Quick classifiers for exploring medium-sized data (redux)

In my last post, I wanted to find out which classifiers are quick enough to allow us to re-train them many times on medium-sized data. A recap of terminology:

• Small Data is the magical and care-free land where you can work on a single computer and do something/anything with "all pairs of samples" (evaluate kernels, compute distances, etc...). It's a mostly mythical place, occupied only by Intro ML students and whoever keeps working on Gaussian Processes.
• If you can't fit your whole dataset on one machine and/or your computation needs to be distributed, then congratulations: you've got Big Data (now go learn Mahout)
• Between Small Data and Big Data is the very common scenario when you are working on a single computer but have too many samples for anything but learning algorithms which scale linearly. Goodbye kernel methods, hello simple boundaries and decision trees.

Previously, I evaluated the runtimes and accuracies of all the scalable classifiers in scikit-learn and found that only Gaussian Naive Bayes and SVM trained by stochastic gradient descent were fast enough. Unfortunately, my evaluation had some obvious shortcomings:

• I used a single synthetic dataset-- who knows if my results generalize to "real world" data?
• More specifically, the dataset was sampled from 10 multivariate Gaussians, giving a significant advantage to models which assume the distribution underlying the data is Gaussian (ahem...Gaussian Naive Bayes).
• I recommended using SGD SVM, but only tried training Logistic Regression with liblinear. However, it's known that linear SVMs and Logistic Regression often get similar performance, so I should have tried SGD Logistic Regression as well.
• I trained the SGD SVM with the recommended ~1 million updates, but where does that heuristic even come from? Would fewer updates still yield good accuracy? If so, SGD classifiers could be trained even faster (hard to believe, given how uncannily quick they already are).
• All of my tree ensembles (random forests and extra trees) had at least 10 base classifiers. They showed good classification accuracy but were too slow for exploratory use. I should have tried fewer trees to see if I get acceptable accuracy with quicker-to-train ensembles.
• Similarly, I never applied regularization to decision trees (either alone or in ensembles) by changing complexity controlling parameters such as the minimum number of samples per leaf. What if I forced the base trees to be simpler? Presumably training would be faster, but would accuracy suffer? I assume that single decision trees might benefit from lowered variance, whereas ensemble methods might do worse.

To get a more complete picture of the speed vs. accuracy landscape, I got three real datasets (UCI's covertype & isolet, as well as some proteomics data) and evaluated scikit-learn's classifiers across a wider swath of parameters. I'll save you from squinting at big tables of numbers and summarize the results first:

• Small ensembles of decision trees work! Yes, people will often recommend hundreds of trees (grown to full height). However, the performance of Random Forests seems to degrade gracefully as you limit both the number of trees in the ensemble and the complexity of each individual tree. Whereas 128 full-grown trees might get 88.7% accuracy (on the forest cover data set), a wimpy little duo of 2 shorter trees can get 80.7%. The difference in training times, however, is ~700 seconds vs. ~10 seconds. Squeezing out that extra 8% of accuracy might be worth it when you're constructing your final classification system but not necessarily when you're exploring large space of feature selection/transformation decisions.
• Gradient boosting ain't so great...and it sometimes seems to fail catastrophically. Perhaps this is a problem with some default parameters in scikit-learn's implementation?
• If you're estimating the variance of a distribution, use smoothing/regularization. Quadratic Discriminant Analysis and Gaussian Naive Bayes break down when a feature appears constant under a particular class label (remember, you're going to be dividing by that zero-variance). Be sure to smooth your variance estimates, either in a rigorous/justified way or by just adding some little epsilon of variance to every feature's estimate.
• It doesn't matter whether you use Linear SVM or Logistic Regression. SVM seems to sometime achieve faster training times (since it presumably only has to update weights when a prediction is wrong), whereas Logistic Regression seems to achieve slightly better generalization. But really, I would need to look at many more datasets before I could claim there is substantive difference.
• Stochastic gradient descent really is blazing fast-- but also delicate. Linear learners can be trained 10X - 100X faster with stochastic gradient descent but choosing the wrong number of training iterations can sometimes ruin your classifier's accuracy. Perhaps using some stopping criterion might work better than just have a fixed number of epochs?

Recommendations

If you only have enough time to use a single type of model, I would recommend a small Random Forest (with 3-7 trees). If even that is too slow, then try a linear SVM trained using stochastic gradient descent (the ~1 million update heuristic seems reasonable). However, since different classifiers seem to excel on different datasets, it would be best to train several diverse models (the two above, as well as regularized Naive Bayes and LDA) and use the best score any of them can achieve. That way, you might avoid tailoring your features too closely to the assumptions of one particular model.

The Numbers (here be dragoons)

Forest Cover Type

Given features such as soil type, elevation, distance to water sources predict which sort of tree is growing (i.e., "Spruce Fir" vs. "Ponderosa Pine").

• Dimensionality: 54
• Training samples: 290,504
• Test samples: 290,508
• Number of classes: 7
• Accuracy if we always predict the most common class: 48.86%
• Source: UCI

Algorithm	Training / Test Time	Training / Test Accuracy
Gaussian Naive Bayes	1.08s / 2.55s	9.66% / 9.69%
Regularized Gaussian Naive Bayes	1.28s / 1.87s	63.08% / 63.11%
LDA	3.97s / 0.24s	67.94% / 68.02%
QDA	2.92s / 8.94s	8.38% / 8.41%
CART
Decision Tree (min samples per leaf = 1)	14.07s / 0.13s	100.00% / 92.58%
Decision Tree (min samples per leaf = log n = 13)	13.38s / 0.13s	97.35% / 91.73%
Decision Tree (min samples per leaf = sqrt n = 538)	10.40s / 0.11s	81.54% / 80.62%
Support Vector Machine
Linear SVM (SGD, ~250k updates = 1 epochs)	1.14s / 0.17s	69.72% / 69.89%
Linear SVM (SGD, ~1m updates = 4 epochs)	4.38s / 0.17s	71.25% / 71.30%
Linear SVM (SGD, ~2m updates = 7 epochs)	7.67s / 0.17s	71.00% / 71.17%
Linear SVM (liblinear)	50.79s / 0.17s	71.21% / 71.34%
Logistic Regression
Logisic Regression (SGD, ~250k updates = 1 epoch)	1.44s / 0.17s	70.06% / 70.26%
Logisic Regression (SGD, ~1m updates = 4 epochs)	5.61s / 0.17s	71.23% / 71.30%
Logisic Regression (SGD, ~2m updates = 7 epochs)	9.81s / 0.17s	71.15% / 71.24%
Logistic Regression (liblinear)	21.56s / 0.17s	71.44% / 71.54%
Extremely Randomized Trees
Extra-Trees (2 trees)	9.30s / 0.26s	75.30% / 74.23%
Extra-Trees (4 trees)	18.39s / 0.54s	78.49% / 77.18%
Extra-Trees (8 trees)	38.67s / 1.04s	80.47% / 78.71%
Extra-Trees (16 trees)	83.77s / 2.26s	80.84% / 78.98%
Extra-Trees (32 trees)	139.69s / 4.00s	79.57% / 77.92%
Extra-Trees (64 trees)	280.71s / 8.01s	79.57% / 77.98%
Extra-Trees (128 trees)	579.19s / 17.25s	79.37% / 77.78%
Gradient Boosting (sample size = 100%)
Gradient Boosting (sample size = 100%, 2 stumps)	21.33s / 0.32s	67.06% / 67.25%
Gradient Boosting (sample size = 100%, 4 stumps)	41.75s / 0.45s	69.28% / 69.41%
Gradient Boosting (sample size = 100%, 8 stumps)	81.61s / 0.68s	70.53% / 70.58%
Gradient Boosting (sample size = 100%, 16 stumps)	163.75s / 1.19s	72.25% / 72.23%
Gradient Boosting (sample size = 100%, 32 stumps)	327.92s / 2.32s	74.28% / 74.15%
Gradient Boosting (sample size = 100%, 64 stumps)	648.88s / 4.14s	76.22% / 76.08%
Gradient Boosting (sample size = 100%, 128 stumps)	1324.65s / 8.23s	77.99% / 77.80%
Gradient Boosting (sample size = 25%)
Gradient Boosting (sample size = 25%, 2 stumps)	12.69s / 0.32s	67.47% / 67.63%
Gradient Boosting (sample size = 25%, 4 stumps)	24.63s / 0.44s	69.32% / 69.43%
Gradient Boosting (sample size = 25%, 8 stumps)	48.25s / 0.69s	70.57% / 70.67%
Gradient Boosting (sample size = 25%, 16 stumps)	97.67s / 1.20s	72.37% / 72.31%
Gradient Boosting (sample size = 25%, 32 stumps)	192.46s / 2.22s	74.53% / 74.45%
Gradient Boosting (sample size = 25%, 64 stumps)	382.37s / 4.15s	76.52% / 76.38%
Gradient Boosting (sample size = 25%, 128 stumps)	770.73s / 8.49s	78.56% / 78.39%
Random Forests (min samples per leaf = 1)
Random Forest (min samples per leaf = 1, 2 trees)	11.19s / 0.25s	85.85% / 80.84%
Random Forest (min samples per leaf = 1, 4 trees)	22.12s / 0.54s	89.96% / 85.40%
Random Forest (min samples per leaf = 1, 8 trees)	46.53s / 1.14s	92.67% / 87.79%
Random Forest (min samples per leaf = 1, 16 trees)	89.64s / 2.16s	92.79% / 88.34%
Random Forest (min samples per leaf = 1, 32 trees)	177.83s / 4.31s	93.01% / 88.53%
Random Forest (min samples per leaf = 1, 64 trees)	352.32s / 8.41s	93.19% / 88.63%
Random Forest (min samples per leaf = 1, 128 trees)	698.02s / 17.02s	93.26% / 88.70%
Random Forests (min samples per leaf = log n)
Random Forest (min samples per leaf = 13, 2 trees)	10.43s / 0.24s	82.73% / 80.70%
Random Forest (min samples per leaf = 13, 4 trees)	19.62s / 0.45s	85.42% / 83.40%
Random Forest (min samples per leaf = 13, 8 trees)	38.41s / 0.85s	86.44% / 84.49%
Random Forest (min samples per leaf = 13, 16 trees)	74.81s / 1.77s	86.61% / 84.65%
Random Forest (min samples per leaf = 13, 32 trees)	153.10s / 3.36s	87.35% / 85.35%
Random Forest (min samples per leaf = 13, 64 trees)	300.55s / 6.69s	87.43% / 85.40%
Random Forest (min samples per leaf = 13, 128 trees)	608.06s / 16.93s	87.45% / 85.48%
Random Forests (min samples per leaf = sqrt n)
Random Forest (min samples per leaf = 538, 2 trees)	7.00s / 0.19s	71.10% / 70.96%
Random Forest (min samples per leaf = 538, 4 trees)	13.48s / 0.37s	72.39% / 72.28%
Random Forest (min samples per leaf = 538, 8 trees)	27.61s / 0.70s	73.62% / 73.54%
Random Forest (min samples per leaf = 538, 16 trees)	53.80s / 1.36s	74.18% / 74.13%
Random Forest (min samples per leaf = 538, 32 trees)	108.25s / 2.76s	74.29% / 74.17%
Random Forest (min samples per leaf = 538, 64 trees)	212.23s / 5.50s	74.23% / 74.09%
Random Forest (min samples per leaf = 538, 128 trees)	423.53s / 10.91s	74.08% / 74.01%

Proteomics

The task is to classify the mass/charge ratio of peptides using some hand-crafted features of their spectra.

• Dimensionality: 109
• Training samples: 25,780
• Test samples: 25,784
• Number of classes: 6
• Accuracy if we always predict the most common class: 49.25%
• Source: Sergey Feldman

Algorithm	Training / Test Time	Training / Test Accuracy
Gaussian Naive Bayes	0.16s / 0.25s	17.18% / 17.40%
Regularized Gaussian Naive Bayes	0.19s / 0.26s	5.51% / 5.38%
LDA	failed
QDA	failed
CART
Decision Tree (min samples per leaf = 1)	3.44s / 0.01s	100.00% / 77.48%
Decision Tree (min samples per leaf = log n = 11)	3.28s / 0.01s	95.65% / 77.39%
Decision Tree (min samples per leaf = sqrt n = 160)	2.69s / 0.01s	83.71% / 79.42%
Support Vector Machine
Linear SVM (SGD, ~250k updates = 10 epochs)	1.05s / 0.06s	36.81% / 36.89%
Linear SVM (SGD, ~1m updates = 39 epochs)	3.98s / 0.06s	61.49% / 61.40%
Linear SVM (SGD, ~2m updates = 78 epochs)	8.04s / 0.06s	47.92% / 46.92%
Linear SVM (liblinear)	109.26s / 0.07s	56.18% / 56.18%
Logistic Regression
Logisic Regression (SGD, ~250k updates = 10 epochs)	1.08s / 0.06s	37.84% / 37.87%
Logisic Regression (SGD, ~1m updates = 39 epochs)	4.12s / 0.06s	37.16% / 37.29%
Logisic Regression (SGD, ~2m updates = 78 epochs)	8.22s / 0.06s	51.12% / 50.48%
Logistic Regression (liblinear)	280.85s / 0.07s	58.66% / 58.18%
Extremely Randomized Trees
Extra-Trees (2 trees)	3.81s / 0.04s	100.00% / 72.67%
Extra-Trees (4 trees)	7.88s / 0.08s	100.00% / 78.80%
Extra-Trees (8 trees)	15.17s / 0.14s	100.00% / 82.09%
Extra-Trees (16 trees)	29.39s / 0.28s	100.00% / 84.15%
Extra-Trees (64 trees)	119.16s / 1.14s	100.00% / 85.53%
Extra-Trees (32 trees)	58.78s / 0.59s	100.00% / 85.09%
Extra-Trees (128 trees)	233.06s / 2.23s	100.00% / 85.89%
Gradient Boosting (sample size = 100%)
Gradient Boosting (sample size = 100%, 2 stumps)	3.99s / 0.03s	58.70% / 58.49%
Gradient Boosting (sample size = 100%, 4 stumps)	7.82s / 0.04s	56.73% / 56.49%
Gradient Boosting (sample size = 100%, 8 stumps)	15.37s / 0.07s	58.73% / 58.62%
Gradient Boosting (sample size = 100%, 16 stumps)	31.09s / 0.11s	43.50% / 43.57%
Gradient Boosting (sample size = 100%, 32 stumps)	61.36s / 0.21s	42.71% / 42.79%
Gradient Boosting (sample size = 100%, 64 stumps)	118.30s / 0.39s	15.08% / 15.31%
Gradient Boosting (sample size = 100%, 128 stumps)	230.85s / 0.78s	20.24% / 20.04%
Gradient Boosting (sample size = 25%)
Gradient Boosting (sample size = 25%, 2 stumps)	2.17s / 0.03s	46.67% / 46.51%
Gradient Boosting (sample size = 25%, 4 stumps)	4.14s / 0.04s	44.78% / 44.85%
Gradient Boosting (sample size = 25%, 8 stumps)	8.10s / 0.07s	43.36% / 43.28%
Gradient Boosting (sample size = 25%, 16 stumps)	16.17s / 0.12s	45.05% / 45.04%
Gradient Boosting (sample size = 25%, 32 stumps)	31.77s / 0.22s	22.70% / 22.88%
Gradient Boosting (sample size = 25%, 64 stumps)	61.40s / 0.42s	32.85% / 33.07%
Gradient Boosting (sample size = 25%, 128 stumps)	100.16s / 0.65s	49.26% / 49.25%
Random Forests (min samples per leaf = 1)
Random Forest (min samples per leaf = 1, 2 trees)	2.57s / 0.03s	90.58% / 73.65%
Random Forest (min samples per leaf = 1, 4 trees)	5.07s / 0.07s	96.53% / 79.84%
Random Forest (min samples per leaf = 1, 8 trees)	10.20s / 0.13s	98.88% / 83.20%
Random Forest (min samples per leaf = 1, 16 trees)	20.47s / 0.27s	99.66% / 84.79%
Random Forest (min samples per leaf = 1, 32 trees)	40.82s / 0.54s	99.95% / 85.78%
Random Forest (min samples per leaf = 1, 64 trees)	82.00s / 1.07s	99.99% / 86.11%
Random Forest (min samples per leaf = 1, 128 trees)	165.42s / 2.17s	100.00% / 86.40%
Random Forests (min samples per leaf = log n)
Random Forest (min samples per leaf = 11, 2 trees)	2.17s / 0.03s	87.35% / 80.06%
Random Forest (min samples per leaf = 11, 4 trees)	4.38s / 0.07s	89.78% / 82.80%
Random Forest (min samples per leaf = 11, 8 trees)	8.75s / 0.13s	91.41% / 84.13%
Random Forest (min samples per leaf = 11, 16 trees)	17.35s / 0.25s	92.00% / 84.94%
Random Forest (min samples per leaf = 11, 32 trees)	35.08s / 0.51s	92.47% / 85.51%
Random Forest (min samples per leaf = 11, 64 trees)	70.10s / 1.02s	92.56% / 85.74%
Random Forest (min samples per leaf = 11, 128 trees)	139.47s / 2.01s	92.60% / 85.84%
Random Forests (min samples per leaf = sqrt n)
Random Forest (min samples per leaf = 160, 2 trees)	1.61s / 0.03s	79.59% / 78.49%
Random Forest (min samples per leaf = 160, 4 trees)	3.22s / 0.05s	81.67% / 80.91%
Random Forest (min samples per leaf = 160, 8 trees)	6.37s / 0.10s	82.60% / 81.99%
Random Forest (min samples per leaf = 160, 16 trees)	12.83s / 0.21s	83.17% / 82.50%
Random Forest (min samples per leaf = 160, 32 trees)	26.01s / 0.42s	83.23% / 82.68%
Random Forest (min samples per leaf = 160, 64 trees)	51.79s / 0.82s	83.41% / 82.90%
Random Forest (min samples per leaf = 160, 128 trees)	102.06s / 1.64s	83.41% / 82.94%

Isolet

Recognize which letter is being spoken from hand-crafted auditory features. I don't really think this dataset is large enough to qualify as "medium-sized", but I still threw it in for comparison.

• Dimensionality: 617
• Training samples: 6,238
• Test samples: 1,559
• Number of classes: 26
• Baseline test accuracy: 3.85%
• Source: UCI

Algorithm	Training / Test Time	Training / Test Accuracy
Gaussian Naive Bayes	0.46s / 0.58s	84.23% / 80.12%
Regularized Gaussian Naive Bayes	0.51s / 0.58s	89.87% / 88.65%
LDA	4.18s / 0.22s	97.52% / 94.16%
QDA	failed
CART
Decision Tree (min samples per leaf = 1)	4.52s / 0.00s	100.00% / 79.92%
Decision Tree (min samples per leaf = log n = 9)	4.39s / 0.00s	95.70% / 79.54%
Decision Tree (min samples per leaf = sqrt n = 78)	4.03s / 0.00s	88.35% / 79.28%
Support Vector Machine
Linear SVM (SGD, ~250k updates / 41 epochs)	9.12s / 0.06s	99.90% / 95.00%
Linear SVM (SGD, ~1 updates / 161 epochs)	35.12s / 0.07s	100.00% / 94.48%
Linear SVM (SGD, ~2m updates / 321 epochs)	70.07s / 0.06s	100.00% / 94.36%
Linear SVM (liblinear)	13.10s / 0.05s	100.00% / 94.55%
Logistic Regression
Logisic Regression (SGD, ~250k updates / 41 epochs)	19.39s / 0.06s	99.98% / 95.00%
Logisic Regression (SGD, ~1m updates / 161 epochs)	74.00s / 0.06s	100.00% / 95.57%
Logisic Regression (SGD, ~2m updates / 321 epochs)	147.53s / 0.06s	100.00% / 95.57%
Logistic Regression (liblinear)	42.67s / 0.05s	99.97% / 95.51%
Extremely Randomized Trees
Extra-Trees (2 trees)	2.52s / 0.01s	100.00% / 66.77%
Extra-Trees (4 trees)	4.75s / 0.02s	100.00% / 79.09%
Extra-Trees (8 trees)	9.34s / 0.03s	100.00% / 87.94%
Extra-Trees (16 trees)	18.39s / 0.06s	100.00% / 92.05%
Extra-Trees (32 trees)	36.38s / 0.11s	100.00% / 93.46%
Extra-Trees (64 trees)	72.55s / 0.22s	100.00% / 94.74%
Extra-Trees (128 trees)	144.06s / 0.44s	100.00% / 95.38%
Gradient Boosting (sample size = 100%)
Gradient Boosting (sample size = 100%, 2 stumps)	20.19s / 0.01s	4.14% / 4.43%
Gradient Boosting (sample size = 100%, 4 stumps)	40.19s / 0.01s	3.91% / 4.87%
Gradient Boosting (sample size = 100%, 8 stumps)	80.14s / 0.02s	4.09% / 4.30%
Gradient Boosting (sample size = 100%, 16 stumps)	160.47s / 0.03s	3.77% / 4.62%
Gradient Boosting (sample size = 100%, 32 stumps)	319.10s / 0.08s	3.77% / 4.43%
Gradient Boosting (sample size = 100%, 64 stumps)	630.17s / 0.13s	3.82% / 4.36%
Gradient Boosting (sample size = 100%, 128 stumps)	1251.35s / 0.25s	3.69% / 3.85%
Gradient Boosting (sample size = 25%)
Gradient Boosting (sample size = 25%, 2 stumps)	10.08s / 0.01s	3.70% / 3.98%
Gradient Boosting (sample size = 25%, 4 stumps)	19.79s / 0.01s	3.72% / 3.85%
Gradient Boosting (sample size = 25%, 8 stumps)	39.54s / 0.02s	3.49% / 3.66%
Gradient Boosting (sample size = 25%, 16 stumps)	63.67s / 0.03s	3.85% / 3.85%
Gradient Boosting (sample size = 25%, 32 stumps)	83.68s / 0.02s	3.85% / 3.85%
Gradient Boosting (sample size = 25%, 64 stumps)	124.81s / 0.04s	3.85% / 3.85%
Gradient Boosting (sample size = 25%, 128 stumps)	203.56s / 0.05s	3.85% / 3.85%
Random Forests (min samples per leaf = 1)
Random Forest (min samples per leaf = 1, 2 trees)	2.60s / 0.01s	88.12% / 66.32%
Random Forest (min samples per leaf = 1, 4 trees)	5.45s / 0.02s	97.74% / 81.46%
Random Forest (min samples per leaf = 1, 8 trees)	10.46s / 0.03s	99.71% / 88.65%
Random Forest (min samples per leaf = 1, 16 trees)	21.04s / 0.05s	99.98% / 92.05%
Random Forest (min samples per leaf = 1, 32 trees)	42.44s / 0.11s	100.00% / 93.52%
Random Forest (min samples per leaf = 1, 64 trees)	83.37s / 0.21s	100.00% / 94.10%
Random Forest (min samples per leaf = 1, 128 trees)	168.74s / 0.43s	100.00% / 93.97%
Random Forests (min samples per leaf = log n)
Random Forest (min samples per leaf = 9, 2 trees)	2.32s / 0.01s	87.48% / 76.78%
Random Forest (min samples per leaf = 9, 4 trees)	4.66s / 0.01s	93.78% / 85.63%
Random Forest (min samples per leaf = 9, 8 trees)	9.32s / 0.03s	96.73% / 89.35%
Random Forest (min samples per leaf = 9, 16 trees)	18.82s / 0.05s	98.12% / 92.62%
Random Forest (min samples per leaf = 9, 32 trees)	37.40s / 0.10s	98.59% / 93.33%
Random Forest (min samples per leaf = 9, 64 trees)	75.41s / 0.21s	98.75% / 94.03%
Random Forest (min samples per leaf = 9, 128 trees)	152.74s / 0.44s	98.86% / 94.16%
Random Forests (min samples per leaf = sqrt n)
Random Forest (min samples per leaf = 78, 2 trees)	2.01s / 0.01s	73.87% / 69.53%
Random Forest (min samples per leaf = 78, 4 trees)	4.02s / 0.01s	82.57% / 78.51%
Random Forest (min samples per leaf = 78, 8 trees)	8.00s / 0.02s	88.14% / 85.05%
Random Forest (min samples per leaf = 78, 16 trees)	15.87s / 0.05s	90.85% / 88.65%
Random Forest (min samples per leaf = 78, 32 trees)	31.94s / 0.11s	91.66% / 90.64%
Random Forest (min samples per leaf = 78, 64 trees)	64.22s / 0.20s	92.29% / 90.38%
Random Forest (min samples per leaf = 78, 128 trees)	128.47s / 0.40s	92.40% / 90.96%

Speeding Up Greedy Feature Selection

In the last post Alex wrote:

"While you can choose and manipulate your features by divination, tea leaf reading, or good old-fashioned gut-following, it's better to actually check whether what you're doing improves performance on a validation set. The trouble is that repeatedly training a classifier can become cumbersome once you leave the cozy world of Small Data. Even worse, if you want to use some sort of automatic feature selection algorithm (which requires training hundreds or thousands of classifiers), you might find yourself waiting for a very very long time."

My favorite variant of feature selection is greedy forward feature selection (GFFS).  First, some notation.  We have $N$ data points $(x_i,y_i)$ where $x_i$ is a $d$-dimensional feature descriptor and $y_i$ is its label, and we want to figure out which of the $d$ dimensions are useful, or maybe rank the dimensions in order of usefulness.  GFFS adds one feature dimension at a time to a set of already selected features, and checks how good that feature is by training & testing classifiers on $k$ cross-validation splits.  The best feature (in terms of average accuracy) is then added to the set of selected features, and the the next iteration begins.

For example, let's say you've already selected feature number $3$.  The algorithm trains and tests the following feature combinations and obtains the corresponding accuracies:

[3 1] - 65.5%
[3 2] - 63.5%
[3 4] - 67.5%
[3 5] - 70.5%    <--- best
...
[3 d] - 62.5%

Feature 3 got the best average hold-out accuracy in combination with feature 5.  So feature 5 is added to the already-chosen set, and the algorithm goes on to the next stage.

The computation time for a fully naive implementation of GFFS is... stupidly big: $O(d^2 k C)$, where $C$ is the complexity of the classifier being used.  In the last post, Alex investigated which classifier is efficient enough to make this feasible.  I found that even using the fastest classifier, the time actual runtime complexity was too ridiculous to be practical (unless you have a cluster).  So... how can we practically speed up GFFS?

Some Ideas

I implemented and tested two ideas:

1. Early stopping - stop the outer loop if there has been no improvement for $p$ epochs  ($p=5$ in all my experiments).
2. Random subsampling - during each epoch, only test $\sqrt{d}$ random dimensions instead of all $d$ dimensions.

Data & Experiments

I tested the above two ideas as follows:

1. Split the data into training and test.  Get test accuracy using all $d$ dimensions
2. Send the training data only into GFFS.
3. Get test accuracy using whichever dimensions are returned from (2).

The best case scenario is that GFFS will get rid of a bunch of noisy dimensions and test accuracy will actually improve.  An acceptable scenario is that GFFS will get rid of a bunch of redundant dimensions and test accuracy will stay the same.

For the classifier, I used quadratic discriminant analysis (QDA), and added $0.1 I$ to the covariance estimates for regularization (chosen without validation).

I tested on three standard datasets: isolet, usps, covertype (with my own training/test split), and two datasets from my research: proteomics and dnasim.

Results

First, the accuracies before and after GFFS.  ES means early stopping.  RS means random subsampling.  A fully naive version of GFFS without early stopping wasn't even close to done running after a whole weekend, so I just stopped it.

Datasets	QDA Test Accuracy	Early Stopping	Early Stopping + Random Subsampling
proteomics	66.8%	79.9%	74.7%
dnasim	64.7%	71.7%	64.8%
isolet	95.9%	94.0%	94.6%
usps	94.0%	92.7%	93.9%
covertype	62.9%	64.0%	62.5%

Using the exhaustive GFFS (with early stopping) can provide some significant gains (proteomics & dnasim), but can also damage performance a bit (isolet and usps).  Adding random subsampling decreases gains somewhat, but not entirely, and can also undo some of the damage caused by feature selection (because more features are usually chosen when using RS).  Looks pretty good, but let's look at the actual runtimes (reported in minutes):

Datasets	Early Stopping	Early Stopping + Random Subsampling
proteomics	5.1 min	0.5 min
dnasim	0.5 min	0.1 min
isolet	698.6 min	41.2 min
usps	115.4 min	4.6 min
covertype	99.5 min	46.0 min

There we go! The time savings range from 50% to 99%. 

In conclusion: greedy forward feature selection is great, but way too expensive. It is still great and much faster if you add (i) early stopping and (ii) random subsampling. 

Next time (maybe):

• Everyone loves PCA.  Is preprocessing your data with PCA actually a good idea in terms of classification performance?
• Feature transforms.  There are a billion of them.  Which ones are a good idea, and when?