Seven Strategies for Optimizing Numerical Code

Transcript

1. Performance Python
   7 Strategies for Optimizing
   Your Numerical Code
   Jake VanderPlas @jakevdp
   PyCon 2018
   

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2. Python is Fast.
   Dynamic, interpreted, & flexible: fast Development
   

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3. Python is Slow.
   CPython has constant overhead per operation
   

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4. Python is Slow.
   CPython has constant overhead per operation
   

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5. Fortran is 100x faster for this simple task!
   Python is Slow.
   CPython has constant overhead per operation
   

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6. The best of both worlds?
   

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7. Seven Strategies
   For Optimizing Your
   Numerical Python Code
   

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8. Example:
   K-means
   Clustering
   

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9. Example:
   K-means
   Clustering
   Algorithm:
   1. Choose some Cluster Centers
   2. Repeat:
   a. Assign points to nearest center
   b. Update center to mean of points
   c. Check if Converged
   

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10. Example:
    K-means
    Clustering
    Algorithm:
    1. Choose some Cluster Centers
    2. Repeat:
    a. Assign points to nearest center
    b. Update center to mean of points
    c. Check if Converged
    

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11. Example:
    K-means
    Clustering
    Algorithm:
    1. Choose some Cluster Centers
    2. Repeat:
    a. Assign points to nearest center
    b. Update center to mean of points
    c. Check if Converged
    

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12. Example:
    K-means
    Clustering
    Algorithm:
    1. Choose some Cluster Centers
    2. Repeat:
    a. Assign points to nearest center
    b. Update center to mean of points
    c. Check if Converged
    

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13. Example:
    K-means
    Clustering
    Algorithm:
    1. Choose some Cluster Centers
    2. Repeat:
    a. Assign points to nearest center
    b. Update center to mean of points
    c. Check if Converged
    

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14. Example:
    K-means
    Clustering
    Algorithm:
    1. Choose some Cluster Centers
    2. Repeat:
    a. Assign points to nearest center
    b. Update center to mean of points
    c. Check if Converged
    

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15. Example:
    K-means
    Clustering
    Algorithm:
    1. Choose some Cluster Centers
    2. Repeat:
    a. Assign points to nearest center
    b. Update center to mean of points
    c. Check if Converged
    

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16. Example:
    K-means
    Clustering
    Algorithm:
    1. Choose some Cluster Centers
    2. Repeat:
    a. Assign points to nearest center
    b. Update center to mean of points
    c. Check if Converged
    

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17. Example:
    K-means
    Clustering
    Algorithm:
    1. Choose some Cluster Centers
    2. Repeat:
    a. Assign points to nearest center
    b. Update center to mean of points
    c. Check if Converged
    

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18. Example:
    K-means
    Clustering
    Algorithm:
    1. Choose some Cluster Centers
    2. Repeat:
    a. Assign points to nearest center
    b. Update center to mean of points
    c. Check if Converged
    

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19. Example:
    K-means
    Clustering
    Algorithm:
    1. Choose some Cluster Centers
    2. Repeat:
    a. Assign points to nearest center
    b. Update center to mean of points
    c. Check if Converged
    

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20. Example:
    K-means
    Clustering
    Algorithm:
    1. Choose some Cluster Centers
    2. Repeat:
    a. Assign points to nearest center
    b. Update center to mean of points
    c. Check if Converged
    

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21. Example:
    K-means
    Clustering
    Algorithm:
    1. Choose some Cluster Centers
    2. Repeat:
    a. Assign points to nearest center
    b. Update center to mean of points
    c. Check if Converged
    

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22. Example:
    K-means
    Clustering
    Algorithm:
    1. Choose some Cluster Centers
    2. Repeat:
    a. Assign points to nearest center
    b. Update center to mean of points
    c. Check if Converged
    

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23. Example:
    K-means
    Clustering
    Algorithm:
    1. Choose some Cluster Centers
    2. Repeat:
    a. Assign points to nearest center
    b. Update center to mean of points
    c. Check if Converged
    

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24. Example:
    K-means
    Clustering
    Algorithm:
    1. Choose some Cluster Centers
    2. Repeat:
    a. Assign points to nearest center
    b. Update center to mean of points
    c. Check if Converged
    

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25. Example:
    K-means
    Clustering
    Algorithm:
    1. Choose some Cluster Centers
    2. Repeat:
    a. Assign points to nearest center
    b. Update center to mean of points
    c. Check if Converged
    

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26. Example:
    K-means
    Clustering
    Algorithm:
    1. Choose some Cluster Centers
    2. Repeat:
    a. Assign points to nearest center
    b. Update center to mean of points
    c. Check if Converged
    

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27. Example:
    K-means
    Clustering
    Algorithm:
    1. Choose some Cluster Centers
    2. Repeat:
    a. Assign points to nearest center
    b. Update center to mean of points
    c. Check if Converged
    

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28. Example:
    K-means
    Clustering
    Algorithm:
    1. Choose some Cluster Centers
    2. Repeat:
    a. Assign points to nearest center
    b. Update center to mean of points
    c. Check if Converged
    

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29. Example:
    K-means
    Clustering
    Algorithm:
    1. Choose some Cluster Centers
    2. Repeat:
    a. Assign points to nearest center
    b. Update center to mean of points
    c. Check if Converged
    

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30. Implementing K Means in Python
    

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31. Python Implementation
    def dist(x, y):
    return sum((xi - yi) ** 2
    for xi, yi in zip(x, y))
    def find_labels(points, centers):
    labels = []
    for point in points:
    distances = [dist(point, center)
    for center in centers]
    labels.append(distances.index(min(distances)))
    return labels
    

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32. Python Implementation
    def dist(x, y):
    return sum((xi - yi) ** 2
    for xi, yi in zip(x, y))
    def find_labels(points, centers):
    labels = []
    for point in points:
    distances = [dist(point, center)
    for center in centers]
    labels.append(distances.index(min(distances)))
    return labels
    

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33. Python Implementation
    def dist(x, y):
    return sum((xi - yi) ** 2
    for xi, yi in zip(x, y))
    def find_labels(points, centers):
    labels = []
    for point in points:
    distances = [dist(point, center)
    for center in centers]
    labels.append(distances.index(min(distances)))
    return labels
    

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34. Python Implementation
    def dist(x, y):
    return sum((xi - yi) ** 2
    for xi, yi in zip(x, y))
    def find_labels(points, centers):
    labels = []
    for point in points:
    distances = [dist(point, center)
    for center in centers]
    labels.append(distances.index(min(distances)))
    return labels
    

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35. Python Implementation
    def compute_centers(points, labels):
    n_centers = len(set(labels))
    n_dims = len(points[0])
    centers = [[0 for i in range(n_dims)]
    for j in range(n_centers)]
    counts = [0 for j in range(n_centers)]
    for label, point in zip(labels, points):
    counts[label] += 1
    centers[label] = [a + b for a, b in
    zip(centers[label], point)]
    return [[x / count for x in center]
    for center, count in zip(centers, counts)]
    

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36. Python Implementation
    def compute_centers(points, labels):
    n_centers = len(set(labels))
    n_dims = len(points[0])
    centers = [[0 for i in range(n_dims)]
    for j in range(n_centers)]
    counts = [0 for j in range(n_centers)]
    for label, point in zip(labels, points):
    counts[label] += 1
    centers[label] = [a + b for a, b in
    zip(centers[label], point)]
    return [[x / count for x in center]
    for center, count in zip(centers, counts)]
    

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37. Python Implementation
    def compute_centers(points, labels):
    n_centers = len(set(labels))
    n_dims = len(points[0])
    centers = [[0 for i in range(n_dims)]
    for j in range(n_centers)]
    counts = [0 for j in range(n_centers)]
    for label, point in zip(labels, points):
    counts[label] += 1
    centers[label] = [a + b for a, b in
    zip(centers[label], point)]
    return [[x / count for x in center]
    for center, count in zip(centers, counts)]
    

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38. Python Implementation
    def compute_centers(points, labels):
    n_centers = len(set(labels))
    n_dims = len(points[0])
    centers = [[0 for i in range(n_dims)]
    for j in range(n_centers)]
    counts = [0 for j in range(n_centers)]
    for label, point in zip(labels, points):
    counts[label] += 1
    centers[label] = [a + b for a, b in
    zip(centers[label], point)]
    return [[x / count for x in center]
    for center, count in zip(centers, counts)]
    

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39. Python Implementation
    def compute_centers(points, labels):
    n_centers = len(set(labels))
    n_dims = len(points[0])
    centers = [[0 for i in range(n_dims)]
    for j in range(n_centers)]
    counts = [0 for j in range(n_centers)]
    for label, point in zip(labels, points):
    counts[label] += 1
    centers[label] = [a + b for a, b in
    zip(centers[label], point)]
    return [[x / count for x in center]
    for center, count in zip(centers, counts)]
    

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40. Python Implementation
    def kmeans(points, n_clusters):
    centers = points[-n_clusters:].tolist()
    while True:
    old_centers = centers
    labels = find_labels(points, centers)
    centers = compute_centers(points, labels)
    if centers == old_centers:
    break
    return labels
    %timeit kmeans(points, 10)
    7.44 s ± 122 ms per loop (mean ± std. dev. of 7 runs,
    1 loop each)
    

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41. Python Implementation
    def kmeans(points, n_clusters):
    centers = points[-n_clusters:].tolist()
    while True:
    old_centers = centers
    labels = find_labels(points, centers)
    centers = compute_centers(points, labels)
    if centers == old_centers:
    break
    return labels
    %timeit kmeans(points, 10)
    7.44 s ± 122 ms per loop (mean ± std. dev. of 7 runs,
    1 loop each)
    

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42. Python Implementation
    def kmeans(points, n_clusters):
    centers = points[-n_clusters:].tolist()
    while True:
    old_centers = centers
    labels = find_labels(points, centers)
    centers = compute_centers(points, labels)
    if centers == old_centers:
    break
    return labels
    %timeit kmeans(points, 10)
    7.44 s ± 122 ms per loop (mean ± std. dev. of 7 runs,
    1 loop each)
    

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43. Python Implementation
    def kmeans(points, n_clusters):
    centers = points[-n_clusters:].tolist()
    while True:
    old_centers = centers
    labels = find_labels(points, centers)
    centers = compute_centers(points, labels)
    if centers == old_centers:
    break
    return labels
    %timeit kmeans(points, 10)
    7.44 s ± 122 ms per loop (mean ± std. dev. of 7 runs,
    1 loop each)
    

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44. Python Implementation
    def kmeans(points, n_clusters):
    centers = points[-n_clusters:].tolist()
    while True:
    old_centers = centers
    labels = find_labels(points, centers)
    centers = compute_centers(points, labels)
    if centers == old_centers:
    break
    return labels
    %timeit kmeans(points, 10)
    7.44 s ± 122 ms per loop (mean ± std. dev. of 7 runs,
    1 loop each)
    

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45. Python Implementation
    def kmeans(points, n_clusters):
    centers = points[-n_clusters:].tolist()
    while True:
    old_centers = centers
    labels = find_labels(points, centers)
    centers = compute_centers(points, labels)
    if centers == old_centers:
    break
    return labels
    %timeit kmeans(points, 10)
    7.44 s ± 122 ms per loop (mean ± std. dev. of 7 runs,
    1 loop each)
    

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46. What to do when Python is too slow?
    

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47. Seven Strategies:
    1. Line Profiling
    

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48. “Premature optimization is the root of all evil”
    - Donald Knuth
    Seven Strategies:
    1. Line Profiling
    

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49. Line Profiling
    %load_ext line_profiler
    %lprun -f kmeans kmeans(points, 10)
    Timer unit: 1e-06 s
    Total time: 11.8153 s
    File:
    Function: kmeans at line 27
    Line # Hits Time Per Hit % Time Line Contents
    ==============================================================
    27 def kmeans(points, n_clusters):
    28 1 16 16.0 0.0 centers = points[-n_clusters:].
    29 1 2 2.0 0.0 while True:
    30 54 55 1.0 0.0 old_centers = centers
    31 54 11012265 203930.8 93.2 labels = find_labels(points
    32 54 802873 14868.0 6.8 centers = compute_centers(p
    labels)
    33 54 116 2.1 0.0 if centers == old_centers:
    34 1 0 0.0 0.0 break
    35 1 1 1.0 0.0 return labels
    

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50. How can we optimize repeated
    operations on arrays?
    

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51. Seven Strategies:
    1. Line Profiling
    2. Numpy Vectorization
    

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52. def dist(x, y):
    return sum((xi - yi) ** 2
    for xi, yi in zip(x, y))
    def find_labels(points, centers):
    labels = []
    for point in points:
    distances = [dist(point, center)
    for center in centers]
    labels.append(distances.index(min(distances)))
    return labels
    Original Code
    

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53. def dist(x, y):
    return sum((xi - yi) ** 2
    for xi, yi in zip(x, y))
    def find_labels(points, centers):
    labels = []
    for point in points:
    distances = [dist(point, center)
    for center in centers]
    labels.append(distances.index(min(distances)))
    return labels
    import numpy as np
    def find_labels(points, centers):
    diff = (points[:, None, :] - centers) ** 2
    distances = diff.sum(-1)
    return distances.argmin(1)
    Original Code
    Numpy Code
    

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54. def dist(x, y):
    return sum((xi - yi) ** 2
    for xi, yi in zip(x, y))
    def find_labels(points, centers):
    labels = []
    for point in points:
    distances = [dist(point, center)
    for center in centers]
    labels.append(distances.index(min(distances)))
    return labels
    import numpy as np
    def find_labels(points, centers):
    diff = (points[:, None, :] - centers) ** 2
    distances = diff.sum(-1)
    return distances.argmin(1)
    Original Code
    Numpy Code
    

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55. def dist(x, y):
    return sum((xi - yi) ** 2
    for xi, yi in zip(x, y))
    def find_labels(points, centers):
    labels = []
    for point in points:
    distances = [dist(point, center)
    for center in centers]
    labels.append(distances.index(min(distances)))
    return labels
    import numpy as np
    def find_labels(points, centers):
    diff = (points[:, None, :] - centers) ** 2
    distances = diff.sum(-1)
    return distances.argmin(1)
    Original Code
    Numpy Code
    

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56. def dist(x, y):
    return sum((xi - yi) ** 2
    for xi, yi in zip(x, y))
    def find_labels(points, centers):
    labels = []
    for point in points:
    distances = [dist(point, center)
    for center in centers]
    labels.append(distances.index(min(distances)))
    return labels
    import numpy as np
    def find_labels(points, centers):
    diff = (points[:, None, :] - centers) ** 2
    distances = diff.sum(-1)
    return distances.argmin(1)
    Original Code
    Numpy Code
    

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57. Timer unit: 1e-06 s
    Total time: 0.960594 s
    File:
    Function: kmeans at line 23
    Line # Hits Time Per Hit % Time Line Contents
    ==============================================================
    23 def kmeans(points, n_clusters):
    24 1 11 11.0 0.0 centers = points[-n_clusters:].
    25 1 0 0.0 0.0 while True:
    26 54 50 0.9 0.0 old_centers = centers
    27 54 87758 1625.1 9.1 labels = find_labels(points
    28 54 872625 16159.7 90.8 centers = compute_centers(p
    29 54 149 2.8 0.0 if centers == old_centers:
    30 1 1 1.0 0.0 break
    31 1 0 0.0 0.0 return labels
    %lprun -f kmeans kmeans(points, 10)
    

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58. %lprun -f kmeans kmeans(points, 10)
    Timer unit: 1e-06 s
    Total time: 0.960594 s
    File:
    Function: kmeans at line 23
    Line # Hits Time Per Hit % Time Line Contents
    ==============================================================
    23 def kmeans(points, n_clusters):
    24 1 11 11.0 0.0 centers = points[-n_clusters:].
    25 1 0 0.0 0.0 while True:
    26 54 50 0.9 0.0 old_centers = centers
    27 54 87758 1625.1 9.1 labels = find_labels(points
    28 54 872625 16159.7 90.8 centers = compute_centers(p
    29 54 149 2.8 0.0 if centers == old_centers:
    30 1 1 1.0 0.0 break
    31 1 0 0.0 0.0 return labels
    

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59. def compute_centers(points, labels):
    n_centers = len(set(labels))
    n_dims = len(points[0])
    centers = [[0 for i in range(n_dims)]
    for j in range(n_centers)]
    counts = [0 for j in range(n_centers)]
    for label, point in zip(labels, points):
    counts[label] += 1
    centers[label] = [a + b for a, b in
    zip(centers[label], point)]
    return [[x / count for x in center]
    for center, count in zip(centers, counts)]
    Original Code
    

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60. def compute_centers(points, labels):
    n_centers = len(set(labels))
    n_dims = len(points[0])
    centers = [[0 for i in range(n_dims)]
    for j in range(n_centers)]
    counts = [0 for j in range(n_centers)]
    for label, point in zip(labels, points):
    counts[label] += 1
    centers[label] = [a + b for a, b in
    zip(centers[label], point)]
    return [[x / count for x in center]
    for center, count in zip(centers, counts)]
    Original Code
    Numpy Code
    def compute_centers(points, labels):
    n_centers = len(set(labels))
    return np.array([points[labels == i].mean(0)
    for i in range(n_centers)])
    

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61. def compute_centers(points, labels):
    n_centers = len(set(labels))
    n_dims = len(points[0])
    centers = [[0 for i in range(n_dims)]
    for j in range(n_centers)]
    counts = [0 for j in range(n_centers)]
    for label, point in zip(labels, points):
    counts[label] += 1
    centers[label] = [a + b for a, b in
    zip(centers[label], point)]
    return [[x / count for x in center]
    for center, count in zip(centers, counts)]
    Original Code
    Numpy Code
    def compute_centers(points, labels):
    n_centers = len(set(labels))
    return np.array([points[labels == i].mean(0)
    for i in range(n_centers)])
    

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62. def compute_centers(points, labels):
    n_centers = len(set(labels))
    n_dims = len(points[0])
    centers = [[0 for i in range(n_dims)]
    for j in range(n_centers)]
    counts = [0 for j in range(n_centers)]
    for label, point in zip(labels, points):
    counts[label] += 1
    centers[label] = [a + b for a, b in
    zip(centers[label], point)]
    return [[x / count for x in center]
    for center, count in zip(centers, counts)]
    Original Code
    Numpy Code
    def compute_centers(points, labels):
    n_centers = len(set(labels))
    return np.array([points[labels == i].mean(0)
    for i in range(n_centers)])
    

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63. %timeit kmeans(points, 10)
    131 ms ± 3.68 ms per loop (mean ± std. dev. of 7 runs,
    10 loops each)
    Down from 7.44 seconds to 0.13 seconds!
    Key: repeated operations pushed into a compiled layer:
    Python overhead per array rather than per array element.
    

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64. Advantages:
    - Python overhead per array rather than per array
    element
    - Compact domain specific language for array
    operations
    - NumPy is widely available
    Disadvantages:
    - Batch operations can lead to excessive memory
    usage
    - Different way of thinking about writing code
    Recommendation: Use NumPy everywhere!
    

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65. Deeper dive into NumPy Vectorization
    “Losing your Loops” / PyCon 2015
    

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66. Seven Strategies:
    1. Line Profiling
    2. Numpy Vectorization
    3. Specialized Data Structures
    Scipy
    

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67. Scipy
    Numpy Code
    import numpy as np
    def find_labels(points, centers):
    diff = (points[:, None, :] - centers) ** 2
    distances = diff.sum(-1)
    return distances.argmin(1)
    

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68. Scipy
    from scipy.spatial import cKDTree
    def find_labels(points, centers):
    distances, labels = cKDTree(centers).query(points, 1)
    return labels
    KD-Tree Code Numpy Code
    import numpy as np
    def find_labels(points, centers):
    diff = (points[:, None, :] - centers) ** 2
    distances = diff.sum(-1)
    return distances.argmin(1)
    KD-Tree:
    Data structure designed for nearest neighbor searches
    

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69. Numpy Code
    def compute_centers(points, labels):
    n_centers = len(set(labels))
    return np.array([points[labels == i].mean(0)
    for i in range(n_centers)])
    

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70. Pandas Code Numpy Code
    import pandas as pd
    def compute_centers(points, labels):
    df = pd.DataFrame(points)
    return df.groupby(labels).mean().values
    def compute_centers(points, labels):
    n_centers = len(set(labels))
    return np.array([points[labels == i].mean(0)
    for i in range(n_centers)])
    Pandas Dataframe:
    Efficient structure for group-wise operations
    

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71. %timeit kmeans(points, 10)
    102 ms ± 2.52 ms per loop (mean ± std. dev. of 7 runs,
    1 loop each)
    Compared to:
    - 7.44 Seconds in Python
    - 131 ms with NumPy
    Scipy
    

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72. Other Useful Data Structures
    scipy.spatial
    for spatial queries like distances, nearest neighbors, etc.
    pandas
    for SQL-like grouping & aggregation
    xarray
    for grouping across multiple dimensions
    scipy.sparse
    sparse matrices for 2-dimensional structured data
    sparse package
    for N-dimensional structured data
    scipy.sparse.csgraph
    for graph-like problems (e.g. finding shortest paths)
    

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73. Advantages:
    - Often fastest possible way to solve a particular
    problem
    Disadvantages:
    - Requires broad & deep understanding of both
    algorithms and their available implementations
    Recommendation: Use whenever possible!
    Scipy
    

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74. Seven Strategies:
    1. Line Profiling
    2. Numpy Vectorization
    3. Specialized Data Structures
    4. Cython
    

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75. def dist(x, y):
    dist = 0
    for i in range(len(x)):
    dist += (x[i] - y[i]) ** 2
    return dist
    def find_labels(points, centers):
    labels = []
    for point in points:
    distances = [dist(point, center) for center in centers]
    labels.append(distances.index(min(distances)))
    return labels
    centers = points[:10]
    %timeit find_labels(points, centers)
    122 ms ± 5.82 ms per loop (mean ± std. dev. of 7
    runs, 10 loops each)
    

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76. %%cython
    cimport numpy as np
    cdef double dist(double[:] x, double[:] y):
    cdef double dist = 0
    for i in range(len(x)):
    dist += (x[i] - y[i]) ** 2
    return dist
    def find_labels(points, centers):
    labels = []
    for point in points:
    distances = [dist(point, center) for center in centers]
    labels.append(distances.index(min(distances)))
    return labels
    centers = points[:10]
    %timeit find_labels(points, centers)
    97.7 ms ± 12.2 ms per loop (mean ± std. dev. of 7
    runs, 10 loops each)
    

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77. %%cython
    cimport numpy as np
    cdef double dist(double[:] x, double[:] y):
    cdef double dist = 0
    for i in range(len(x)):
    dist += (x[i] - y[i]) ** 2
    return dist
    def find_labels(points, centers):
    labels = []
    for point in points:
    distances = [dist(point, center) for center in centers]
    labels.append(distances.index(min(distances)))
    return labels
    centers = points[:10]
    %timeit find_labels(points, centers)
    97.7 ms ± 12.2 ms per loop (mean ± std. dev. of 7
    runs, 10 loops each)
    

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78. def find_labels(points, centers):
    labels = []
    for point in points:
    distances = [dist(point, center) for center in centers]
    labels.append(distances.index(min(distances)))
    return labels
    centers = points[:10]
    %timeit find_labels(points, centers)
    97.7 ms ± 12.2 ms per loop (mean ± std. dev. of 7
    runs, 10 loops each)
    

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79. def find_labels(double[:, :] points, double[:, :] centers):
    cdef int n_points = points.shape[0]
    cdef int n_centers = centers.shape[0]
    cdef double[:] labels = np.zeros(n_points)
    cdef double distance, nearest_distance
    cdef int nearest_index
    for i in range(n_points):
    nearest_distance = np.inf
    for j in range(n_centers):
    distance = dist(points[i], centers[j])
    if distance < nearest_distance:
    nearest_distance = distance
    nearest_index = j
    labels[i] = nearest_index
    return np.asarray(labels)
    centers = points[:10]
    %timeit find_labels(points, centers)
    1.72 ms ± 27.3 µs per loop (mean ± std. dev. of 7
    runs, 1000 loops each)
    

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80. def find_labels(double[:, :] points, double[:, :] centers):
    cdef int n_points = points.shape[0]
    cdef int n_centers = centers.shape[0]
    cdef double[:] labels = np.zeros(n_points)
    cdef double distance, nearest_distance
    cdef int nearest_index
    for i in range(n_points):
    nearest_distance = np.inf
    for j in range(n_centers):
    distance = dist(points[i], centers[j])
    if distance < nearest_distance:
    nearest_distance = distance
    nearest_index = j
    labels[i] = nearest_index
    return np.asarray(labels)
    centers = points[:10]
    %timeit find_labels(points, centers)
    1.72 ms ± 27.3 µs per loop (mean ± std. dev. of 7
    runs, 1000 loops each)
    

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81. def find_labels(double[:, :] points, double[:, :] centers):
    cdef int n_points = points.shape[0]
    cdef int n_centers = centers.shape[0]
    cdef double[:] labels = np.zeros(n_points)
    cdef double distance, nearest_distance
    cdef int nearest_index
    for i in range(n_points):
    nearest_distance = np.inf
    for j in range(n_centers):
    distance = dist(points[i], centers[j])
    if distance < nearest_distance:
    nearest_distance = distance
    nearest_index = j
    labels[i] = nearest_index
    return np.asarray(labels)
    centers = points[:10]
    %timeit find_labels(points, centers)
    1.72 ms ± 27.3 µs per loop (mean ± std. dev. of 7
    runs, 1000 loops each)
    

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82. Advantages:
    - Python-like code at C-like speeds!
    Disadvantages:
    - Explicit type annotation can be cumbersome
    - Often requires restructuring code
    - Code build becomes more complicated
    Recommendation: use for operations that can’t easily
    be expressed in NumPy
    

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83. Seven Strategies:
    1. Line Profiling
    2. Numpy Vectorization
    3. Specialized Data Structures
    4. Cython
    5. Numba
    

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84. def dist(x, y):
    dist = 0
    for i in range(len(x)):
    dist += (x[i] - y[i]) ** 2
    return dist
    def find_labels(points, centers):
    labels = []
    min_dist = np.inf
    min_label = 0
    for i in range(len(points)):
    for j in range(len(centers)):
    distance = dist(points[i], centers[j])
    if distance < min_dist:
    min_dist
    , min_label = distance, j
    labels.append(min_label)
    return labels
    centers = points[:10]
    %timeit find_labels(points, centers)
    97.7 ms ± 12.2 ms per loop (mean ± std. dev. of 7
    runs, 10 loops each)
    

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85. import numba
    @numba.jit(nopython=True)
    def dist(x, y):
    dist = 0
    for i in range(len(x)):
    dist += (x[i] - y[i]) ** 2
    return dist
    @numba.jit(nopython=True)
    def find_labels(points, centers):
    labels = []
    min_dist = np.inf
    min_label = 0
    for i in range(len(points)):
    for j in range(len(centers)):
    distance = dist(points[i], centers[j])
    if distance < min_dist:
    min_dist
    , min_label = distance, j
    labels.append(min_label)
    return labels
    centers = points[:10]
    %timeit find_labels(points, centers)
    1.47 ms ± 14.2 µs per loop (mean ± std. dev. of 7
    runs, 1000 loops each)
    

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86. Advantages:
    - Python code JIT-compiled to fortran speeds!
    Disadvantages:
    - Heavy dependency chain (LLVM)
    - Some Python constructs not supported
    - Still a bit finnicky
    Recommendation: use for analysis scripts where
    dependencies are not a concern.
    See my blog post Optimizing Python in the Real World: NumPy, Numba, and the NUFFT
    http://jakevdp.github.io/blog/2015/02/24/optimizing-python-with-numpy-and-numba/
    

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87. Seven Strategies:
    1. Line Profiling
    2. Numpy Vectorization
    3. Specialized Data Structures
    4. Cython
    5. Numba
    6. Dask
    

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88. Parallel Computation:
    http://dask.pydata.org/
    import numpy as np
    a = np.random.randn(1000)
    b = a * 4
    b_min = b.min()
    print(b_min)
    -13.2982888603
    Typical data manipulation with NumPy:
    

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89. Parallel Computation:
    http://dask.pydata.org/
    import dask.array as da
    a2 = da.from_array(a, chunks=200)
    b2 = a2 * 4
    b2_min = b2.min()
    print(b2_min)
    dask.arraydtype=float64, chunksize=()>
    Same operation with dask
    

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90. Parallel Computation:
    http://dask.pydata.org/
    import dask.array as da
    a2 = da.from_array(a, chunks=200)
    b2 = a2 * 4
    b2_min = b2.min()
    print(b2_min)
    dask.arraydtype=float64, chunksize=()>
    Same operation with dask
    “Task Graph”
    

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91. Parallel Computation:
    http://dask.pydata.org/
    import dask.array as da
    a2 = da.from_array(a, chunks=200)
    b2 = a2 * 4
    b2_min = b2.min()
    print(b2_min)
    dask.arraydtype=float64, chunksize=()>
    Same operation with dask
    b2_min.compute()
    -13.298288860312757
    

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92. def find_labels(points, centers):
    diff = (points[:, None, :] - centers) ** 2
    distances = diff.sum(-1)
    return distances.argmin(1)
    labels = find_labels(points, centers)
    

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93. from dask import array as da
    def find_labels(points, centers):
    diff = (points[:, None, :] - centers) ** 2
    distances = diff.sum(-1)
    return distances.argmin(1)
    points = da.from_array(points, chunks=1000)
    centers = da.from_array(centers, chunks=5)
    labels = find_labels(points, centers)
    

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94. from dask import array as da
    def find_labels(points, centers):
    diff = (points[:, None, :] - centers)
    distances = (diff ** 2).sum(-1)
    return distances.argmin(1)
    points_dask = da.from_array(points, chunks=1000)
    centers_dask = da.from_array(centers, chunks=5)
    labels = find_labels(points_dask, centers_dask)
    

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95. def find_labels(points, centers):
    diff = (points[:, None, :] - centers)
    distances = (diff ** 2).sum(-1)
    return distances.argmin(1)
    def compute_centers(points, labels):
    points_df = dd.from_dask_array(points)
    labels_df = dd.from_dask_array(labels)
    return points_df.groupby(labels_df).mean()
    def kmeans(points, n_clusters):
    centers = points[-n_clusters:]
    points = da.from_array(points, 1000)
    while True:
    old_centers = centers
    labels = find_labels(points, da.from_array(centers, 5))
    centers = compute_centers(points, labels)
    centers = centers.compute().values
    if np.all(centers == old_centers):
    break
    return labels.compute()
    %timeit kmeans(points, 10)
    3.28 s ± 192 ms per loop (mean ± std. dev. of 7 runs)
    Full, Parallelized K-Means
    

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96. def find_labels(points, centers):
    diff = (points[:, None, :] - centers)
    distances = (diff ** 2).sum(-1)
    return distances.argmin(1)
    def compute_centers(points, labels):
    points_df = dd.from_dask_array(points)
    labels_df = dd.from_dask_array(labels)
    return points_df.groupby(labels_df).mean()
    def kmeans(points, n_clusters):
    centers = points[-n_clusters:]
    points = da.from_array(points, 1000)
    while True:
    old_centers = centers
    labels = find_labels(points, da.from_array(centers, 5))
    centers = compute_centers(points, labels)
    centers = centers.compute().values
    if np.all(centers == old_centers):
    break
    return labels.compute()
    %timeit kmeans(points, 10)
    3.28 s ± 192 ms per loop (mean ± std. dev. of 7 runs)
    Full, Parallelized K-Means
    

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97. def find_labels(points, centers):
    diff = (points[:, None, :] - centers)
    distances = (diff ** 2).sum(-1)
    return distances.argmin(1)
    def compute_centers(points, labels):
    points_df = dd.from_dask_array(points)
    labels_df = dd.from_dask_array(labels)
    return points_df.groupby(labels_df).mean()
    def kmeans(points, n_clusters):
    centers = points[-n_clusters:]
    points = da.from_array(points, 1000)
    while True:
    old_centers = centers
    labels = find_labels(points, da.from_array(centers, 5))
    centers = compute_centers(points, labels)
    centers = centers.compute().values
    if np.all(centers == old_centers):
    break
    return labels.compute()
    %timeit kmeans(points, 10)
    3.28 s ± 192 ms per loop (mean ± std. dev. of 7 runs)
    Full, Parallelized K-Means
    

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98. Advantages:
    - Easy distributed coding, often with no change to
    NumPy or Pandas code!
    - Even works locally on out-of-core data
    Disadvantages:
    - High overhead, so not suitable for smaller problems
    Recommendation: use when data size or
    computation time warrants
    See my blog post Out of Core Dataframes in Python: Dask and OpenStreetMap
    http://jakevdp.github.io/blog/2015/08/14/out-of-core-dataframes-in-python/
    

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99. Seven Strategies:
    1. Line Profiling
    2. Numpy Vectorization
    3. Specialized Data Structures
    4. Cython
    5. Numba
    6. Dask
    

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100. Seven Strategies:
     1. Line Profiling
     2. Numpy Vectorization
     3. Specialized Data Structures
     4. Cython
     5. Numba
     6. Dask
     

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101. Seven Strategies:
     1. Line Profiling
     2. Numpy Vectorization
     3. Specialized Data Structures
     4. Cython
     5. Numba
     6. Dask
     7. Find an Existing Implementation!
     

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102. from sklearn.cluster import KMeans
     %timeit KMeans(4).fit_predict(points)
     28.5 ms ± 701 µs per loop (mean ± std.
     dev. of 7 runs, 10 loops each)
     from dask_ml.cluster import KMeans
     %timeit KMeans(4).fit(points).predict(points)
     8.7 s ± 202 ms per loop (mean ± std.
     dev. of 7 runs, 1 loop each)
     ML
     Recommendation: resist the urge to reinvent the wheel.
     

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103. You can implement it yourself . . .
     you can make your numerical code fast!
     But the community is Python’s greatest strength.
     

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104. Email: [email protected]
     Twitter: @jakevdp
     Github: jakevdp
     Web: http://vanderplas.com/
     Blog: http://jakevdp.github.io/
     Thank You!
     Slides available at https://speakerdeck.com/jakevdp/
     Notebook with code from this talk: http:goo.gl/d8ZWwp
     

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