Normalization 與訓練診斷

輸出對原始輸入的導數 = 0.25^20
                       = 9.094947017729282e-13

Var(output) ≈ fan_in * s² * v

mu_batch[c]  = mean_n X[n,c]
var_batch[c] = mean_n (X[n,c]-mu_batch[c])²
X_hat[n,c]   = (X[n,c]-mu_batch[c])/sqrt(var_batch[c]+epsilon)
Y[n,c]       = gamma[c]*X_hat[n,c] + beta[c]

X = [[1, 2],
     [3, 0],
     [5, 4]]
gamma = [2, 0.5]
beta  = [-1, 1]
epsilon = 1e-5

Y_train = [[-3.449485150, 1.000000000],
           [-1.000000000, 0.387628712],
           [ 1.449485150, 1.612371288]]

running_mean_new = 0.9*running_mean_old + 0.1*batch_mean
                 = [0.3, 0.2]
running_var_new  = 0.9*running_var_old + 0.1*batch_unbiased_var
                 = [1.3, 1.3]

Y_eval = gamma*(X-running_mean)/sqrt(running_var+epsilon) + beta

import numpy as np
import torch

print(f"numpy={np.__version__} torch={torch.__version__} device=cpu dtype=float64")
X = np.array([[1., 2.], [3., 0.], [5., 4.]])
gamma = np.array([2., .5])
beta = np.array([-1., 1.])
eps, momentum = 1e-5, .1
mean = X.mean(axis=0)
biased_var = X.var(axis=0, ddof=0)
unbiased_var = X.var(axis=0, ddof=1)
expected_train = (X - mean) / np.sqrt(biased_var + eps) * gamma + beta
bn = torch.nn.BatchNorm1d(2, eps=eps, momentum=momentum, affine=True,
                          track_running_stats=True, dtype=torch.float64)
with torch.no_grad():
    bn.weight.copy_(torch.tensor(gamma))
    bn.bias.copy_(torch.tensor(beta))
x = torch.tensor(X, dtype=torch.float64)
bn.train()
with torch.no_grad():
    actual_train = bn(x).numpy()
assert np.allclose(actual_train, expected_train, atol=1e-12, rtol=0)
expected_running_mean = momentum * mean
expected_running_var = (1 - momentum) * np.ones(2) + momentum * unbiased_var
assert np.allclose(bn.running_mean.numpy(), expected_running_mean, atol=1e-12, rtol=0)
assert np.allclose(bn.running_var.numpy(), expected_running_var, atol=1e-12, rtol=0)
assert bn.num_batches_tracked.item() == 1
bn.eval()
expected_eval = (X - expected_running_mean) / np.sqrt(expected_running_var + eps) * gamma + beta
with torch.no_grad():
    actual_eval = bn(x).numpy()
    alone_eval = bn(x[:1]).numpy()
    mixed_eval = bn(torch.cat([x[:1], torch.tensor([[100., -100.]], dtype=torch.float64)])).numpy()
assert np.allclose(actual_eval, expected_eval, atol=1e-12, rtol=0)
assert np.array_equal(alone_eval[0], mixed_eval[0])
print("batch_mean=", mean.tolist())
print("batch_variance_biased=", np.round(biased_var, 9).tolist())
print("batch_variance_unbiased=", unbiased_var.tolist())
print("BN_train_output=", np.round(actual_train, 9).tolist())
print("BN_train_column_mean_after_affine=", np.round(actual_train.mean(axis=0), 9).tolist())
print("BN_running_mean_after_one_batch=", bn.running_mean.tolist())
print("BN_running_variance_after_one_batch=", bn.running_var.tolist())
print("BN_eval_output=", np.round(actual_eval, 9).tolist())
print("BN_eval_first_sample_batch_composition_invariant=", True)
ln = torch.nn.LayerNorm(2, eps=eps, elementwise_affine=False, dtype=torch.float64)
row_mean = X.mean(axis=1, keepdims=True)
row_var = X.var(axis=1, ddof=0, keepdims=True)
expected_ln = (X - row_mean) / np.sqrt(row_var + eps)
with torch.no_grad():
    actual_ln = ln(x).numpy()
    alone_ln = ln(x[:1]).numpy()
assert np.allclose(actual_ln, expected_ln, atol=1e-12, rtol=0)
assert np.array_equal(alone_ln[0], actual_ln[0])
print("LayerNorm_rowwise_output=", np.round(actual_ln, 9).tolist())
print("LayerNorm_first_sample_batch_composition_invariant=", True)
# A constructed scalar chain: preactivations all zero, weights1, fixed biases.
u = torch.tensor(0., dtype=torch.float64, requires_grad=True)
h = u
for layer in range(20):
    h = torch.sigmoid(h - (0. if layer == 0 else .5))
h.backward()
expected_chain = .25 ** 20
assert np.isclose(u.grad.item(), expected_chain, atol=0., rtol=1e-12)
print(f"constructed20sigmoid_chain_input_gradient={u.grad.item():.15e}")
print("scope=one CPU BN batch with biased forward/unbiased running variance, fixed affine; eval/LN checks and constructed scalar chain only; no training fit, benchmark, or general vanishing-gradient guarantee")

numpy=1.26.3 torch=2.3.1+cu118 device=cpu dtype=float64
batch_mean= [3.0, 2.0]
batch_variance_biased= [2.666666667, 2.666666667]
batch_variance_unbiased= [4.0, 4.0]
BN_train_output= [[-3.44948515, 1.0], [-1.0, 0.387628712], [1.44948515, 1.612371288]]
BN_train_column_mean_after_affine= [-1.0, 1.0]
BN_running_mean_after_one_batch= [0.30000000000000004, 0.2]
BN_running_variance_after_one_batch= [1.3, 1.3]
BN_eval_output= [[0.227876504, 1.789349181], [3.736095089, 0.912294535], [7.244313673, 2.666403827]]
BN_eval_first_sample_batch_composition_invariant= True
LayerNorm_rowwise_output= [[-0.999980001, 0.999980001], [0.999997778, -0.999997778], [0.999980001, -0.999980001]]
LayerNorm_first_sample_batch_composition_invariant= True
constructed20sigmoid_chain_input_gradient=9.094947017729282e-13
scope=one CPU BN batch with biased forward/unbiased running variance, fixed affine; eval/LN checks and constructed scalar chain only; no training fit, benchmark, or general vanishing-gradient guarantee

The MIT License (MIT)

Copyright (c) 2015 Andrej Karpathy

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SOFTWARE.

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Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the “Software”), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED “AS IS”, WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.