# Neural Network

## Activation Functions

The activation ops provide different types of nonlinearities for use in neural networks. These include smooth nonlinearities (sigmoid, tanh, elu, selu, softplus, and softsign), continuous but not everywhere differentiable functions (relu, relu6, crelu and relu_x), and random regularization (dropout).

All activation ops apply componentwise, and produce a tensor of the same shape as the input tensor.

## Convolution

The convolution ops sweep a 2-D filter over a batch of images, applying the filter to each window of each image of the appropriate size. The different ops trade off between generic vs. specific filters:

• conv2d: Arbitrary filters that can mix channels together.
• depthwise_conv2d: Filters that operate on each channel independently.
• separable_conv2d: A depthwise spatial filter followed by a pointwise filter.

Note that although these ops are called "convolution", they are strictly speaking "cross-correlation" since the filter is combined with an input window without reversing the filter. For details, see the properties of cross-correlation.

The filter is applied to image patches of the same size as the filter and strided according to the strides argument. strides = [1, 1, 1, 1] applies the filter to a patch at every offset, strides = [1, 2, 2, 1] applies the filter to every other image patch in each dimension, etc.

Ignoring channels for the moment, assume that the 4-D input has shape [batch, in_height, in_width, ...] and the 4-D filter has shape [filter_height, filter_width, ...]. The spatial semantics of the convolution ops depend on the padding scheme chosen: 'SAME' or 'VALID'. Note that the padding values are always zero.

First, consider the 'SAME' padding scheme. A detailed explanation of the reasoning behind it is given in these notes. Here, we summarize the mechanics of this padding scheme. When using 'SAME', the output height and width are computed as:

out_height = ceil(float(in_height) / float(strides[1]))
out_width  = ceil(float(in_width) / float(strides[2]))

The total padding applied along the height and width is computed as:

if (in_height % strides[1] == 0):
pad_along_height = max(filter_height - strides[1], 0)
else:
pad_along_height = max(filter_height - (in_height % strides[1]), 0)
if (in_width % strides[2] == 0):
pad_along_width = max(filter_width - strides[2], 0)
else:
pad_along_width = max(filter_width - (in_width % strides[2]), 0)

Finally, the padding on the top, bottom, left and right are:

Note that the division by 2 means that there might be cases when the padding on both sides (top vs bottom, right vs left) are off by one. In this case, the bottom and right sides always get the one additional padded pixel. For example, when pad_along_height is 5, we pad 2 pixels at the top and 3 pixels at the bottom. Note that this is different from existing libraries such as cuDNN and Caffe, which explicitly specify the number of padded pixels and always pad the same number of pixels on both sides.

For the 'VALID' scheme, the output height and width are computed as:

out_height = ceil(float(in_height - filter_height + 1) / float(strides[1]))
out_width  = ceil(float(in_width - filter_width + 1) / float(strides[2]))

Given the output size and the padding, the output can be computed as

output[b, i, j, :] =
sum_{di, dj} input[b, strides[1] * i + di - pad_top,
strides[2] * j + dj - pad_left, ...] *
filter[di, dj, ...]

where any value outside the original input image region are considered zero ( i.e. we pad zero values around the border of the image).

Since input is 4-D, each input[b, i, j, :] is a vector. For conv2d, these vectors are multiplied by the filter[di, dj, :, :] matrices to produce new vectors. For depthwise_conv_2d, each scalar component input[b, i, j, k] is multiplied by a vector filter[di, dj, k], and all the vectors are concatenated.

## Pooling

The pooling ops sweep a rectangular window over the input tensor, computing a reduction operation for each window (average, max, or max with argmax). Each pooling op uses rectangular windows of size ksize separated by offset strides. For example, if strides is all ones every window is used, if strides is all twos every other window is used in each dimension, etc.

In detail, the output is

output[i] = reduce(value[strides * i:strides * i + ksize])

## Morphological filtering

Morphological operators are non-linear filters used in image processing.

Greyscale morphological dilation is the max-sum counterpart of standard sum-product convolution:

output[b, y, x, c] =
max_{dy, dx} input[b,
strides[1] * y + rates[1] * dy,
strides[2] * x + rates[2] * dx,
c] +
filter[dy, dx, c]

The filter is usually called structuring function. Max-pooling is a special case of greyscale morphological dilation when the filter assumes all-zero values (a.k.a. flat structuring function).

Greyscale morphological erosion is the min-sum counterpart of standard sum-product convolution:

output[b, y, x, c] =
min_{dy, dx} input[b,
strides[1] * y - rates[1] * dy,
strides[2] * x - rates[2] * dx,
c] -
filter[dy, dx, c]

Dilation and erosion are dual to each other. The dilation of the input signal f by the structuring signal g is equal to the negation of the erosion of -f by the reflected g, and vice versa.

Striding and padding is carried out in exactly the same way as in standard convolution. Please refer to the Convolution section for details.

## Normalization

Normalization is useful to prevent neurons from saturating when inputs may have varying scale, and to aid generalization.

## Losses

The loss ops measure error between two tensors, or between a tensor and zero. These can be used for measuring accuracy of a network in a regression task or for regularization purposes (weight decay).

## Embeddings

TensorFlow provides library support for looking up values in embedding tensors.

## Recurrent Neural Networks

TensorFlow provides a number of methods for constructing Recurrent Neural Networks. Most accept an RNNCell-subclassed object (see the documentation for tf.contrib.rnn).

## Evaluation

The evaluation ops are useful for measuring the performance of a network. They are typically used at evaluation time.

## Candidate Sampling

Do you want to train a multiclass or multilabel model with thousands or millions of output classes (for example, a language model with a large vocabulary)? Training with a full Softmax is slow in this case, since all of the classes are evaluated for every training example. Candidate Sampling training algorithms can speed up your step times by only considering a small randomly-chosen subset of contrastive classes (called candidates) for each batch of training examples.

### Sampled Loss Functions

TensorFlow provides the following sampled loss functions for faster training.

### Candidate Samplers

TensorFlow provides the following samplers for randomly sampling candidate classes when using one of the sampled loss functions above.

## Notes on SAME Convolution Padding

In these notes, we provide more background on the use of the 'SAME' padding scheme for convolution operations.

Tensorflow uses the smallest possible padding to achieve the desired output size. To understand what is done, consider the $$1$$-dimensional case. Denote $$n_i$$ and $$n_o$$ the input and output sizes, respectively, and denote the kernel size $$k$$ and stride $$s$$. As discussed in the Convolution section, for 'SAME', $$n_o = \left \lceil{\frac{n_i}{s}}\right \rceil$$.

To achieve a desired output size $$n_o$$, we need to pad the input such that the output size after a 'VALID' convolution is $$n_o$$. In other words, we need to have padding $$p_i$$ such that:

$$\left \lceil{\frac{n_i + p_i - k + 1}{s}}\right \rceil = n_o \label{eq:tf_pad_1}$$

What is the smallest $$p_i$$ that we could possibly use? In general, $$\left \lceil{\frac{x}{a}}\right \rceil = b$$ (with $$a > 0$$) means that $$b-1 < \frac{x}{a} \leq b$$, and the smallest integer $$x$$ we can choose to satisfy this is $$x = a\cdot (b-1) + 1$$. The same applies to our problem; we need $$p_i$$ such that:

$$n_i + p_i - k + 1 = s\cdot (n_o - 1) + 1 \label{eq:tf_pad_2}$$

$$p_i = s\cdot (n_o - 1) + k - n_i \label{eq:tf_pad_3}$$

Note that this might lead to negative $$p_i$$, since in some cases we might already have more input samples than we actually need. Thus,

$$p_i = max(s\cdot (n_o - 1) + k - n_i, 0) \label{eq:tf_pad_4}$$

Remember that, for 'SAME' padding, $$n_o = \left \lceil{\frac{n_i}{s}}\right \rceil$$, as mentioned above. We need to analyze in detail two cases:

• $$n_i \text{ mod } s = 0$$

In this simple case, $$n_o = \frac{n_i}{s}$$, and the expression for $$p_i$$ becomes:

$$p_i = max(k - s, 0) \label{eq:tf_pad_5}$$

• $$n_i \text{ mod } s \neq 0$$

This case is more involved to parse. First, we write:

$$n_i = s\cdot\left \lceil{\frac{n_i}{s}}\right \rceil - s \left(\left \lceil{\frac{n_i}{s}}\right \rceil - \left \lfloor{\frac{n_i}{s}}\right \rfloor\right) + (n_i \text{ mod } s) \label{eq:tf_pad_6}$$

For the case where $$(n_i \text{ mod } s) \neq 0$$, we have $$\left \lceil{\frac{n_i}{s}}\right \rceil -\left \lfloor{\frac{n_i}{s}}\right \rfloor = 1$$, leading to:

$$n_i = s\cdot\left \lceil{\frac{n_i}{s}}\right \rceil - s + (n_i \text{ mod } s) \label{eq:tf_pad_7}$$

We can use this expression to substitute $$n_o = \left \lceil{\frac{n_i}{s}}\right \rceil$$ and get:

\begin{align} p_i &= max\left(s\cdot \left(\frac{n_i + s - (n_i \text{ mod } s)}{s} - 1\right) + k - n_i, 0\right) \nonumber\\ &= max(n_i + s - (n_i \text{ mod } s) - s + k - n_i,0) \nonumber \\ &= max(k - (n_i \text{ mod } s),0) \label{eq:tf_pad_8} \end{align}

### Final expression

Putting all together, the total padding used by tensorflow's convolution with 'SAME' mode is:

\begin{align} p_i = \begin{cases} max(k - s, 0), & \text{if (n_i \text{ mod } s) = 0} \\ max(k - (n_i \text{ mod } s),0), & \text{if (n_i \text{ mod } s) \neq 0} \end{cases} \label{eq:tf_pad_9} \end{align}

This expression is exactly equal to the ones presented for pad_along_height and pad_along_width in the Convolution section.