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 1x2 matrix, and W is a 2x3 matrix. Let Y be the matrix multiplication of the two matrices:

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}

{'W':
 [[ 3.  4.  5.]
  [ 6.  7.  8.]]
 <NDArray 2x3 @cpu(0)>, 'X':
 [[ 1.  0.]]
 <NDArray 1x2 @cpu(0)>, 'Y':
 [[ 3.  4.  5.]]
 <NDArray 1x3 @cpu(0)>}

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:

grad_W = mx.nd.dot(X, mx.nd.ones_like(Y), transpose_a=True)
grad_W

[[ 1.  1.  1.]
 [ 0.  0.  0.]]
<NDArray 2x3 @cpu(0)>

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 and Adam 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

• Jupyter pip install jupyter

• Basic knowledge of NDArray in MXNet. See the detailed tutorial for NDArray in NDArray - Imperative tensor operations on CPU/GPU

• Understanding of automatic differentiation with autograd

• 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 of shape [LARGE0, D1, .. , Dn] 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:

[[ 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.

data = [[1, 2, 3], [4, 0, 5]]
indices = [0, 2]

RowSparseNDArray supports multidimensional arrays. Given this 3D tensor:

[[[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):

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:

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=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=shape)
{'a':a, 'b':b}

{'a':  <RowSparseNDArray 6x2 @cpu(0)>, 'b':   <RowSparseNDArray 6x2 @cpu(0)>}

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.

# 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)

(numpy.float32, numpy.float16)

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.

a.asnumpy()

array([[ 0.,  0.],
       [ 1.,  2.],
       [ 0.,  0.],
       [ 0.,  0.],
       [ 3.,  4.],
       [ 0.,  0.]], dtype=float32)

You can inspect the internal storage of a RowSparseNDArray by accessing attributes such as indices and data:

# Access data array
data = a.data
# Access indices array
indices = a.indices
{'a.stype': a.stype, 'data':data, 'indices':indices}

{'a.stype': 'row_sparse', 'data':
 [[ 1.  2.]
  [ 3.  4.]]
 <NDArray 2x2 @cpu(0)>, 'indices':
 [1 4]
 <NDArray 2 @cpu(0)>}

Storage Type Conversion

You can convert an NDArray to a RowSparseNDArray and vice versa by using the tostype function:

# 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}

{'dense':
 [[ 1.  1.]
  [ 1.  1.]]
 <NDArray 2x2 @cpu(0)>, 'rsp':
 <RowSparseNDArray 2x2 @cpu(0)>}

You can also convert the storage type by using the cast_storage operator:

# 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}

{'dense':
 [[ 1.  1.]
  [ 1.  1.]]
 <NDArray 2x2 @cpu(0)>, 'rsp':
 <RowSparseNDArray 2x2 @cpu(0)>}

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.

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()}

{'b is a': False, 'b.asnumpy()': array([[ 1.,  1.],
        [ 1.,  1.]], dtype=float32), 'c.asnumpy()': array([[ 1.,  1.],
        [ 1.,  1.]], dtype=float32), 'd.asnumpy()': array([[ 1.,  1.],
        [ 1.,  1.]], dtype=float32)}

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.

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}

{'e.stype': 'row_sparse', 'f.stype': 'row_sparse', 'g.stype': 'default'}

Retain Row Slices

You can retain a subset of row slices from a RowSparseNDArray specified by their row indices.

data = [[1, 2], [3, 4], [5, 6]]
indices = [0, 2, 3]
rsp = mx.nd.sparse.row_sparse_array((data, indices), shape=(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()}

{'rsp.asnumpy()': array([[ 1.,  2.],
        [ 0.,  0.],
        [ 3.,  4.],
        [ 5.,  6.],
        [ 0.,  0.]], dtype=float32), 'rsp_retained':
 <RowSparseNDArray 5x2 @cpu(0)>, 'rsp_retained.asnumpy()': array([[ 1.,  2.],
        [ 0.,  0.],
        [ 0.,  0.],
        [ 0.,  0.],
        [ 0.,  0.]], dtype=float32)}

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 to find what sparse operators are available.

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, indices, indptr), shape=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()}

{'transpose_dot':
 <RowSparseNDArray 5x2 @cpu(0)>, 'transpose_dot.asnumpy()': array([[ 7.,  7.],
        [ 9.,  9.],
        [ 8.,  8.],
        [ 0.,  0.],
        [ 0.,  0.]], dtype=float32)}

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:

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}

{'b.stype': 'row_sparse', 'c.stype': 'default'}

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.

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

{'a.stype': 'row_sparse', 'd.stype': 'default', 'e.stype': 'row_sparse'}

Note that warning messages will be printed when such a storage fallback event happens. If you are using jupyter notebook, the warning message will be printed in your terminal console.

Sparse Optimizers

In MXNet, sparse gradient updates are applied when gradient is in row_sparse storage and the optimizer is created with lazy_update=True. 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

However, with sparse gradient the SGD optimizer uses the following lazy update by default:

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]

This means that the lazy update leads to different optimization results if weight_decay or momentum is non-zero. To disable lazy update, please set lazy_update to be False when creating the optimizer.

# 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=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()}

{'grad.asnumpy()': array([[ 0.,  0.],
        [ 1.,  2.],
        [ 4.,  5.],
        [ 0.,  0.]], dtype=float32), 'momentum.asnumpy()': array([[ 0.,  0.],
        [ 0.,  0.],
        [ 0.,  0.],
        [ 0.,  0.]], dtype=float32), 'weight.asnumpy()': array([[ 1.,  1.],
        [ 1.,  1.],
        [ 1.,  1.],
        [ 1.,  1.]], dtype=float32)}

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()}

{'momentum.asnumpy()': array([[ 0.  ,  0.  ],
        [-0.01, -0.02],
        [-0.04, -0.05],
        [ 0.  ,  0.  ]], dtype=float32),
 'weight.asnumpy()': array([[ 1.        ,  1.        ],
        [ 0.99000001,  0.98000002],
        [ 0.95999998,  0.94999999],
        [ 1.        ,  1.        ]], dtype=float32)}

Note that only mxnet.optimizer.SGD, mxnet.optimizer.Adam, and mxnet.optimizer.AdaGrad support sparse updates in MXNet.

Advanced Topics

GPU Support

By default, RowSparseNDArray operators are executed on CPU. 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 on a cpu, set gpu_device to mx.cpu().

import sys
gpu_device=mx.gpu() # Change this to mx.cpu() in absence of GPUs.
try:
    a = mx.nd.sparse.zeros('row_sparse', (100, 100), ctx=gpu_device)
    a
except mx.MXNetError as err:
    sys.stderr.write(str(err))

Next

Train a Linear Regression Model with Sparse Symbols

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