KMeans as a classifier for the WIFI and MNIST datasets – III – KMeans as a classifier for the WIFI-example

In the previous articles of this series

KMeans as a classifier for the WIFI and MNIST datasets – I – Cluster analysis of the WIFI example
KMeans as a classifier for the WIFI and MNIST datasets – II – PCA in combination with KMeans for the WIFI-example

I applied KMeans to the „WIFI“ dataset – a small and rather simple training set of the UCI Irvine. The 7 dimensional feature space for this example is defined by the strength of seven different WLAN-signals. The data samples are labeled by numbers specifying four different rooms. We just have 2000 samples. But „simple“ does not mean that one cannot learn something of it.

An elbow and a silhouette analysis indicated that the data can well be described by 4 to 5 clusters.
A detailed PCA analysis helped us to understand that the data could basically be described by two primary components whose axes were defined by a diagonal in a two-dimensional sub-space of the original feature space (defined by the features „WLAN-0“ and „WLAN-3“) plus an orthogonal axis (defined by „WLAN-4“). As the data points segregated into well separated clusters in the coordinate system of the primary components we could expect that they also segregate well when projected onto two 2-dim sub-spaces of the feature space defined by the signal combinations {WLAN-0/WLAN-4} and {WLAN-3/WLAN-4}.

So we projected the results of a KMeans analysis into a 2-dim sub-space of the 7-dim feature space spanned by two selected „features“ (WLAN signals WLAN-0 and WLAN-4). And indeed: For 2 special signal- or feature-combinations we saw a field with 4 to 5 well separated clusters.

As we have well separated clusters – either in sub-planes of the feature space or the space defined by the dominant two PCA components – a legitimate question is: Can we use the cluster information for classifying?

Clusters and classification

In the special example of the WIFI data the data points can obviously be divided into different „groups“ filled by data points which belong to a certain „label“ or class (in the WIFI case: a room):

In the plots above the colorization of the data points is given by their label in the plot above. However:
Does this mean that these spatially separated „groups“ coincide with the clusters detected by KMeans?

In the next plots I superimposed the data points – first colorized by class and then by cluster with different colors and a slight spatial deviation.

We see that the border of the class related points on average overlaps well with the border of the clusters. Only a few points show a major deviation.

Is this always the case? Of course not!

We see this already when we we just use 2 clusters and colorize the data points according to cluster membership in one of the two clusters:

Oops, all the beauty is gone! Members of the previously visual dark-red, green and pink „clusters“ would now certainly be members of the first defined „green“ cluster. And members of the visual pink and orange clusters would now be members of the „defined“ blue cluster. So: Labels may but do strong>not always define clusters.

An important thing we learned by this consideration is that we should at least use a number of cluster greater or equal the number of labels.

But even then the separation may not work. Let us imagine three rooms: In room „A“ we only have male persons, in room „B“ only female persons and in the third room „C“ female persons on one side of the room and male persons on the other – like in some old fashioned dancing courses. If we used the persons‘ coordinates to define features and the „male“/“female“ attributes as labels and tried a cluster analysis in the feature space we would clearly identify three clusters. However, in the cluster for room „C“ we would find a mixture of samples with different labels. Spatial vicinity in the feature space does not mean class identity and a label does not necessarily mean a big distance in the feature space.

Hyperplanes and clusters

The only thing you can hope for with respect to a solvable ML problem is that the data points for different labels may be distributed such that a complicated and curved hyperplane can be found in the multidimensional feature space which separates groups of data points with the same label sufficiently well. But this hyperplane does not necessarily coincide with fictitious borders of some clusters identified by KMeans. We are lucky if the topologically closed surface of a cluster lies completely in a region of the hyperspace separated by complex and topologically open hyperplanes separating data points belonging to one class from other feature space regions which contain data points with different labels.

Not all data ensembles in ML will fall apart in extended clusters each of which dominated by some specific label. Ring like distributions of data samples with identical labels may pose a problem to cluster algorithms – mot only in 2 dimensions.

On the other side: If the labeled data – after some useful transformation – are not well separable in the feature space, then the posed ML task is problematic anyway and the chosen feature definitions may not be appropriate.

Granularity: Hope for label dominance in sufficiently fine grained clusters

What does a label-cluster correlation depend on? Well, if there is an important factor regarding the position of the centroids of KMeans then it is the number of clusters. In the extreme case of as many clusters as data points a super-well defined cluster/label-association exists – but it is not of much use for ML-task. It just represents the most extreme case of overfitting you can think of. It will be useless for new unknown samples.

But the example with the distribution of male/female samples gives you an indication of what we should try: With a growing number of cluster the chances for a fine grained separation into clusters residing on one side of a hyperplane separating samples of different labels rises and with it the chance for a clearer dominance of one label per cluster. This is even true for ring like data distributions: With more then 4 clusters we may even describe a ring like data distribution quite well.

Therefore: If the sample distribution has some reasonable separation hyperplane at all there is a chance that you may find a number of clusters for which each cluster is dominated by a specific label.

In the case of the WIFI example this is pretty obvious. The following first plot shows the data points colorized according to their labels:

The next plot shows four cluster – with data points colorized according to their cluster membership:

The colors are different (we have no control of it) – but the different data point „clouds“ in the pictures coincide rather well. Only some data points do not behave well. We can again use the trick we iused in 2 dimensions and superimpose data points with colorization according to cluster and class:

How do we define a classifier by the help of KMeans?

We need a predict()-function which predicts a label from a predicted cluster membership. The recipe is rather simple:

Define the number of cluster you want to use.

  • Use KMeans for a cluster analysis of a training set of samples.
  • Predict the cluster-membership of a sample with the help of the fitted kmeans-object and its predict()-function.
  • Get the labels of the samples belonging to a specific cluster.
  • Find out what the amount of samples with a certain label for a cluster is and compared the data.
  • Find the label which contributes most samples to a cluster. Check the relative amount of deviating data points with other labels. Should be sufficiently small…
  • Build a dictionary which associates the cluster number with its dominant label.
  • Build a prediction function for yet unknown data points. It first predicts the cluster and then the associated label.

You should then test the accuracy of your new cluster-based classifier model on test data.

More clusters than labels?

What will happen to such an algorithm if you use more clusters than labels? Short answer: Nothing – as long as each cluster is really dominated by a specific label. More precisely: A vast majority of your clusters should exhibit contributions from data points with one specific label to an amount significantly far beyond the statistical average. Having more clusters means that under reasonable conditions we just have more than one cluster predicting a certain label. For the case of the WIFI example this means that we can work with five clusters without getting nervous.

Application to the WIFI example

As the clusters represent the labels quite well we expect very good values for the recall values. The experiment is pretty simple. I just present the resulting confusion matrices for the four labels (identifying rooms) and different numbers of clusters „nclus“. I first show you a presentation with seaborn, which explains, how to interpret rows and columns:

nclus = 5

Here I used five (5) clusters and included all samples in the calculation. I.e. I did not separate a training set from a test set of data points.

Confusion matrices for different numbers of clusters

nclus = 4

[[496   0   4   0]
 [  0 425  75   0]
 [  2   0 492   6]
 [  2   0   2 496]]
num of wrongly predicted samples:  91  :: avg_err =  0.0455

The „avg_err“ gives you the number of wrongly predicted samples divided by the total number of samples. We see that 4 clusters have a problem with the differentiation of the data points for the room „Diele“.

nclus = 5

[[495   0   5   0]
 [  0 455  45   0]
 [  2   0 493   5]
 [  2   0   2 496]]
num of wrongly predicted samples:  61  :: avg_err =  0.0305

nclus = 6

[[499   0   1   0]
 [  0 452  48   0]
 [  5   0 490   5]
 [  2   0   2 496]]
num of wrongly predicted samples:  63  :: avg_err =  0.0315

nclus = 7

[[499   0   1   0]
 [  0 492   8   0]
 [  4  38 455   3]
 [  2   0   3 495]]
 num of wrongly predicted samples:  59  :: avg_err =  0.0295

nclus =

num wrongly predicted samples:  58  :: avg_err =  0.029

nclus = 9

[[499   0   1   0]
 [  0 484  16   0]
 [  4  16 476   4]
 [  2   0   2 496]]
num of wrongly predicted samples:  45  :: avg_err =  0.0225

All data above were derived for the same random_state-variable to KMeans.

However, this result depends on the initial shuffling of the samples, the random_state parameter and the initial statistical distribution of the centroids by KMeans. And other hyperparameters … The next plot shows a slightly different result for nclus = 9:

nclus = 9

This means: Nine cluster give us a reasonably fine grained resolution in the case of the WIFI example.

But: Note that the problem areas in the confusion matrix have changed. „Diele“ is handled a bit better, but „Wohnzimmer“ is not so sharply separated as before. This is not astonishing as we saw a significant mix of data of different labels for different clusters above already:

After nclus = 9 we do not get much better – and dance around an average error of 0.025 => 2.5%:

nclus = 10

num of wrongly predicted samples:  55  :: avg_err =  0.0275

nclus = 11

num of wrongly predicted samples:  52  :: avg_err =  0.026

nclus = 25

num of wrongly predicted samples:  40  :: avg_err =  0.02

Being conservative we can say that our simple cluster based classifier approaches an accuracy around 97.4%. Not so bad regarding the crudeness of our approach! A random forest algorithm reaches something above 98.2%. This is not so much better.

Results after a separation of test data samples

I divided the data set than into 1500 train and 500 test samples.

For 9 clusters I got:

nclus = 9

[[374   0   1   0]
 [  0 362  13   0]
 [  3  17 352   3]
 [  2   0   2 371]]
num wrongly predicted train samples:  41  :: avg_err =  0.027333333333333334
num wrongly predicted train samples:  8  :: avg_err =  0.016

But these numbers depend strongly on the splitting – even if we split stratified. Another run gives:

nclus = 9

[[374   0   1   0]
 [  0 352  23   0]
 [  3   0 369   3]
 [  1   0   2 372]]
num wrongly predicted train samples:  33  :: avg_err =  0.022 
num wrongly predicted test samples:  11  :: avg_err =  0.022

Other runs may even give higher average error values. The variation depend upon of how many critical data points in the intermixing zone were omitted in the train samples. At least the results are not too different from the ones named above.

Conclusion

In this article I presented a very simple method by which a cluster algorithm can be used as a classifier. When we applied the approach to the rather simple WIFI example we saw that this worked pretty well.

What we in addition should try is to combine the classifier with a dimension reduction based on a PCA-analysis. From the results of a previous post we would expect that a cluster classifier should work well on the WIFI data after a transformation and projection of the samples‘ data points into a 3-dimensional space spanned by the most important orthogonal main component axes. This is the topic of the next post in this series:

KMeans as a classifier for the WIFI and MNIST datasets – IV – KMeans on PCA transformed data

Ceterum censeo: The worst fascist today who must be denazified is the Putler.

 

A simple Python program for an ANN to cover the MNIST dataset – XIV – cluster detection in feature space

We extend our studies of a program for a Multilayer perceptron and gradient descent in combination with the MNIST dataset:

A simple Python program for an ANN to cover the MNIST dataset – XIII – the impact of regularization
A simple Python program for an ANN to cover the MNIST dataset – XII – accuracy evolution, learning rate, normalization
A simple Python program for an ANN to cover the MNIST dataset – XI – confusion matrix
A simple Python program for an ANN to cover the MNIST dataset – X – mini-batch-shuffling and some more tests
A simple Python program for an ANN to cover the MNIST dataset – IX – First Tests
A simple Python program for an ANN to cover the MNIST dataset – VIII – coding Error Backward Propagation
A simple Python program for an ANN to cover the MNIST dataset – VII – EBP related topics and obstacles
A simple Python program for an ANN to cover the MNIST dataset – VI – the math behind the „error back-propagation“
A simple Python program for an ANN to cover the MNIST dataset – V – coding the loss function
A simple Python program for an ANN to cover the MNIST dataset – IV – the concept of a cost or loss function
A simple Python program for an ANN to cover the MNIST dataset – III – forward propagation
A simple Python program for an ANN to cover the MNIST dataset – II – initial random weight values
A simple Python program for an ANN to cover the MNIST dataset – I – a starting point

In this article we shall work a bit on the following topic: How can we reduce the computational time required for gradient descent runs of our MLP?

Readers who followed my last articles will have noticed that I sometimes used 1800 epochs in a gradient descent run. The computational time including

  • costly intermediate print outs into Jupyter cells,
  • a full determination of the reached accuracy both on the full training and the test dataset at every epoch

lay in a region of 40 to 45 minutes for our MLP with two hidden layers and roughly 58000 weights. Using an Intel I7 standard CPU with OpenBlas
support. And I plan to work with bigger MLPs – not on MNIST but other data sets. Believe me: Everything beyond 10 minutes is a burden. So, I have a natural interest in accelerating things on a very basic level already before turning to GPUs or arrays of them.

Factors for CPU-time

This introductory question leads to another one: What basic factors beyond technical capabilities of our Linux system and badly written parts of my Python code influence the consumption of computational time? Four points come to my mind; you probably find even more:

  • One factor is certainly the extra forward propagation run which we apply to all samples of both the test and training data seat the end of each epoch. We perform this propagation to make predictions and to get data on the evolution of the accuracy, the total loss and the ratio of the regularization term to the real costs. We could do this in the future at every 2nd or 5th epoch to save some time. But this will reduce CPU-time only by less than 22%. 76% of the CPU-time of an epoch is spent in batch-handling with a dominant part in error backward propagation and weight corrections.
  • The learning rate has a direct impact on the number of required epochs. We could enlarge the learning rate in combination with input data normalization; see the last article. This could reduce the number of required epochs significantly. Depending on the parameter choices before by up to 40% or 50%. But it requires a bit of experimenting ….
  • Two other, more important factors are the frequent number of matrix operations during error back-propagation and the size of the involved matrices. These operations depend directly on the number of nodes involved. We could therefore reduce the number of nodes of our MLP to a minimum compatible with the required accuracy and precision. This leads directly to the next point.
  • The dominant weight matrix is of course the one which couples layer L0 and layer L1. In our case its shape is 784 x 70; it has almost 55000 elements. The matrix for the next pair of layers has only 70×30 = 2100 elements – it is much, much smaller. To reduce CPU time for forward propagation we should try to make this matrix smaller. During error back propagation we must perform multiple matrix multiplications; the matrix dimensions depend on the number of samples in a mini-batch AND on the number of nodes in the involved layers. The dimensions of the the result matrix correspond to the those of the weight matrix. So once again: A reduction of the nodes in the first 2 layers would be extremely helpful for the expensive backward propagation. See: The math behind EBP.

We shall mainly concentrate on the last point in this article.

Reduction of the dimensions of the dominant matrix“requires a reduction of input features

The following numbers show typical CPU times spend for matrix operations during error back propagation [EBP] between different layers of our MLP and for two different batches at the beginning of gradient descent:

Time_CPU for BW layer operations (to L2) 0.00029015699965384556
Time_CPU for BW layer operations (to L1) 0.0008645610000712622
Time_CPU for BW layer operations (to L0) 0.006551215999934357

Time_CPU for BW layer operations (to L2) 0.00029157400012991275
Time_CPU for BW layer operations (to L1) 0.0009575330000188842
Time_CPU for BW layer operations (to L0) 0.007488838999961445

The operations involving layer L0 cost a factor of 7 more CPU time than the other operations! Therefore, a key to the reduction of the number of mathematical operations is obviously the reduction of the number of nodes in the input layer! We cannot reduce the numbers in the hidden layers much, if we
do not want to hamper the accuracy properties of our MLP too much. So the basic question is

Can we reduce the number of input nodes somehow?

Yes, maybe we can! Input nodes correspond to „features„. In case of the MNIST dataset the relevant features are given by the gray-values for the 784 pixels of each image. A first idea is that there are many pixels within each MNIST image which are probably not used at all for classification – especially pixels at the outer image borders. So, it would be helpful to chop them off or to ignore them by some appropriate method. In addition, special significant pixel areas may exist to which the MLP, i.e. its weight optimization, reacts during training. For example: The digits 3, 5, 6, 8, 9 all have a bow within the lower 30% of an image, but in other regions, e.g. to the left and the right, they are rather different.

If we could identify suitable image areas in which dark pixels have a higher probability for certain digits then, maybe, we could use this information to discriminate the represented digits? But a „higher density of dark pixels in an image area“ is nothing else than a description of a „cluster“ of (dark) pixels in certain image areas. Can we use pixel clusters at numerous areas of an image to learn about the represented digits? Is the combination of (averaged) feature values in certain clusters of pixels representative for a handwritten digit in the MNIST dataset?

If the number of such pixel clusters could be reduced below lets say 100 then we could indeed reduce the number of input features significantly!

Cluster detection

To be able to use relevant „clusters“ of pixels – if they exist in a usable form in MNIST images at all – we must first identify them. Cluster identification and discrimination is a major discipline of Machine Learning. This discipline works in general with unlabeled data. In the MNIST case we would not use the labels in the „y“-data at all to identify clusters; we would only use the „X“-data. A nice introduction to the mechanisms of cluster identification is given in the book of Paul Wilcott (see Machine Learning – book recommendations for the reference). The most fundamental method – called „kmeans“ – iterates over 3 major steps [I simplify a bit :-)]:

  • We assume that K clusters exist and start with random initial positions of their centers (called „centroids“) in the multidimensional feature space
  • We measure the distance of all data points to he centroids and associate a point with that centroid to which the distance is smallest
  • We determine the „center of mass“ (according to some distance metric) of the identified data point groups and assume it as a new position of the centroids and move the old positions (a bit) in this direction.

We iterate over these steps until the centroids‘ positions hopefully get stable. Pretty simple. But there is a major drawback: You must make an assumption on the number „K“ of clusters. To make such an assumption can become difficult in the complex case of a feature space with hundreds of dimensions.

You can compensate this by executing multiple cluster runs and comparing the results. By what? Regarding the closure or separation of clusters in terms of an appropriate norm. One such norm is called „cluster inertia„; it measures the mean squared distance to the center for all points of a cluster. The theory is that the sum of the inertias for all clusters drops significantly with the number of clusters until an optimal number is reached and the inertia curve flattens out. The point where this happens in a plot of inertia vs. number of clusters is called „elbow„.
Identifying this „elbow“ is one of the means to find an optimal number of clusters. However, this recipe does not work under all circumstances. As the number of clusters get big we may be confronted with a smooth decline of the inertia sum.

What data do we use for gradient descent after cluster detection?

How could we measure whether an image shows certain clusters? We could e.g. measure distances (with some appropriate metric) of all image points to the clusters. The „fit_transform()“-method of KMeans and MiniBatchKMeans provide us with with some distance measure of each image to the identified clusters. This means our images are transformed into a new feature space – namely into a „cluster-distance space“. This is a quite complex space, too. But it has less dimensions than the original feature space!

Note: We would of course normalize the resulting distance data in the new feature space before applying gradient descent.

Application of „KMeansBatch“ to MNIST

There are multiple variants of „KMeans“. We shall use one which is provided by SciKit-Learn and which is optimized for large datasets: „MiniBatchKMeans„. It operates batch-wise without loosing too much of accuracy and convergence properties in comparison to KMeans (or a comparison see here). „MiniBatchKMeans“has some parameters you can play with.

We could be tempted to use 10 clusters as there are 10 digits to discriminate between. But remember: A digit can be written in very many ways. So, it is much more probable that we need a significant larger number of clusters. But again: How to determine on which K-values we should invest a bit more time? „Kmeans“ and methods alike offer another quantity called „silhouette“ coefficient. It measures how well the data points are within, at or outside the borders of a cluster. See the book of Geron referenced at the link given above on more information.

Variation of CPU time, inertia and average silhouette coefficients with the number of clusters „K“

Let us first have a look at the evolution of CPU time, total inertia and averaged silhouette with the number of clusters „K“ for two different runs. The following code for a Jupyter cell gives us the data:

    
# *********************************************************
# Pre-Clustering => Searching for the elbow 
# *********************************************************
from sklearn.cluster import KMeans
from sklearn.cluster import MiniBatchKMeans
from sklearn.preprocessing import StandardScaler
from sklearn.metrics import silhouette_score
X = np.concatenate((ANN._X_train, ANN._X_test), axis=0)
y = np.concatenate((ANN._y_train, ANN._y_test), axis=0)
print("X-shape = ", X.shape, "y-shape = ", y.shape)
num = X.shape[0]

li_n = []
li_inertia = []
li_CPU = []
li_sil1 = []

# Loop over the number "n" of assumed clusters 
rg_n = range(10,171,10)
for n in rg_n:
    print("\nNumber of clusters: ", n)
    start = time.perf_counter()
    kmeans = MiniBatchKMeans(n_clusters=n, n_init=500, max_iter=1000, batch_size=500 )  
    X_clustered = kmeans.fit_transform(X)
    sil1 = silhouette_score(X, kmeans.labels_)
    #sil2 = silhouette_score(X_clustered, kmeans.labels_)
    end = time.perf_counter()
    dtime = end - start
    print('Inertia = ', kmeans.inertia_)
    print('Time_CPU = ', dtime)
    print('sil1 score = ', sil1)
    li_n.append(n)    
    li_inertia.append(kmeans.inertia_)    
    li_CPU.append(dtime)    
    li_sil1.append(sil1)    

    
# Plots         
# ******
fig_size = plt.rcParams["figure.figsize"]
fig_size[
0] = 14
fig_size[1] = 5
fig1 = plt.figure(1)
fig2 = plt.figure(2)

ax1_1 = fig1.add_subplot(121)
ax1_2 = fig1.add_subplot(122)

ax1_1.plot(li_n, li_CPU)
ax1_1.set_xlabel("num clusters K")
ax1_1.set_ylabel("CPU time")

ax1_2.plot(li_n, li_inertia)
ax1_2.set_xlabel("num clusters K")
ax1_2.set_ylabel("inertia")

ax2_1 = fig2.add_subplot(121)
ax2_2 = fig2.add_subplot(122)

ax2_1.plot(li_n, li_sil1)
ax2_1.set_xlabel("num clusters K")
ax2_1.set_ylabel("silhoutte 1")

 
You see that I allowed for large numbers of initial centroid positions and iterations to be on the safe side. Before you try it yourself: Such runs for a broad variation of K-values are relatively costly. The CPU time rises from around 32 seconds for 30 clusters to a little less than 1 minute for 180 clusters. These times add up to a significant sum after a while …

Here are some plots:

The second run was executed with a higher resolution of K_(n+1) – K_n 5 = 5.

We see that the CPU time to determine the centroids‘ positions varies fairly linear with „K“. And even for 170 clusters it does not take more than a minute! So, CPU-time for cluster identification is not a major limitation.

Unfortunately, we do not see a clear elbow in the inertia curve! What you regard as a reasonable choice for the number K depends a lot on where you say the curve starts to flatten. You could say that this happens around K = 60 to 90. But the results for the silhouette-quantity indicate for our parameter setting that K=40, K=70, K=90 are interesting points. We shall look at these points a bit closer with higher resolution later on.

Reduction of the regularization factor (for Ridge regularization)

Now, I want to discuss an important point which I did not find in the literature:
In my last article we saw that regularization plays a significant but also delicate role in reaching top accuracy values for the test dataset. We saw that Lambda2 = 0.2 was a good choice for a normalized input of the MNIST data. It corresponded to a certain ratio of the regularization term to average batch costs.
But when we reduce the number of input nodes we also reduce the number of total weights. So the weight values themselves will automatically become bigger if we want to get to similar good values at the second layer. But as the regularization term depends in a quadratic way on the weights we may assume that we roughly need a linear reduction of Lambda2. So, for K=100 clusters we may shrink Lambda2 to (0.2/784*100) = 0.025 instead of 0.2. In general:

Lambda2_cluster = Lambda2_std * K / (number of input nodes)

I applied this rule of a thumb successfully throughout experiments with clustering befor gradient descent.

Reference run without clustering

We saw at the end of article XII that we could reach an accuracy of around 0.975 after 500 epochs under optimal circumstances. But in the case I presented ten I was extremely lucky with the statistical initial weight distribution and the batch composition. In other runs with the same parameter setup I got smaller accuracy values. So, let us take an ad hoc run with the following parameters and results:
Parameters: learn_rate = 0.001, decrease_rate = 0.00001, mom_rate = 0.00005, n_size_mini_batch = 500, n_epochs = 600, Lambda2 = 0.2, weights at
all layers in [-2*1.0/sqrt(num_nodes_layer), 2*1.0/sqrt(num_nodes_layer)]
Results: acc_train: 0.9949 , acc_test: 0.9735, convergence after ca. 550-600 epochs

The next plot shows (from left to right and the down) the evolution of the costs per batch, the averaged error of the last mini-batch during an epoch, the ratio of regularization to batch costs and the total costs of the training set, respectively .

The following plot summarizes the evolution of the total costs of the traaining set (including the regularization contribution) and the evolution of the accuracy on the training and the test data sets (in orange and blue, respectively).

The required computational time for the 600 epochs was roughly 18,2 minutes.

Results of gradient descent based on a prior cluster identification

Before we go into a more detailed discussion of code adaption and test runs with things like clusters in unnormalized and normalized feature spaces, I want to show what we – without too much effort – can get out of using cluster detection ahead of gradient descent. The next plot shows the evolution of a run for K=70 clusters in combination with a special normalization:

and the total cost and accuracy evolution

The dotted line marks an accuracy of 97.8%! This is 0.5% bigger then our reference value of 97.3%. The total gain of %gt; 0.5% means however 18.5% of the remaining difference of 2.7% to 100% and we past a value of 97.8% already at epoch 600 of the run.

What were the required computational times?

If we just wanted 97.4% as accuracy we need around 150 epochs. And a total CPU time of 1.3 minutes to get to the same accuracy as our reference run. This is a factor of roughly 14 in required CPU time. For a stable 97.73% after epoch 350 we were still a factor of 5.6 better. For a stable accuracy beyond 97.8% we needed around 600 epochs – and still were by a factor of 3.3 faster than our reference run! So, clustering really brings some big advantages with it.

Conclusion

In
this article I discussed the idea of introducing cluster identification in the (unnormalized or normalized) feature space ahead of gradient descent as a possible means to save computational time. A preliminary trial run showed that we indeed can become significantly faster by at least a factor of 3 up to 5 and even more. This is just due to the point that we reduced the number of input nodes and thus the number of mathematical calculations during matrix operations.

In the next article we shall have a more detailed look at clustering techniques in combination with normalization.