Difference on performance between numpy and matlab

It would be wrong to say “Matlab is always faster than NumPy” or vice
versa. Often their performance is comparable. When using NumPy, to get good
performance you have to keep in mind that NumPy’s speed comes from calling
underlying functions written in C/C++/Fortran. It performs well when you apply
those functions to whole arrays. In general, you get poorer performance when you call those NumPy function on smaller arrays or scalars in a Python loop.

What’s wrong with a Python loop you ask? Every iteration through the Python loop is
a call to a next method. Every use of [] indexing is a call to a
__getitem__ method. Every += is a call to __iadd__. Every dotted attribute
lookup (such as in like np.dot) involves function calls. Those function calls
add up to a significant hinderance to speed. These hooks give Python
expressive power — indexing for strings means something different than indexing
for dicts for example. Same syntax, different meanings. The magic is accomplished by giving the objects different __getitem__ methods.

But that expressive power comes at a cost in speed. So when you don’t need all
that dynamic expressivity, to get better performance, try to limit yourself to
NumPy function calls on whole arrays.

So, remove the for-loop; use “vectorized” equations when possible. For example, instead of

for i in range(m):
    delta3 = -(x[i,:]-a3[i,:])*a3[i,:]* (1 - a3[i,:])    

you can compute delta3 for each i all at once:

delta3 = -(x-a3)*a3*(1-a3)

Whereas in the for-loop delta3 is a vector, using the vectorized equation delta3 is a matrix.

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Some of the computations in the for-loop do not depend on i and therefore should be lifted outside the loop. For example, sum2 looks like a constant:

sum2 = sparse.beta*(-float(sparse.rho)/rhoest + float(1.0 - sparse.rho) / (1.0 - rhoest) )

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Here is a runnable example with an alternative implementation (alt) of your code (orig).

My timeit benchmark shows a 6.8x improvement in speed:

In [52]: %timeit orig()
1 loops, best of 3: 495 ms per loop

In [53]: %timeit alt()
10 loops, best of 3: 72.6 ms per loop

---

import numpy as np


class Bunch(object):
    """ http://code.activestate.com/recipes/52308 """
    def __init__(self, **kwds):
        self.__dict__.update(kwds)

m, n, p = 10 ** 4, 64, 25

sparse = Bunch(
    theta1=np.random.random((p, n)),
    theta2=np.random.random((n, p)),
    b1=np.random.random((p, 1)),
    b2=np.random.random((n, 1)),
)

x = np.random.random((m, n))
a3 = np.random.random((m, n))
a2 = np.random.random((m, p))
a1 = np.random.random((m, n))
sum2 = np.random.random((p, ))
sum2 = sum2[:, np.newaxis]

def orig():
    partial_j1 = np.zeros(sparse.theta1.shape)
    partial_j2 = np.zeros(sparse.theta2.shape)
    partial_b1 = np.zeros(sparse.b1.shape)
    partial_b2 = np.zeros(sparse.b2.shape)
    delta3t = (-(x - a3) * a3 * (1 - a3)).T
    for i in range(m):
        delta3 = delta3t[:, i:(i + 1)]
        sum1 = np.dot(sparse.theta2.T, delta3)
        delta2 = (sum1 + sum2) * a2[i:(i + 1), :].T * (1 - a2[i:(i + 1), :].T)
        partial_j1 += np.dot(delta2, a1[i:(i + 1), :])
        partial_j2 += np.dot(delta3, a2[i:(i + 1), :])
        partial_b1 += delta2
        partial_b2 += delta3
        # delta3: (64, 1)
        # sum1: (25, 1)
        # delta2: (25, 1)
        # a1[i:(i+1),:]: (1, 64)
        # partial_j1: (25, 64)
        # partial_j2: (64, 25)
        # partial_b1: (25, 1)
        # partial_b2: (64, 1)
        # a2[i:(i+1),:]: (1, 25)
    return partial_j1, partial_j2, partial_b1, partial_b2


def alt():
    delta3 = (-(x - a3) * a3 * (1 - a3)).T
    sum1 = np.dot(sparse.theta2.T, delta3)
    delta2 = (sum1 + sum2) * a2.T * (1 - a2.T)
    # delta3: (64, 10000)
    # sum1: (25, 10000)
    # delta2: (25, 10000)
    # a1: (10000, 64)
    # a2: (10000, 25)
    partial_j1 = np.dot(delta2, a1)
    partial_j2 = np.dot(delta3, a2)
    partial_b1 = delta2.sum(axis=1)
    partial_b2 = delta3.sum(axis=1)
    return partial_j1, partial_j2, partial_b1, partial_b2

answer = orig()
result = alt()
for a, r in zip(answer, result):
    try:
        assert np.allclose(np.squeeze(a), r)
    except AssertionError:
        print(a.shape)
        print(r.shape)
        raise

---

Tip: Notice that I left in the comments the shape of all the intermediate arrays. Knowing the shape of the arrays helped me understand what your code was doing. The shape of the arrays can help guide you toward the right NumPy functions to use. Or at least, paying attention to the shapes can help you know if an operation is sensible. For example, when you compute

np.dot(A, B)

and A.shape = (n, m) and B.shape = (m, p), then np.dot(A, B) will be an array of shape (n, p).

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It can help to build the arrays in C_CONTIGUOUS-order (at least, if using np.dot). There might be as much as a 3x speed up by doing so:

Below, x is the same as xf except that x is C_CONTIGUOUS and
xf is F_CONTIGUOUS — and the same relationship for y and yf.

import numpy as np

m, n, p = 10 ** 4, 64, 25
x = np.random.random((n, m))
xf = np.asarray(x, order='F')

y = np.random.random((m, n))
yf = np.asarray(y, order='F')

assert np.allclose(x, xf)
assert np.allclose(y, yf)
assert np.allclose(np.dot(x, y), np.dot(xf, y))
assert np.allclose(np.dot(x, y), np.dot(xf, yf))

%timeit benchmarks show the difference in speed:

In [50]: %timeit np.dot(x, y)
100 loops, best of 3: 12.9 ms per loop

In [51]: %timeit np.dot(xf, y)
10 loops, best of 3: 27.7 ms per loop

In [56]: %timeit np.dot(x, yf)
10 loops, best of 3: 21.8 ms per loop

In [53]: %timeit np.dot(xf, yf)
10 loops, best of 3: 33.3 ms per loop

---

Regarding benchmarking in Python:

It can be misleading to use the difference in pairs of time.time() calls to benchmark the speed of code in Python.
You need to repeat the measurement many times. It’s better to disable the automatic garbage collector. It is also important to measure large spans of time (such as at least 10 seconds worth of repetitions) to avoid errors due to poor resolution in the clock timer and to reduce the significance of time.time call overhead. Instead of writing all that code yourself, Python provides you with the timeit module. I’m essentially using that to time the pieces of code, except that I’m calling it through an IPython terminal for convenience.

I’m not sure if this is affecting your benchmarks, but be aware it could make a difference. In the question I linked to, according to time.time two pieces of code differed by a factor of 1.7x while benchmarks using timeit showed the pieces of code ran in essentially identical amounts of time.