Does PyTorch support dict-like objects, through which we can backpropagate gradients, like Tensors in PyTorch?
My goal is to compute gradients with respect to a few (1%) elements of a large matrix. But if I use PyTorch's standard Tensors to store the matrix, I need to keep the whole matrix in my GPU, which causes problems due to limited GPU memory available during training. So I was thinking whether I could store the matrix as a dict instead, indexing only the relevant elements of the matrix, and computing gradients and backpropagating w.r.t those select elements only.
So far, I have tried using Tensors only, but it's causing memory issues for the above reasons. So I searched extensively for alternate options like dicts in PyTorch but couldn't find any such information on Google.
It sounds like you want your parameter to be a torch.sparse tensor.
This interface allows you to have tensors that are mostly zeros, with only a few non-zero elements in known locations. Sparse tensors should allow you to significantly reduce the memory footprint of your model.
Note that this interface is still "under construction": not all operations are supported for sparse tensors. However, it is being constantly improving.
Related
Let's say I have a network model made in TensorFlow/Keras/Caffe etc.
I can use CoreML Converters API to get a CoreML model file (.mlmodel) from it.
Now, as I have a .mlmodel file, and know input shape and output shape, how can a maximum RAM footprint be estimated?
I know that a model сan have a lot of layers, their size can be much bigger than input/output shape.
So the questions are:
Can be a maximal mlmodel memory footprint be known with some formula/API, without compiling and running an app?
Is a maximal footprint closer to a memory size of the biggest intermediate layer, or is it closer to a sum of the all layer's sizes?
Any advice is appreciated.
As I am new to CoreML, you may give any feedback and I'll try to improve the question if needed.
IMHO, whatever formula you come up with at the end of the day must be based on the number of trainable parameters of the network.
For classifying networks it can be found by iterating or the existing API can be used.
In keras.
import keras.applications.resnet50 as resnet
model =resnet.ResNet50(include_top=True, weights=None, input_tensor=None, input_shape=None, pooling=None, classes=2)
print model.summary()
Total params: 23,591,810
Trainable params: 23,538,690
Non-trainable params: 53,120
Pytorch:
def count_parameters(model):
return sum(p.numel() for p in model.parameters() if p.requires_grad)
For the detectors, you probably need to do the same for all the important parts of the network, including the backbone, rpn, etc., whatever your network consists of.
The second important parameter is the precision of the network. You must be heard about quantization. It changes the precision of floats for all or some layers and can be static (when the network is trained in desired precision and calibrated) or dynamic when the network is converted after the training. The simplest dynamic quantization replaces floats to some kind of ints on linear layers. Maskrcnn in pytorch results in 30% smaller file size and a substantial reduction in memory consumption with the same number of trainable parameters.
So the final equation is like size = number_of_trainable_parameters * precision * X, where X is some factor you have to find out for your particular network and coreml specifics )
I wrote a blog post a few years ago that goes into some of this: https://machinethink.net/blog/how-fast-is-my-model/
However, keep in mind that Core ML's actual behavior is not known. It will most likely try to be as efficient as possible (i.e. reuse the memory for tensors that are no longer needed) but it's a black box, so who knows. The only way to find out is to try your models on an actual device.
I am new to machine learning field and right now trying to get a grasp of how the most common learning algorithms work and understand when to apply each one of them. At the moment I am learning on how Support Vector Machines work and have a question on custom kernel functions.
There is plenty of information on the web on more standard (linear, RBF, polynomial) kernels for SVMs. I, however, would like to understand when it is reasonable to go for a custom kernel function. My questions are:
1) What are other possible kernels for SVMs?
2) In which situation one would apply custom kernels?
3) Can custom kernel substantially improve prediction quality of SVM?
1) What are other possible kernels for SVMs?
There are infinitely many of these, see for example list of ones implemented in pykernels (which is far from being exhaustive)
https://github.com/gmum/pykernels
Linear
Polynomial
RBF
Cosine similarity
Exponential
Laplacian
Rational quadratic
Inverse multiquadratic
Cauchy
T-Student
ANOVA
Additive Chi^2
Chi^2
MinMax
Min/Histogram intersection
Generalized histogram intersection
Spline
Sorensen
Tanimoto
Wavelet
Fourier
Log (CPD)
Power (CPD)
2) In which situation one would apply custom kernels?
Basically in two cases:
"simple" ones give very bad results
data is specific in some sense and so - in order to apply traditional kernels one has to degenerate it. For example if your data is in a graph format, you cannot apply RBF kernel, as graph is not a constant-size vector, thus you need a graph kernel to work with this object without some kind of information-loosing projection. also sometimes you have an insight into data, you know about some underlying structure, which might help classifier. One such example is a periodicity, you know that there is a kind of recuring effect in your data - then it might be worth looking for a specific kernel etc.
3) Can custom kernel substantially improve prediction quality of SVM?
Yes, in particular there always exists a (hypothethical) Bayesian optimal kernel, defined as:
K(x, y) = 1 iff arg max_l P(l|x) == arg max_l P(l|y)
in other words, if one has a true probability P(l|x) of label l being assigned to a point x, then we can create a kernel, which pretty much maps your data points onto one-hot encodings of their most probable labels, thus leading to Bayes optimal classification (as it will obtain Bayes risk).
In practise it is of course impossible to get such kernel, as it means that you already solved your problem. However, it shows that there is a notion of "optimal kernel", and obviously none of the classical ones is not of this type (unless your data comes from veeeery simple distributions). Furthermore, each kernel is a kind of prior over decision functions - closer you get to the actual one with your induced family of functions - the more probable is to get a reasonable classifier with SVM.
If the data to cluster are literally points (either 2D (x, y) or 3D (x, y,z)), it would be quite intuitive to choose a clustering method. Because we can draw them and visualize them, we somewhat know better which clustering method is more suitable.
e.g.1 If my 2D data set is of the formation shown in the right top corner, I would know that K-means may not be a wise choice here, whereas DBSCAN seems like a better idea.
However, just as the scikit-learn website states:
While these examples give some intuition about the algorithms, this
intuition might not apply to very high dimensional data.
AFAIK, in most of the piratical problems we don't have such simple data. Most probably, we have high-dimensional tuples, which cannot be visualized like such, as data.
e.g.2 I wish to cluster a data set where each data is represented as a 4-D tuple <characteristic1, characteristic2, characteristic3, characteristic4>. I CANNOT visualize it in a coordinate system and observes its distribution like before. So I will NOT be able to say DBSCAN is superior to K-means in this case.
So my question:
How does one choose the suitable clustering method for such an "invisualizable" high-dimensional case?
"High-dimensional" in clustering probably starts at some 10-20 dimensions in dense data, and 1000+ dimensions in sparse data (e.g. text).
4 dimensions are not much of a problem, and can still be visualized; for example by using multiple 2d projections (or even 3d, using rotation); or using parallel coordinates. Here's a visualization of the 4-dimensional "iris" data set using a scatter plot matrix.
However, the first thing you still should do is spend a lot of time on preprocessing, and finding an appropriate distance function.
If you really need methods for high-dimensional data, have a look at subspace clustering and correlation clustering, e.g.
Kriegel, Hans-Peter, Peer Kröger, and Arthur Zimek. Clustering high-dimensional data: A survey on subspace clustering, pattern-based clustering, and correlation clustering. ACM Transactions on Knowledge Discovery from Data (TKDD) 3.1 (2009): 1.
The authors of that survey also publish a software framework which has a lot of these advanced clustering methods (not just k-means, but e.h. CASH, FourC, ERiC): ELKI
There are at least two common, generic approaches:
One can use some dimensionality reduction technique in order to actually visualize the high dimensional data, there are dozens of popular solutions including (but not limited to):
PCA - principal component analysis
SOM - self-organizing maps
Sammon's mapping
Autoencoder Neural Networks
KPCA - kernel principal component analysis
Isomap
After this one goes back to the original space and use some techniques that seems resonable based on observations in the reduced space, or performs clustering in the reduced space itself.First approach uses all avaliable information, but can be invalid due to differences induced by the reduction process. While the second one ensures that your observations and choice is valid (as you reduce your problem to the nice, 2d/3d one) but it loses lots of information due to transformation used.
One tries many different algorithms and choose the one with the best metrics (there have been many clustering evaluation metrics proposed). This is computationally expensive approach, but has a lower bias (as reducting the dimensionality introduces the information change following from the used transformation)
It is true that high dimensional data cannot be easily visualized in an euclidean high dimensional data but it is not true that there are no visualization techniques for them.
In addition to this claim I will add that with just 4 features (your dimensions) you can easily try the parallel coordinates visualization method. Or simply try a multivariate data analysis taking two features at a time (so 6 times in total) to try to figure out which relations intercour between the two (correlation and dependency generally). Or you can even use a 3d space for three at a time.
Then, how to get some info from these visualizations? Well, it is not as easy as in an euclidean space but the point is to spot visually if the data clusters in some groups (eg near some values on an axis for a parallel coordinate diagram) and think if the data is somehow separable (eg if it forms regions like circles or line separable in the scatter plots).
A little digression: the diagram you posted is not indicative of the power or capabilities of each algorithm given some particular data distributions, it simply highlights the nature of some algorithms: for instance k-means is able to separate only convex and ellipsoidail areas (and keep in mind that convexity and ellipsoids exist even in N-th dimensions). What I mean is that there is not a rule that says: given the distributiuons depicted in this diagram, you have to choose the correct clustering algorithm consequently.
I suggest to use a data mining toolbox that lets you explore and visualize the data (and easily transform them since you can change their topology with transformations, projections and reductions, check the other answer by lejlot for that) like Weka (plus you do not have to implement all the algorithms by yourself.
In the end I will point you to this resource for different cluster goodness and fitness measures so you can compare the results rfom different algorithms.
I would also suggest soft subspace clustering, a pretty common approach nowadays, where feature weights are added to find the most relevant features. You can use these weights to increase performance and improve the BMU calculation with euclidean distance, for example.
For clustering, Mahout input needs to be in vector form. There are two types of vector implementations. One is Sparse Vector and another is Dense Vector.
What is difference between two ?
Usage scenarios for Sparse and Dense ?
Concept-wise, most of the values in a sparse vector are zero, in a dense vector they are not. Same for dense and sparse matrices. The terms sparse and dense generally describe these properties, not only in Mahout.
In Mahout the DenseVector assumes not too many zero entries and therefore "Implements vector as an array of doubles" (org.apache.mahout.math.DenseVector). In contrast, the sparse vector implementations of AbstractVector, e.g. RandomAccessSparseVector and SequentialAccessSparseVector, use different data structures which don't store the zero values at all.
Which one to take depends on the data you want to store in the vector. If you expect mostly zero values, a sparse vector implementation would be more space efficient, however if you use it for data with just a few zero values you introduce a lot of data structure overhead which could cause worse performance.
The choice of dense vs. sparse vector does not affect your calculation results on the vectors, only memory usage and calculation speed.
I have some problems with understanding the kernels for non-linear SVM.
First what I understood by non-linear SVM is: using kernels the input is transformed to a very high dimension space where the transformed input can be separated by a linear hyper-plane.
Kernel for e.g: RBF:
K(x_i, x_j) = exp(-||x_i - x_j||^2/(2*sigma^2));
where x_i and x_j are two inputs. here we need to change the sigma to adapt to our problem.
(1) Say if my input dimension is d, what will be the dimension of the
transformed space?
(2) If the transformed space has a dimension of more than 10000 is it
effective to use a linear SVM there to separate the inputs?
Well it is not only a matter of increasing the dimension. That's the general mechanism but not the whole idea, if it were true that the only goal of the kernel mapping is to increase the dimension, one could conclude that all kernels functions are equivalent and they are not.
The way how the mapping is made would make possible a linear separation in the new space.
Talking about your example and just to extend a bit what greeness said, RBF kernel would order the feature space in terms of hyperspheres where an input vector would need to be close to an existing sphere in order to produce an activation.
So to answer directly your questions:
1) Note that you don't work on feature space directly. Instead, the optimization problem is solved using the inner product of the vectors in the feature space, so computationally you won't increase the dimension of the vectors.
2) It would depend on the nature of your data, having a high dimensional pattern would somehow help you to prevent overfitting but not necessarily will be linearly separable. Again, the linear separability in the new space would be achieved because the way the map is made and not only because it is in a higher dimension. In that sense, RBF would help but keep in mind that it might not perform well on generalization if your data is not locally enclosed.
The transformation usually increases the number of dimensions of your data, not necessarily very high. It depends. The RBF Kernel is one of the most popular kernel functions. It adds a "bump" around each data point. The corresponding feature space is a Hilbert space of infinite dimensions.
It's hard to tell if a transformation into 10000 dimensions is effective or not for classification without knowing the specific background of your data. However, choosing a good mapping (encoding prior knowledge + getting right complexity of function class) for your problem improves results.
For example, the MNIST database of handwritten digits contains 60K training examples and 10K test examples with 28x28 binary images.
Linear SVM has ~8.5% test error.
Polynomial SVM has ~ 1% test error.
Your question is a very natural one that almost everyone who's learned about kernel methods has asked some variant of. However, I wouldn't try to understand what's going on with a non-linear kernel in terms of the implied feature space in which the linear hyperplane is operating, because most non-trivial kernels have feature spaces that it is very difficult to visualise.
Instead, focus on understanding the kernel trick, and think of the kernels as introducing a particular form of non-linear decision boundary in input space. Because of the kernel trick, and some fairly daunting maths if you're not familiar with it, any kernel function satisfying certain properties can be viewed as operating in some feature space, but the mapping into that space is never performed. You can read the following (fairly) accessible tutorial if you're interested: from zero to Reproducing Kernel Hilbert Spaces in twelve pages or less.
Also note that because of the formulation in terms of slack variables, the hyperplane does not have to separate points exactly: there's an objective function that's being maximised which contains penalties for misclassifying instances, but some misclassification can be tolerated if the margin of the resulting classifier on most instances is better. Basically, we're optimising a classification rule according to some criteria of:
how big the margin is
the error on the training set
and the SVM formulation allows us to solve this efficiently. Whether one kernel or another is better is very application-dependent (for example, text classification and other language processing problems routinely show best performance with a linear kernel, probably due to the extreme dimensionality of the input data). There's no real substitute for trying a bunch out and seeing which one works best (and make sure the SVM hyperparameters are set properly---this talk by one of the LibSVM authors has the gory details).