Given D=(x,y), y=F(x), it seems most machine learning methods only outputs y as a univariate, either a label or a real value. But I am facing a situation that x vector may only have 5~9 dimensions while I need y to be a multinomial distribution vector which can have up to 800 dimensions. This makes the problem really tricky.
I looked into a lot of things in multitask machine learning methods, where I can train all these y_i at the same time. And of course, another stupid way is that I can also train all these dimensions separately without considering the linkage between tasks. But the problem is, after reviewing many papers, seem that most MTL experiments only deal with 10~30 tasks, which means 800 tasks can be crazy and bad to train. Maybe clustering could be a solution, but I am really curious that can anyone give some suggestions about other ways to deal with this problem, not from a MTL perspective.
When the input is so "small" and the output so big, I would expect there to be a different representation of those output values. You could analyze if they are a linear or nonlinear combination of some sort, so to estimate the "function parameters" instead of the values itself. Example: We once have estimated a time series which could be "reduced" to a weighted sum of normal distributions, so we just had to estimate the weights and parameters.
In the end you will reach only a 6-to-12-dimensional subspace in some sense (not linear, probably) when you have only 6 input parameters. They can of course be a bit complicated, but to avoid the chaos in a 800-dim space I would really look into parametrizing the result.
And as I commented the machine learning that I know produce vectors. http://en.wikipedia.org/wiki/Bayes_estimator
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tl;dr - I use an autoencoder to try to reduce input dimensions for a reinforcement-learning (RL) agent to learn how to play Atari-KungFu. But it fails at encoding/decoding thrown knives, because they are only a couple pixels and getting them wrong probably has negligible impact on the autoencoder MSE loss (see green arrows in bottom left of image). This will probably permanently hobble the results. I want to figure out if there is a way to solve this -- preferably with a generalized solution, but I'd be happy for now with something specific to this problem.
Background:
I am working on Week5 of the "Practical Reinforcement Learning" course on Coursera (National Research University HSE), and I decided to spend extra time trying to expand performance on the Atari-KungFu assignment using Actor-Critic architecture. This post is not about actor-critic, but more about an interesting sub-problem I ran into related to autoencoders.
I create an encoder which outputs a tanh-64-neuron layer, which is used as a common input to the decoder, policy learner (actor), and value learner (critic). During training, the simulator returns batches of four sequential frames (64 x 144 x 4) and rewards from the last action. Then images are first used to train the autoencoder, then used with the rewards to train the actor & critic branches.
I display some metrics and example frames every 25000 iterations to see how it's doing. If the reconstructed images are accurate, then the inputs to the actor & critic branches should be getting good distilled information for efficient learning.
You can see below that the autoencoder is pretty good except for the thrown knives (see bottom-left). Arguably this is because missing those couple pixels minimally increases the MSE loss of the reconstructed image, so it has little incentive to learn it (and also there's not a lot of frames that have knives). Yet, seeing those knives is critical for the RL agent to learn to how to survive.
I haven't seen this kind of problem addressed before. A tiny artifact in the input images is crucial for learning, but is unlikely to be learned by the autoencoder. Can we fix/improve this?
IMO your problem is loss specific, some things which would probably help autoencoder reconstruct knife as well:
Find knives in input image using image processing techniques. Regions where knives are present should have higher loss value in MSE, say 10 times more. One way to find those semi-automatically could probably be convolution with big kernel; White pixels at the strict center would give more weight and only zeros around it would give it more weight as well. Something along these lines should find a region where only knives are located (throwing guys wouldn't, as they contain too many white pixels and holes). Using some threshold found empirically for the value of this kernel should be enough to correctly find them.
Lower loss for images when no knive was found, say divided by half. This would focus autoencoder harder on rarely seen cases when knive is seen.
On the downside - I suppose it could introduce some artifacts. In such case you may think about usage of pretrained encoder (like some version of ResNet) and increase model's capabilities.
Purpose:
I am trying to build a model to classify multiple inputs to a single output class, which is something like this:
{x_i1, x_i2, x_i3, ..., x_i16} (features) to y_i (class)
I am using a SVM to make the classification, but the 0/1-loss was bad (half of the data a misclassified), which leads me to the conclusion that the data might be non-linear. This is why I played around with polynomial basis function. I transformed each coefficient such that I get any combinations of polynomials up to degree 4, in the hope that my features are linear in the transformed space. my new transformed input looks like this:
{x_i1, ..., x_i16, x_i1^2, ..., x_i16^2, ... x_i1^4, ..., x_i16^4, x_i1^3, ..., x_i16^3, x_i1*x_i2, ...}
The loss was minimized but still not quite where I want to go. Since with the number of polynomial degree the chance of overfitting rises, i added regularization in order to counter balance that. I also added a forward greedy algorithm in order to pick up the coefficients which leads to minimal cross-validation error, but with no great improvement.
Question:
Is there a systematic way to figure out which transform leads to linear feature behaviour in the transformed space? Seems little odd to me that I have to try out every polynomial until it "fits". Are there perhaps better basis functions except polynomials? I understand that in low dimensional feature space, one can simply plot the data out and estimate the transform visually, but how can I do it in high dimensional space?
Maybe a little off topic but I also informed myself about PCA in order to throw away the components which doesnt provide much informations in the first place. Is this worth a try?
Thank you for your help.
Did you try other kernel functions such as RBF other than linear and polynomial? Since different dataset may have different characteristics, some kernel functions may work better than others do, especially in non-linear cases.
I don't know which tools you are using, but the following one also provides a guide for beginners on how to build SVM models:
https://www.csie.ntu.edu.tw/~cjlin/libsvm/
It is always a good idea to have a feature selection step first, especially for high-dimensional data. Those noisy or irrelevant features should be taken away, leading to a better performance and higher efficiency.
I have a dataset of approx. 4800 rows with 22 attributes, all numerical, describing mostly the geometry of rock / minerals, and 3 different classes.
I tried out a cross validation with k-nn Model inside it, with k= 7 and Numerical Measure -> Camberra Distance as parameters set..and I got a performance of 82.53% and 0.673 kappa. Is that result representative for the dataset? I mean 82% is quite ok..
Before doing this, I evaluated the best subset of attributes with a decision table, I got out 6 different attributes for that.
the problem is, you still don't learn much from that kind of models, like instance-based k-nn. Can I get any more insight from knn? I don't know how to visualize the clusters in that high dimensional space in Rapidminer, is that somehow possible?
I tried decision tree on the data, but I got too much branches (300 or so) and it looked all too messy, the problem is, all numerical attributes have about the same mean and distribution, therefore its hard to get a distinct subset of meaningful attributes...
ideally, the staff wants to "Learn" something about the data, but my impression is, that you cannot learn much meaningful of that data, all that works best is "Blackbox" Learning models like Neural Nets, SVM, and those other instance-based models...
how should I proceed?
Welcome to the world of machine learning! This sounds like a classic real-world case: we want to make firm conclusions, but the data rows don't cooperate. :-)
Your goal is vague: "learn something"? I'm taking this to mean that you're investigating, hoping to find quantitative discriminations among the three classes.
First of all, I highly recommend Principal Component Analysis (PCA): find out whether you can eliminate some of these attributes by automated matrix operations, rather than a hand-built decision table. I expect that the messy branches are due to unfortunate choice of factors; decision trees work very hard at over-fitting. :-)
How clean are the separations of the data sets? Since you already used Knn, I'm hopeful that you have dense clusters with gaps. If so, perhaps a spectral clustering would help; these methods are good at classifying data based on gaps between the clusters, even if the cluster shapes aren't spherical. Interpretation depends on having someone on staff who can read eigenvectors, to interpret what the values mean.
Try a multi-class SVM. Start with 3 classes, but increase if necessary until your 3 expected classes appear. (Sometimes you get one tiny outlier class, and then two major ones get combined.) The resulting kernel functions and the placement of the gaps can teach you something about your data.
Try the Naive Bayes family, especially if you observe that the features come from a Gaussian or Bernoulli distribution.
As a holistic approach, try a neural net, but use something to visualize the neurons and weights. Letting the human visual cortex play with relationships can help extract subtle relationships.
I'm using WEKA/LibSVM to train a classifier for a term extraction system. My data is not linearly separable, so I used an RBF kernel instead of a linear one.
I followed the guide from Hsu et al. and iterated over several values for both c and gamma. The parameters which worked best for classifying known terms (test and training material differ of course) are rather high, c=2^10 and gamma=2^3.
So far the high parameters seem to work ok, yet I wonder if they may cause any problems further on, especially regarding overfitting. I plan to do another evaluation by extracting new terms, yet those are costly as I need human judges.
Could anything still be wrong with my parameters, even if both evaluation turns out positive? Do I perhaps need another kernel type?
Thank you very much!
In general you have to perform cross validation to answer whether the parameters are all right or do they lead to the overfitting.
From the "intuition" perspective - it seems like highly overfitted model. High value of gamma means that your Gaussians are very narrow (condensed around each poinT) which combined with high C value will result in memorizing most of the training set. If you check out the number of support vectors I would not be surprised if it would be the 50% of your whole data. Other possible explanation is that you did not scale your data. Most ML methods, especially SVM, requires data to be properly preprocessed. This means in particular, that you should normalize (standarize) the input data so it is more or less contained in the unit sphere.
RBF seems like a reasonable choice so I would keep using it. A high value of gamma is not necessary a bad thing, it would depends on the scale where your data lives. While a high C value can lead to overfitting, it would also be affected by the scale so in some cases it might be just fine.
If you think that your dataset is a good representation of the whole data, then you could use crossvalidation to test your parameters and have some peace of mind.
How should I approach a situtation when I try to apply some ML algorithm (classification, to be more specific, SVM in particular) over some high dimensional input, and the results I get are not quite satisfactory?
1, 2 or 3 dimensional data can be visualized, along with the algorithm's results, so you can get the hang of what's going on, and have some idea how to aproach the problem. Once the data is over 3 dimensions, other than intuitively playing around with the parameters I am not really sure how to attack it?
What do you do to the data? My answer: nothing. SVMs are designed to handle high-dimensional data. I'm working on a research problem right now that involves supervised classification using SVMs. Along with finding sources on the Internet, I did my own experiments on the impact of dimensionality reduction prior to classification. Preprocessing the features using PCA/LDA did not significantly increase classification accuracy of the SVM.
To me, this totally makes sense from the way SVMs work. Let x be an m-dimensional feature vector. Let y = Ax where y is in R^n and x is in R^m for n < m, i.e., y is x projected onto a space of lower dimension. If the classes Y1 and Y2 are linearly separable in R^n, then the corresponding classes X1 and X2 are linearly separable in R^m. Therefore, the original subspaces should be "at least" as separable as their projections onto lower dimensions, i.e., PCA should not help, in theory.
Here is one discussion that debates the use of PCA before SVM: link
What you can do is change your SVM parameters. For example, with libsvm link, the parameters C and gamma are crucially important to classification success. The libsvm faq, particularly this entry link, contains more helpful tips. Among them:
Scale your features before classification.
Try to obtain balanced classes. If impossible, then penalize one class more than the other. See more references on SVM imbalance.
Check the SVM parameters. Try many combinations to arrive at the best one.
Use the RBF kernel first. It almost always works best (computationally speaking).
Almost forgot... before testing, cross validate!
EDIT: Let me just add this "data point." I recently did another large-scale experiment using the SVM with PCA preprocessing on four exclusive data sets. PCA did not improve the classification results for any choice of reduced dimensionality. The original data with simple diagonal scaling (for each feature, subtract mean and divide by standard deviation) performed better. I'm not making any broad conclusion -- just sharing this one experiment. Maybe on different data, PCA can help.
Some suggestions:
Project data (just for visualization) to a lower-dimensional space (using PCA or MDS or whatever makes sense for your data)
Try to understand why learning fails. Do you think it overfits? Do you think you have enough data? Is it possible there isn't enough information in your features to solve the task you are trying to solve? There are ways to answer each of these questions without visualizing the data.
Also, if you tell us what the task is and what your SVM output is, there may be more specific suggestions people could make.
You can try reducing the dimensionality of the problem by PCA or the similar technique. Beware that PCA has two important points. (1) It assumes that the data it is applied to is normally distributed and (2) the resulting data looses its natural meaning (resulting in a blackbox). If you can live with that, try it.
Another option is to try several parameter selection algorithms. Since SVM's were already mentioned here, you might try the approach of Chang and Li (Feature Ranking Using Linear SVM) in which they used linear SVM to pre-select "interesting features" and then used RBF - based SVM on the selected features. If you are familiar with Orange, a python data mining library, you will be able to code this method in less than an hour. Note that this is a greedy approach which, due to its "greediness" might fail in cases where the input variables are highly correlated. In that case, and if you cannot solve this problem with PCA (see above), you might want to go to heuristic methods, which try to select best possible combinations of predictors. The main pitfall of this kind of approaches is the high potential of overfitting. Make sure you have a bunch "virgin" data that was not seen during the entire process of model building. Test your model on that data only once, after you are sure that the model is ready. If you fail, don't use this data once more to validate another model, you will have to find a new data set. Otherwise you won't be sure that you didn't overfit once more.
List of selected papers on parameter selection:
Feature selection for high-dimensional genomic microarray data
Oh, and one more thing about SVM. SVM is a black box. You better figure out what is the mechanism that generate the data and model the mechanism and not the data. On the other hand, if this would be possible, most probably you wouldn't be here asking this question (and I wouldn't be so bitter about overfitting).
List of selected papers on parameter selection
Feature selection for high-dimensional genomic microarray data
Wrappers for feature subset selection
Parameter selection in particle swarm optimization
I worked in the laboratory that developed this Stochastic method to determine, in silico, the drug like character of molecules
I would approach the problem as follows:
What do you mean by "the results I get are not quite satisfactory"?
If the classification rate on the training data is unsatisfactory, it implies that either
You have outliers in your training data (data that is misclassified). In this case you can try algorithms such as RANSAC to deal with it.
Your model(SVM in this case) is not well suited for this problem. This can be diagnozed by trying other models (adaboost etc.) or adding more parameters to your current model.
The representation of the data is not well suited for your classification task. In this case preprocessing the data with feature selection or dimensionality reduction techniques would help
If the classification rate on the test data is unsatisfactory, it implies that your model overfits the data:
Either your model is too complex(too many parameters) and it needs to be constrained further,
Or you trained it on a training set which is too small and you need more data
Of course it may be a mixture of the above elements. These are all "blind" methods to attack the problem. In order to gain more insight into the problem you may use visualization methods by projecting the data into lower dimensions or look for models which are suited better to the problem domain as you understand it (for example if you know the data is normally distributed you can use GMMs to model the data ...)
If I'm not wrong, you are trying to see which parameters to the SVM gives you the best result. Your problem is model/curve fitting.
I worked on a similar problem couple of years ago. There are tons of libraries and algos to do the same. I used Newton-Raphson's algorithm and a variation of genetic algorithm to fit the curve.
Generate/guess/get the result you are hoping for, through real world experiment (or if you are doing simple classification, just do it yourself). Compare this with the output of your SVM. The algos I mentioned earlier reiterates this process till the result of your model(SVM in this case) somewhat matches the expected values (note that this process would take some time based your problem/data size.. it took about 2 months for me on a 140 node beowulf cluster).
If you choose to go with Newton-Raphson's, this might be a good place to start.