anirudh2290 commented on a change in pull request #7921: Add three sparse
tutorials
URL: https://github.com/apache/incubator-mxnet/pull/7921#discussion_r141507608
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File path: docs/tutorials/sparse/row_sparse.md
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+
+# RowSparseNDArray - NDArray for Sparse Gradient Updates
+
+## Motivation
+
+Many real world datasets deal with high dimensional sparse feature vectors.
When learning
+the weights of models with sparse datasets, the derived gradients of the
weights could be sparse.
+
+Let's say we perform a matrix multiplication of ``X`` and ``W``, where ``X``
is a 2x2 matrix, and ``W`` is a 2x1 matrix. Let ``Y`` be the matrix
multiplication of the two matrices:
+
+```python
+import mxnet as mx
+X = mx.nd.array([[1,0]])
+W = mx.nd.array([[3,4,5], [6,7,8]])
+Y = mx.nd.dot(X, W)
+{'X': X, 'W': W, 'Y': Y}
+```
+
+As you can see,
+
+```
+Y[0][0] = X[0][0] * W[0][0] + X[0][1] * W[1][0] = 1 * 3 + 0 * 6 = 3
+Y[0][1] = X[0][0] * W[0][1] + X[0][1] * W[1][1] = 1 * 4 + 0 * 7 = 4
+Y[0][2] = X[0][0] * W[0][2] + X[0][1] * W[1][2] = 1 * 5 + 0 * 8 = 5
+```
+
+What about dY / dW, the gradient for ``W``? Let's call it ``grad_W``. To start
with, the shape of ``grad_W`` is the same as that of ``W`` as we are taking the
derivatives with respect to ``W``, which is 2x3. Then we calculate each entry
in ``grad_W`` as follows:
+
+```
+grad_W[0][0] = X[0][0] = 1
+grad_W[0][1] = X[0][0] = 1
+grad_W[0][2] = X[0][0] = 1
+grad_W[1][0] = X[0][1] = 0
+grad_W[1][1] = X[0][1] = 0
+grad_W[1][2] = X[0][1] = 0
+```
+
+As a matter of fact, you can calculate ``grad_W`` by multiplying the transpose
of ``X`` with a matrix of ones:
+
+```python
+grad_W = mx.nd.dot(X, mx.nd.ones_like(Y), transpose_a=True)
+grad_W
+```
+
+As you can see, row 0 of ``grad_W`` contains non-zero values while row 1 of
``grad_W`` does not. Why did that happen?
+If you look at how ``grad_W`` is calculated, notice that since column 1 of
``X`` is filled with zeros, row 1 of ``grad_W`` is filled with zeros too.
+
+In the real world, gradients for parameters that interact with sparse inputs
ususally have gradients where many row slices are completely zeros. Storing and
manipulating such sparse matrices with many row slices of all zeros in the
default dense structure results in wasted memory and processing on the zeros.
More importantly, many gradient based optimization methods such as SGD,
[AdaGrad](https://stanford.edu/~jduchi/projects/DuchiHaSi10_colt.pdf) and
[Adam](https://arxiv.org/pdf/1412.6980.pdf)
+take advantage of sparse gradients and prove to be efficient and effective.
+**In MXNet, the ``RowSparseNDArray`` stores the matrix in ``row sparse``
format, which is designed for arrays of which most row slices are all zeros.**
+In this tutorial, we will describe what the row sparse format is and how to
use RowSparseNDArray for sparse gradient updates in MXNet.
+
+## Prerequisites
+
+To complete this tutorial, we need:
+
+- MXNet. See the instructions for your operating system in [Setup and
Installation](https://mxnet.io/get_started/install.html)
+- [Jupyter](http://jupyter.org/)
+ ```
+ pip install jupyter
+ ```
+- Basic knowledge of NDArray in MXNet. See the detailed tutorial for NDArray
in [NDArray - Imperative tensor operations on
CPU/GPU](https://mxnet.incubator.apache.org/tutorials/basic/ndarray.html)
+- Understanding of [automatic differentiation with
autograd](http://gluon.mxnet.io/chapter01_crashcourse/autograd.html)
+- GPUs - A section of this tutorial uses GPUs. If you don't have GPUs on your
+machine, simply set the variable `gpu_device` (set in the GPUs section of this
+tutorial) to `mx.cpu()`
+
+## Row Sparse Format
+
+A RowSparseNDArray represents a multidimensional NDArray using two separate 1D
arrays:
+`data` and `indices`.
+
+- data: an NDArray of any dtype with shape `[D0, D1, ..., Dn]`.
+- indices: a 1D int64 NDArray with shape `[D0]` with values sorted in
ascending order.
+
+The ``indices`` array stores the indices of the row slices with non-zeros,
+while the values are stored in ``data`` array. The corresponding NDArray
`dense` represented by RowSparseNDArray `rsp` has
+
+``dense[rsp.indices[i], :, :, :, ...] = rsp.data[i, :, :, :, ...]``
+
+A RowSparseNDArray is typically used to represent non-zero row slices of a
large NDArray of shape [LARGE0, D1, .. , Dn] where LARGE0 >> D0 and most row
slices are zeros.
+
+Given this two-dimension matrix:
+
+
+```python
+[[ 1, 2, 3],
+ [ 0, 0, 0],
+ [ 4, 0, 5],
+ [ 0, 0, 0],
+ [ 0, 0, 0]]
+```
+
+The row sparse representation would be:
+- `data` array holds all the non-zero row slices of the array.
+- `indices` array stores the row index for each row slice with non-zero
elements.
+
+
+
+```python
+data = [[1, 2, 3], [4, 0, 5]]
+indices = [0, 2]
+```
+
+`RowSparseNDArray` supports multidimensional arrays. Given this 3D tensor:
+
+
+```python
+[[[1, 0],
+ [0, 2],
+ [3, 4]],
+
+ [[5, 0],
+ [6, 0],
+ [0, 0]],
+
+ [[0, 0],
+ [0, 0],
+ [0, 0]]]
+```
+
+The row sparse representation would be (with `data` and `indices` defined the
same as above):
+
+
+```python
+data = [[[1, 0], [0, 2], [3, 4]], [[5, 0], [6, 0], [0, 0]]]
+indices = [0, 1]
+```
+
+``RowSparseNDArray`` is a subclass of ``NDArray``. If you query **stype** of a
RowSparseNDArray,
+the value will be **"row_sparse"**.
+
+## Array Creation
+
+You can create a `RowSparseNDArray` with data and indices by using the
`row_sparse_array` function:
+
+
+```python
+import mxnet as mx
+import numpy as np
+# Create a RowSparseNDArray with python lists
+shape = (6, 2)
+data_list = [[1, 2], [3, 4]]
+indices_list = [1, 4]
+a = mx.nd.sparse.row_sparse_array(data_list, indices_list, shape)
+# Create a RowSparseNDArray with numpy arrays
+data_np = np.array([[1, 2], [3, 4]])
+indices_np = np.array([1, 4])
+b = mx.nd.sparse.row_sparse_array(data_np, indices_np, shape)
+{'a':a, 'b':b}
+```
+
+## Function Overview
+
+Similar to `CSRNDArray`, the are several functions with `RowSparseNDArray`
that behave the same way. In the code blocks below you can try out these common
functions:
+
+- **.dtype** - to set the data type
+- **.asnumpy** - to cast as a numpy array for inspecting it
+- **.data** - to access the data array
+- **.indices** - to access the indices array
+- **.tostype** - to set the storage type
+- **.cast_storage** - to convert the storage type
+- **.copy** - to copy the array
+- **.copyto** - to copy to deep copy an existing array
+
+
+## Setting Type
+
+You can create a `RowSparseNDArray` from another specifying the element data
type with the option `dtype`, which accepts a numpy type. By default, `float32`
is used.
+
+
+```python
+# Float32 is used by default
+c = mx.nd.sparse.array(a)
+# Create a 16-bit float array
+d = mx.nd.array(a, dtype=np.float16)
+(c.dtype, d.dtype)
+```
+
+## Inspecting Arrays
+
+As with `CSRNDArray`, you can inspect the contents of a `RowSparseNDArray` by
filling
+its contents into a dense `numpy.ndarray` using the `asnumpy` function.
+
+
+```python
+a.asnumpy()
+```
+
+You can inspect the internal storage of a RowSparseNDArray by accessing
attributes such as `indices` and `data`:
+
+
+```python
+# Access data array
+data = a.data
+# Access indices array
+indices = a.indices
+{'a.stype': a.stype, 'data':data, 'indices':indices}
+```
+
+## Storage Type Conversion
+
+You can convert an NDArray to a RowSparseNDArray and vice versa by using the
`tostype` function:
+
+
+```python
+# Create a dense NDArray
+ones = mx.nd.ones((2,2))
+# Cast the storage type from `default` to `row_sparse`
+rsp = ones.tostype('row_sparse')
+# Cast the storage type from `row_sparse` to `default`
+dense = rsp.tostype('default')
+{'rsp':rsp, 'dense':dense}
+```
+
+You can also convert the storage type by using the `cast_storage` operator:
+
+
+```python
+# Create a dense NDArray
+ones = mx.nd.ones((2,2))
+# Cast the storage type to `row_sparse`
+rsp = mx.nd.sparse.cast_storage(ones, 'row_sparse')
+# Cast the storage type to `default`
+dense = mx.nd.sparse.cast_storage(rsp, 'default')
+{'rsp':rsp, 'dense':dense}
+```
+
+## Copies
+
+You can use the `copy` method which makes a deep copy of the array and its
data, and returns a new array.
+We can also use the `copyto` method or the slice operator `[]` to deep copy to
an existing array.
+
+
+```python
+a = mx.nd.ones((2,2)).tostype('row_sparse')
+b = a.copy()
+c = mx.nd.sparse.zeros('row_sparse', (2,2))
+c[:] = a
+d = mx.nd.sparse.zeros('row_sparse', (2,2))
+a.copyto(d)
+{'b is a': b is a, 'b.asnumpy()':b.asnumpy(), 'c.asnumpy()':c.asnumpy(),
'd.asnumpy()':d.asnumpy()}
+```
+
+If the storage types of source array and destination array do not match,
+the storage type of destination array will not change when copying with
`copyto` or the slice operator `[]`. The source array will be temporarily
converted to desired storage type before the copy.
+
+
+```python
+e = mx.nd.sparse.zeros('row_sparse', (2,2))
+f = mx.nd.sparse.zeros('row_sparse', (2,2))
+g = mx.nd.ones(e.shape)
+e[:] = g
+g.copyto(f)
+{'e.stype':e.stype, 'f.stype':f.stype, 'g.stype':g.stype}
+```
+
+## Retain Row Slices
+
+You can retain a subset of row slices from a RowSparseNDArray specified by
their row indices.
+
+
+```python
+data = [[1, 2], [3, 4], [5, 6]]
+indices = [0, 2, 3]
+rsp = mx.nd.sparse.row_sparse_array(data, indices, (5, 2))
+# Retain row 0 and row 1
+rsp_retained = mx.nd.sparse.retain(rsp, mx.nd.array([0, 1]))
+{'rsp.asnumpy()': rsp.asnumpy(), 'rsp_retained': rsp_retained,
'rsp_retained.asnumpy()': rsp_retained.asnumpy()}
+```
+
+## Sparse Operators and Storage Type Inference
+
+Operators that have specialized implementation for sparse arrays can be
accessed in ``mx.nd.sparse``. You can read the [mxnet.ndarray.sparse API
documentation](https://mxnet.io/versions/master/api/python/ndarray/sparse.html)
to find what sparse operators are available.
+
+
+```python
+shape = (3, 5)
+data = [7, 8, 9]
+indptr = [0, 2, 2, 3]
+indices = [0, 2, 1]
+# A csr matrix as lhs
+lhs = mx.nd.sparse.csr_matrix(data, indptr, indices, shape)
+# A dense matrix as rhs
+rhs = mx.nd.ones((3, 2))
+# row_sparse result is inferred from sparse operator dot(csr.T, dense) based
on input stypes
+transpose_dot = mx.nd.sparse.dot(lhs, rhs, transpose_a=True)
+{'transpose_dot': transpose_dot, 'transpose_dot.asnumpy()':
transpose_dot.asnumpy()}
+```
+
+For any sparse operator, the storage type of output array is inferred based on
inputs. You can either read the documentation or inspect the `stype` attribute
of output array to know what storage type is inferred:
+
+
+```python
+a = transpose_dot.copy()
+b = a * 2 # b will be a RowSparseNDArray since zero multiplied by 2 is still
zero
+c = a + mx.nd.ones((5, 2)) # c will be a dense NDArray
+{'b.stype':b.stype, 'c.stype':c.stype}
+```
+
+For operators that don't specialize in sparse arrays, you can still use them
with sparse inputs with some performance penalty.
+In MXNet, dense operators require all inputs and outputs to be in the dense
format.
+
+If sparse inputs are provided, MXNet will convert sparse inputs into dense
ones temporarily so that the dense operator can be used.
+
+If sparse outputs are provided, MXNet will convert the dense outputs generated
by the dense operator into the provided sparse format.
+
+For operators that don't specialize in sparse arrays, you can still use them
with sparse inputs with some performance penalty.
+
+
+```python
+e = mx.nd.sparse.zeros('row_sparse', a.shape)
+d = mx.nd.log(a) # dense operator with a sparse input
+e = mx.nd.log(a, out=e) # dense operator with a sparse output
+{'a.stype':a.stype, 'd.stype':d.stype, 'e.stype':e.stype} # stypes of a and e
will be not changed
+```
+
+Note that warning messages will be printed when such a storage fallback event
happens (work in progress).
+
+## Sparse Optimizers
+
+In MXNet, sparse gradient updates are applied when weight, state and gradient
are all in `row_sparse` storage.
+The sparse optimizers only update the row slices of the weight and the states
whose indices appear
+in `gradient.indices`. For example, the default update rule for SGD optimizer
is:
+
+```
+rescaled_grad = learning_rate * rescale_grad * clip(grad, clip_gradient) +
weight_decay * weight
+state = momentum * state + rescaled_grad
+weight = weight - state
+```
+
+Meanwhile, the sparse update rule for SGD optimizer is:
+
+```
+for row in grad.indices:
+ rescaled_grad[row] = learning_rate * rescale_grad * clip(grad[row],
clip_gradient) + weight_decay * weight[row]
+ state[row] = momentum[row] * state[row] + rescaled_grad[row]
+ weight[row] = weight[row] - state[row]
+```
+
+
+```python
+# Create weight
+shape = (4, 2)
+weight = mx.nd.ones(shape).tostype('row_sparse')
+# Create gradient
+data = [[1, 2], [4, 5]]
+indices = [1, 2]
+grad = mx.nd.sparse.row_sparse_array(data, indices, shape)
+sgd = mx.optimizer.SGD(learning_rate=0.01, momentum=0.01)
+# Create momentum
+momentum = sgd.create_state(0, weight)
+# Before the update
+{"grad.asnumpy()":grad.asnumpy(), "weight.asnumpy()":weight.asnumpy(),
"momentum.asnumpy()":momentum.asnumpy()}
+```
+
+
+```python
+sgd.update(0, weight, grad, momentum)
+# Only row 0 and row 2 are updated for both weight and momentum
+{"weight.asnumpy()":weight.asnumpy(), "momentum.asnumpy()":momentum.asnumpy()}
+```
+
+Note that both
[mxnet.optimizer.SGD](https://mxnet.incubator.apache.org/api/python/optimization.html#mxnet.optimizer.SGD)
+and
[mxnet.optimizer.Adam](https://mxnet.incubator.apache.org/api/python/optimization.html#mxnet.optimizer.Adam)
support sparse updates in MXNet.
+
+## Advanced Topics
+
+### GPU Support
+
+By default, RowSparseNDArray operators are executed on CPU. In MXNet, GPU
support for RowSparseNDArray is experimental
+with only a few sparse operators such as cast_storage and dot.
+
+To create a RowSparseNDArray on gpu, we need to explicitly specify the context:
+
+**Note** If a GPU is not available, an error will be reported in the following
section. In order to execute it a cpu, set gpu_device to mx.cpu().
Review comment:
In order to execute it a cpu -> In order to execute it on a cpu
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