## 5. Hyperparameter tuning

#### 5.1 Tuning process

Hyperparameters:

$\alpha$, $\beta$, $\beta_1,\beta_2, \epsilon$, layers, hidden units, learning rate decay, mini-batch size.

The learning rate is the most important hyperparameter to tune. $\beta$, mini-batch size and hidden units is second in importance to tune.

Try random values: Don’t use a grid. Corarse to fine.

#### 5.2 Using an appropriate scale to pick hyperparameters

Appropriate scale to hyperparameters:

$\alpha = [0.0001, 1]$, r = -4 * np.random.rand(), $\alpha = 10^r$.

If $\alpha = [10^a, 10^b]$, random pick from [a, b] uniformly, and set $\alpha = 10^r$.

Hyperparameters for exponentially weighted average

$\beta = [0.9, 0.999]$, don’t random pick from $[0.9, 0.999]$. Use $1-\beta = [0.001, 0.1]$, use similar method lik $\alpha$.

Why don’t use linear pick? Because when $\beta$ is close one, even if a little change, it will have a huge impact on algorithm.

#### 5.3 Hyperparameters tuning in practice: Pandas vs Caviar

• Re-test hyperparamters occasionally

• Babysitting one model(Pandas)

• Training many models in parallel(Caviar)

## 6. Batch Normalization

#### 6.1 Normalizing activations in a network

In logistic regression, normalizing inputs to speed up learning.

1. compute means$\mu = \frac {1} {m} \sum_{i=1}^n x^{(i)}$
2. subtract off the means from training set $x = x - \mu$\
3. compute the variances $\sigma ^2 = \frac {1} {m} \sum_{i=1}^n {x^{(i)}}^2$
4. normalize training set $X = \frac {X} {\sigma ^2}$

Similarly, in order to speed up training neural network, we can normalize intermediate values in layers（z in hidden layer）, it is called Batch Normalization or Batch Norm.

Implementing Batch Norm

1. Given some intermediate value in neural network, $z^{(1)}, z^{(2)},…,z^{(m)}$
2. compute means $\mu = \frac {1} {m} \sum_{i=1} z^{(i)}$
3. compute the variances $\sigma ^2 = \frac {1} {m} \sum_{i=1} (z^{(i)} - \mu)^2$
4. normalize $z$, $z^{(i)} = \frac {z^{(i)} - \mu} {\sqrt {(\sigma ^2 + \epsilon)}}$
5. compute $\hat z$, $\hat z = \gamma z^{(i)} + \beta$.

Now we have normalized Z to have mean zero and standard unit variance. But maybe it makes sense for hidden units to have a different distribution. So we use $\hat z$ instead of $z$, $\gamma$ and $\beta$ are learnable parameters of your model.

#### 6.2 Fitting Batch Norm into a neural network

Add Batch Norm to a network

$X \rightarrow Z^{[1]} \rightarrow {\hat Z^{[1]}} \rightarrow {a^{[1]}} \rightarrow Z^{[2]} \rightarrow {\hat Z^{[2]}} \rightarrow {a^{[2]}}…$

Parameters:
$W^{[1]}, b^{[1]}$, $W^{[2]}, b^{[2]}…$
$\gamma^{[1]}, \beta^{[1]}$, $\gamma^{[2]}, \beta^{[2]}…$

If you use Batch Norm, you need to computing means and subtracting means, so $b^{[i]}$ is useless, so we can set $b^{[i]} = 0$ permanently.

#### 6.3 Why does Batch Norm work?

Covariate Shift: You have learned a function from $x \rightarrow y$, it works well. If the distribution of $x$ changes, you need to learn a new function to make it work well.

Hidden unit values change all the time, and so it’s suffering from the problem of covariate.

Batch Norm as regularization

• Each mini-batch is scaled by the mean/variance computed on just that mini-batch.
• This adds some noise to the values $z^{[l]}$ within that mini-batch. So similar to dropout, it adds some noise to each hidden layer’s activations.
• This has a slight regularization effect.

#### 6.4 Batch Norm at test time

In order to apply neural network at test time, come up with some seperate estimate of mu and sigma squared.

Hard max.

Loss function.

• Caffe/Caffe2
• TensorFlow
• Torch
• Theano
• mxnet