What is the best way to iterate da.linalg.inv over a multi-dimensional dask array?
I have a dask array of shape (4, 4, 8, 8), and need to compute the inverse of the last two dimensions. With numpy, np.linalg.inv loops over all dimensions except the last two, so in the following example, I can just call np.linalg.inv(A).
I have chosen to use a for loop, but I have read about gufuncs in dask (the documentation seems a little outdated). However, I'm not sure how to implement the it, particularly the "signature" bit,
import dask.array as da
import numpy as np
A = da.random.random((4,4,8,8))
A2 = A.reshape((-1,) + A.shape[-2:])
B = [da.linalg.inv(a) for a in A2]
B2 = da.asarray(B)
B3 = B2.reshape(A.shape)
np.testing.assert_array_almost_equal(
np.linalg.inv(A.compute()),
B3
)
My attempt at a gufunc leads to an error:
def foo(x):
return da.linalg.inv(x)
gufoo = da.gufunc(foo, signature="()->()", output_dtypes=float, vectorize=True)
gufoo(A2).compute() # IndexError: tuple index out of range
I think that you want to apply the numpy function np.linalg.inv over your Dask array rather than the dask array function.
If np.linalg.inv is already a gufunc then it might work as expected today
np.linalg.inv(A)
i'm still new in Julia and in machine learning in general, but I'm quite eager to learn. In the current project i'm working on I have a problem about dimensions mismatch, and can't figure what to do.
I have two arrays as follow:
x_array:
9-element Array{Array{Int64,N} where N,1}:
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 72, 73]
[11, 12, 13, 14, 15, 16, 17, 72, 73]
[18, 12, 19, 20, 21, 22, 72, 74]
[23, 24, 12, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 72, 74]
[36, 37, 38, 39, 40, 38, 41, 42, 72, 73]
[43, 44, 45, 46, 47, 48, 72, 74]
[49, 50, 51, 52, 14, 53, 72, 74]
[54, 55, 41, 56, 57, 58, 59, 60, 61, 62, 63, 62, 64, 72, 74]
[65, 66, 67, 68, 32, 69, 70, 71, 72, 74]
y_array:
9-element Array{Int64,1}
75
76
77
78
79
80
81
82
83
and the next model using Flux:
model = Chain(
LSTM(10, 256),
LSTM(256, 128),
LSTM(128, 128),
Dense(128, 9),
softmax
)
I zip both arrays, and then feed them into the model using Flux.train!
data = zip(x_array, y_array)
Flux.train!(loss, Flux.params(model), data, opt)
and immediately throws the next error:
ERROR: DimensionMismatch("matrix A has dimensions (1024,10), vector B has length 9")
Now, I know that the first dimension of matrix A is the sum of the hidden layers (256 + 256 + 128 + 128 + 128 + 128) and the second dimension is the input layer, which is 10. The first thing I did was change the 10 for a 9, but then it only throws the error:
ERROR: DimensionMismatch("dimensions must match")
Can someone explain to me what dimensions are the ones that mismatch, and how to make them match?
Introduction
First off, you should know that from an architectural standpoint, you are asking something very difficult from your network; softmax re-normalizes outputs to be between 0 and 1 (weighted like a probability distribution), which means that asking your network to output values like 77 to match y will be impossible. That's not what is causing the dimension mismatch, but it's something to be aware of. I'm going to drop the softmax() at the end to give the network a fighting chance, especially since it's not what's causing the problem.
Debugging shape mismatches
Let's walk through what actually happens inside of Flux.train!(). The definition is actually surprisingly simple. Ignoring everything that doesn't matter to us, we are left with:
for d in data
gs = gradient(ps) do
loss(d...)
end
end
Therefore, let's start by pulling the first element out of your data, and splatting it into your loss function. You didn't specify your loss function or optimizer in the question. Although softmax usually means you should use crossentropy loss, your y values are very much not probabilities, and so if we drop the softmax we can just use the dead-simple mse() loss. For optimizer, we'll default to good old ADAM:
model = Chain(
LSTM(10, 256),
LSTM(256, 128),
LSTM(128, 128),
Dense(128, 9),
#softmax, # commented out for now
)
loss(x, y) = Flux.mse(model(x), y)
opt = ADAM(0.001)
data = zip(x_array, y_array)
Now, to simulate the first run of Flux.train!(), we take first(data) and splat that into loss():
loss(first(data)...)
This gives us the error message you've seen before; ERROR: DimensionMismatch("matrix A has dimensions (1024,10), vector B has length 12"). Looking at our data, we see that yes, indeed, the first element of our dataset has a length of 12. And so we will change our model to instead expect 12 values instead of 10:
model = Chain(
LSTM(12, 256),
LSTM(256, 128),
LSTM(128, 128),
Dense(128, 9),
)
And now we re-run:
julia> loss(first(data)...)
50595.52542674723 (tracked)
Huzzah! It worked! We can run this again:
julia> loss(first(data)...)
50578.01417593167 (tracked)
The value changes because the RNN holds memory within itself which gets updated each time we run the network, otherwise we would expect the network to give the same answer for the same inputs!
The problem comes, however, when we try to run the second training instance through our network:
julia> loss([d for d in data][2]...)
ERROR: DimensionMismatch("matrix A has dimensions (1024,12), vector B has length 9")
Understanding LSTMs
This is where we run into Machine Learning problems more than programming problems; the issue here is that we have promised to feed that first LSTM network a vector of length 10 (well, 12 now) and we are breaking that promise. This is a general rule of deep learning; you always have to obey the contracts you sign about the shape of the tensors that are flowing through your model.
Now, the reasons you're using LSTMs at all is probably because you want to feed in ragged data, chew it up, then do something with the result. Maybe you're processing sentences, which are all of variable length, and you want to do sentiment analysis, or somesuch. The beauty of recurrent architectures like LSTMs is that they are able to carry information from one execution to another, and they are therefore able to build up an internal representation of a sequence when applied upon one time point after another.
When building an LSTM layer in Flux, you are therefore declaring not the length of the sequence you will feed in, but rather the dimensionality of each time point; imagine if you had an accelerometer reading that was 1000 points long and gave you X, Y, Z values at each time point; to read that in, you would create an LSTM that takes in a dimensionality of 3, then feed it 1000 times.
Writing our own training loop
I find it very instructive to write our own training loop and model execution function so that we have full control over everything. When dealing with time series, it's often easy to get confused about how to call LSTMs and Dense layers and whatnot, so I offer these simple rules of thumb:
When mapping from one time series to another (E.g. constantly predict future motion from previous motion), you can use a single Chain and call it in a loop; for every input time point, you output another.
When mapping from a time series to a single "output" (E.g. reduce sentence to "happy sentiment" or "sad sentiment") you must first chomp all the data up and reduce it to a fixed size; you feed many things in, but at the end, only one comes out.
We're going to re-architect our model into two pieces; first the recurrent "pacman" section, where we chomp up a variable-length time sequence into an internal state vector of pre-determined length, then a feed-forward section that takes that internal state vector and reduces it down to a single output:
pacman = Chain(
LSTM(1, 128), # map from timepoint size 1 to 128
LSTM(128, 256), # blow it up even larger to 256
LSTM(256, 128), # bottleneck back down to 128
)
reducer = Chain(
Dense(128, 9),
#softmax, # keep this commented out for now
)
The reason we split it up into two pieces like this is because the problem statement wants us to reduce a variable-length input series to a single number; we're in the second bullet point above. So our code naturally must take this into account; we will write our loss(x, y) function to, instead of calling model(x), it will instead do the pacman dance, then call the reducer on the output. Note that we also must reset!() the RNN state so that the internal state is cleared for each independent training example:
function loss(x, y)
# Reset internal RNN state so that it doesn't "carry over" from
# the previous invocation of `loss()`.
Flux.reset!(pacman)
# Iterate over every timepoint in `x`
for x_t in x
y_hat = pacman(x_t)
end
# Take the very last output from the recurrent section, reduce it
y_hat = reducer(y_hat)
# Calculate reduced output difference against `y`
return Flux.mse(y_hat, y)
end
Feeding this into Flux.train!() actually trains, albeit not very well. ;)
Final observations
Although your data is all Int64's, it's pretty typical to use floating point numbers with everything except embeddings (an embedding is a way to take non-numeric data such as characters or words and assign numbers to them, kind of like ASCII); if you're dealing with text, you're almost certainly going to be working with some kind of embedding, and that embedding will dictate what the dimensionality of your first LSTM is, whereupon your inputs will all be "one-hot" encoded.
softmax is used when you want to predict probabilities; it's going to ensure that for each input, the outputs are all between [0...1] and moreover that they sum to 1.0, like a good little probability distribution should. This is most useful when doing classification, when you want to wrangle your wild network output values of [-2, 5, 0.101] into something where you can say "we have 99.1% certainty that the second class is correct, and 0.7% certainty it's the third class."
When training these networks, you're often going to want to batch multiple time series at once through your network for hardware efficiency reasons; this is both simple and complex, because on one hand it just means that instead of passing a single Sx1 vector through (where S is the size of your embedding) you're instead going to be passing through an SxN matrix, but it also means that the number of timesteps of everything within your batch must match (because the SxN must remain the same across all timesteps, so if one time series ends before any of the others in your batch you can't just drop it and thereby reduce N halfway through a batch). So what most people do is pad their timeseries all to the same length.
Good luck in your ML journey!
I have a distributed dask array with shape (2400,2400) with chunksize (100,100). I thought I could use topk(-n) to find the smallest n values. However, it appears to return an array of shape (2400,n), so it looks like it finds the smallest n in each row.Is there a way to use topk to get the smallest n values across all rows (entire array)?
One idea is to call topk twice, once for each axis.
>>> dist
dask.array<pow, shape=(2400, 2400), dtype=float64, chunksize=(100, 100)>
>>> dist.topk(-5,axis=0).topk(-5,axis=1).compute()
array([[ 0. , 2620.09503644, 2842.15200157, 2955.08409356,
3163.49458669],
[3660.67698657, 3670.4457495 , 3700.09837707, 3717.09052889,
4002.86497399],
[4125.89820524, 4139.44658137, 4250.50420539, 4331.01304547,
4402.14606754],
[4328.22966119, 4378.25193428, 4507.94409903, 4522.4913488 ,
4555.06860541],
[4441.58755402, 4560.95625938, 4576.39333974, 4682.06215251,
4765.11531865]])
One idea is to call topk twice, once for each axis.
Sounds good to me!
You might consider flattening the array first, but I can't see an advantage to this to what you've already found.
x.flatten().topk(...)
Suppose wanted to train a machine learning algorithm on some dataset including some categorical parameters. (New to machine learning, but my thinking is...) Even if converted all the categorical data to 1-hot-encoded vectors, how will this encoding map be "remembered" after training?
Eg. converting the initial dataset to use 1-hot encoding before training, say
universe of categories for some column c is {"good","bad","ok"}, so convert rows to
[1, 2, "good"] ---> [1, 2, [1, 0, 0]],
[3, 4, "bad"] ---> [3, 4, [0, 1, 0]],
...
, after training the model, all future prediction inputs would need to use the same encoding scheme for column c.
How then during future predictions will data inputs remember that mapping (where "good" maps to index 0, etc.) (Specifically, when planning on using a keras RNN or LSTM model)? Do I need to save it somewhere (eg. python pickle)(if so, how do I get the explicit mapping)? Or is there a way to have the model automatically handle categorical inputs internally so can just input the original label data during training and future use?
If anything in this question shows any serious confusion on my part about something, please let me know (again, very new to ML).
** Wasn't sure if this belongs in https://stats.stackexchange.com/, but posted here since specifically wanted to know how to deal with the actual code implementation of this problem.
What I've been doing is the following:
After you use StringIndexer.fit(), you can save its metadata (includes the actual encoder mapping, like "good" being the first column)
This is the following code I use (using java, but can be adjusted to python):
StringIndexerModel sim = new StringIndexer()
.setInputCol(field)
.setOutputCol(field + "_INDEX")
.setHandleInvalid("skip")
.fit(dataset);
sim.write().overwrite().save("IndexMappingModels/" + field + "_INDEX");
and later, when trying to make predictions on a new dataset, you can load the stored metadata:
StringIndexerModel sim = StringIndexerModel.load("IndexMappingModels/" + field + "_INDEX");
dataset = sim.transform(dataset);
I imagine you have already solved this issue, since it was posted in 2018, but I've not found this solution anywhere else, so I believe its worth sharing.
My thought would be to do something like this on the training/testing dataset D (using a mix of python and plain psudo-code):
Do something like
# Before: D.schema == {num_col_1: int, cat_col_1: str, cat_col_2: str, ...}
# assign unique index for each distinct label for categorical column annd store in a new column
# http://spark.apache.org/docs/latest/ml-features.html#stringindexer
label_indexer = StringIndexer(inputCol="cat_col_i", outputCol="cat_col_i_index").fit(D)
D = label_indexer.transform(D)
# After: D.schema == {num_col_1: int, cat_col_1: str, cat_col_2: str, ..., cat_col_1_index: int, cat_col_2_index: int, ...}
for all the categorical columns
Then for all of these categorical name and index columns in D, make a map of form
map = {}
for all categorical column names colname in D:
map[colname] = []
# create mapping dict for all categorical values for all
# see https://spark.apache.org/docs/latest/sql-programming-guide.html#untyped-dataset-operations-aka-dataframe-operations
for all rows r in D.select(colname, '%s_index' % colname).drop_duplicates():
enc_from = r['%s' % colname]
enc_to = r['%s_index' % colname]
map[colname].append((enc_from, enc_to))
# for cats that may appear later that have yet to be seen
# (IDK if this is best practice, may be another way, see https://medium.com/#vaibhavshukla182/how-to-solve-mismatch-in-train-and-test-set-after-categorical-encoding-8320ed03552f)
map[colname].append(('NOVEL_CAT', map[colname].len))
# sort by index encoding
map[colname].sort(key = lamdba pair: pair[1])
to end up with something like
{
'cat_col_1': [('orig_label_11', 0), ('orig_label_12', 1), ...],
'cat_col_2': [(), (), ...],
...
'cat_col_n': [(orig_label_n1, 0), ...]
}
which can then be used to generate 1-hot-encoded vectors for each categorical column in any later data sample row ds. Eg.
for all categorical column names colname in ds:
enc_from = ds[colname]
# make zero vector for 1-hot for category
col_onehot = zeros.(size = map[colname].len)
for label, index in map[colname]:
if (label == enc_from):
col_onehot[index] = 1
# make new column in sample for 1-hot vector
ds['%s_onehot' % colname] = col_onehot
break
Can then save this structure as pickle pickle.dump( map, open( "cats_map.pkl", "wb" ) ) to use to compare against categorical column values when making actual predictions later.
** There may be a better way, but I think would need to better understand this article (https://medium.com/#satnalikamayank12/on-learning-embeddings-for-categorical-data-using-keras-165ff2773fc9). Will update answer if anything.
In dask what is the difference between
df.col.unique()
and
df.col.drop_duplicates()
Both return a series containing the unique elements of df.col.
There is a difference in the index, unique result is indexed by 1..N while drop_duplicates indexed by an arbitrary looking sequence of numbers.
What is the significance of the index returned by drop_duplicates?
Is there any reason to use one over the other if the index is not important?
Dask.dataframe has both because Pandas has both, and dask.dataframe mostly copies the Pandas API. Unique is a holdover from Pandas' history with Numpy.
In [1]: import pandas as pd
In [2]: df = pd.DataFrame({'x': [1, 2, 1], 'y': [1., 2., 3.]}, index=pd.Index(['a', 'b', 'A'], name='I'))
In [3]: df.x.drop_duplicates()
Out[3]:
I
a 1
b 2
Name: x, dtype: int64
In [4]: df.x.unique()
Out[4]: array([1, 2])
In dask.dataframe we deviate slightly and choose to use a dask.dataframe.Series rather than a dask.array.Array because one can't precompute the length of the array and so can't act lazily.
In practice there is little reason to use unique over drop_duplicates