch.nn.Linear.forward??
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torch.nn.functional.linear??
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torch.nn.Conv2d??
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torch.nn.modules.conv._ConvNd.reset_parameters??
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# what if...?
def relu(x): return x.clamp_min(0.) - 0.5
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# kaiming init / he init for relu
w1 = torch.randn(m,nh)*math.sqrt(2./m )
t1 = relu(lin(x_valid, w1, b1))
t1.mean(),t1.std()
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def model(xb):
l1 = lin(xb, w1, b1)
l2 = relu(l1)
l3 = lin(l2, w2, b2)
return l3
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%timeit -n 10 _=model(x_valid)
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assert model(x_valid).shape==torch.Size([x_valid.shape[0],1])
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model(x_valid).shape
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We need squeeze() to get rid of that trailing (,1), in order to use mse. (Of course, mse is not a suitable loss function for multi-class classification; we'll use a better loss function soon. We'll use mse for now to keep things simple.)
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#export
def mse(output, targ): return (output.squeeze(-1) - targ).pow(2).mean()
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y_train,y_valid = y_train.float(),y_valid.float()
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preds = model(x_train)
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preds.shape
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mse(preds, y_train)
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def mse_grad(inp, targ):
# grad of loss with respect to output of previous layer
inp.g = 2. * (inp.squeeze() - targ).unsqueeze(-1) / inp.shape[0]
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def relu_grad(inp, out):
# grad of relu with respect to input activations
inp.g = (inp>0).float() * out.g
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def lin_grad(inp, out, w, b):
# grad of matmul with respect to input
inp.g = out.g @ w.t()
w.g = (inp.unsqueeze(-1) * out.g.unsqueeze(1)).sum(0)
b.g = out.g.sum(0)
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def forward_and_backward(inp, targ):
# forward pass:
l1 = inp @ w1 + b1
l2 = relu(l1)
out = l2 @ w2 + b2
# we don't actually need the loss in backward!
loss = mse(out, targ)
# backward pass:
mse_grad(out, targ)
lin_grad(l2, out, w2, b2)
relu_grad(l1, l2)
lin_grad(inp, l1, w1, b1)
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forward_and_backward(x_train, y_train)
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# Save for testing against later
w1g = w1.g.clone()
w2g = w2.g.clone()
b1g = b1.g.clone()
b2g = b2.g.clone()
ig = x_train.g.clone()
We cheat a little bit and use PyTorch autograd to check our results.
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xt2 = x_train.clone().requires_grad_(True)
w12 = w1.clone().requires_grad_(True)
w22 = w2.clone().requires_grad_(True)
b12 = b1.clone().requires_grad_(True)
b22 = b2.clone().requires_grad_(True)
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def forward(inp, targ):
# forward pass:
l1 = inp @ w12 + b12
l2 = relu(l1)
out = l2 @ w22 + b22
# we don't actually need the loss in backward!
return mse(out, targ)
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loss = forward(xt2, y_train)
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loss.backward()
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test_near(w22.grad, w2g)
test_near(b22.grad, b2g)
test_near(w12.grad, w1g)
test_near(b12.grad, b1g)
test_near(xt2.grad, ig )
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class Relu():
def __call__(self, inp):
self.inp = inp
self.out = inp.clamp_min(0.)-0.5
return self.out
def backward(self): self.inp.g = (self.inp>0).float() * self.out.g
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class Lin():
def __init__(self, w, b): self.w,self.b = w,b
def __call__(self, inp):
self.inp = inp
self.out = inp@self.w + self.b
return self.out
def backward(self):
self.inp.g = self.out.g @ self.w.t()
# Creating a giant outer product, just to sum it, is inefficient!
self.w.g = (self.inp.unsqueeze(-1) * self.out.g.unsqueeze(1)).sum(0)
self.b.g = self.out.g.sum(0)
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class Mse():
def __call__(self, inp, targ):
self.inp = inp
self.targ = targ
self.out = (inp.squeeze() - targ).pow(2).mean()
return self.out
def backward(self):
self.inp.g = 2. * (self.inp.squeeze() - self.targ).unsqueeze(-1) / self.targ.shape[0]
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class Model():
def __init__(self, w1, b1, w2, b2):
self.layers = [Lin(w1,b1), Relu(), Lin(w2,b2)]
self.loss = Mse()
def __call__(self, x, targ):
for l in self.layers: x = l(x)
return self.loss(x, targ)
def backward(self):
self.loss.backward()
for l in reversed(self.layers): l.backward()
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w1.g,b1.g,w2.g,b2.g = [None]*4
model = Model(w1, b1, w2, b2)
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%time loss = model(x_train, y_train)
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%time model.backward()
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test_near(w2g, w2.g)
test_near(b2g, b2.g)
test_near(w1g, w1.g)
test_near(b1g, b1.g)
test_near(ig, x_train.g)
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class Module():
def __call__(self, *args):
self.args = args
self.out = self.forward(*args)
return self.out
def forward(self): raise Exception('not implemented')
def backward(self): self.bwd(self.out, *self.args)
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class Relu(Module):
def forward(self, inp): return inp.clamp_min(0.)-0.5
def bwd(self, out, inp): inp.g = (inp>0).float() * out.g
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class Lin(Module):
def __init__(self, w, b): self.w,self.b = w,b
def forward(self, inp): return inp@self.w + self.b
def bwd(self, out, inp):
inp.g = out.g @ self.w.t()
self.w.g = torch.einsum("bi,bj->ij", inp, out.g)
self.b.g = out.g.sum(0)
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class Mse(Module):
def forward (self, inp, targ): return (inp.squeeze() - targ).pow(2).mean()
def bwd(self, out, inp, targ): inp.g = 2*(inp.squeeze()-targ).unsqueeze(-1) / targ.shape[0]
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class Model():
def __init__(self):
self.layers = [Lin(w1,b1), Relu(), Lin(w2,b2)]
self.loss = Mse()
def __call__(self, x, targ):
for l in self.layers: x = l(x)
return self.loss(x, targ)
def backward(self):
self.loss.backward()
for l in reversed(self.layers): l.backward()
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w1.g,b1.g,w2.g,b2.g = [None]*4
model = Model()
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%time loss = model(x_train, y_train)
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%time model.backward()
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test_near(w2g, w2.g)
test_near(b2g, b2.g)
test_near(w1g, w1.g)
test_near(b1g, b1.g)
test_near(ig, x_train.g)
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class Lin(Module):
def __init__(self, w, b): self.w,self.b = w,b
def forward(self, inp): return inp@self.w + self.b
def bwd(self, out, inp):
inp.g = out.g @ self.w.t()
self.w.g = inp.t() @ out.g
self.b.g = out.g.sum(0)
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w1.g,b1.g,w2.g,b2.g = [None]*4
model = Model()
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%time loss = model(x_train, y_train)
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%time model.backward()
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test_near(w2g, w2.g)
test_near(b2g, b2.g)
test_near(w1g, w1.g)
test_near(b1g, b1.g)
test_near(ig, x_train.g)
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#export
from torch import nn
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class Model(nn.Module):
def __init__(self, n_in, nh, n_out):
super().__init__()
self.layers = [nn.Linear(n_in,nh), nn.ReLU(), nn.Linear(nh,n_out)]
self.loss = mse
def __call__(self, x, targ):
for l in self.layers: x = l(x)
return self.loss(x.squeeze(), targ)
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model = Model(m, nh, 1)
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%time loss = model(x_train, y_train)
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%time loss.backward()
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!./notebook2script.py 02_fully_connected.ipynb
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