[WIP] Add large svd draft

_posts/2019-07-02-very-large-svds.md

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+---
+layout: post
+title: Very Large SVDs
+tagline: Dask + CuPy + Zarr + Genomics
+author: Matthew Rocklin & Alistair Miles
+tags: [GPU, array, cupy]
+draft: true
+theme: twitter
+---
+{% include JB/setup %}
+
+Summary
+-------
+
+We perform Singular Value Decomposition (SVD) calculations on large datasets.
+
+We modify the computation both by using fully precise and approximate methods,
+and by using both CPUs and GPUs.
+
+In the end we compute an approximate SVD of 200GB of simulated data and using a mutli-GPU machine in 15-20 seconds.  Then we run this from a dataset stored in the cloud I/O is, predictably, a major bottleneck.
+
+
+SVD - The simple case
+---------------------
+
+Dask arrays contain a relatively sophisticated SVD algorithm that works in the
+tall-and-skinny or short-and-fat cases, but not so well in the roughly-square
+case.  It works by taking QR decompositions of each block of the array,
+combining the R matrices somehow, doing another smaller SVD on those, and then
+performing some matrix multiplication to get back to the full result.  It's
+numerically stable and decently fast, assuming that the intermediate R
+matrices of the QR decompositions mostly fit in memory.
+
+The memory constraints here are that if you have an `n` by `m` tall and
+skinny array (`n >> m`) cut into `k` blocks then you need to have about `m**2 *
+k` space.  This is true in many cases, including typical PCA machine learning
+workloads, where you have tabular data with a few columns (hundreds at most)
+and many rows.
+
+It's easy to use and quite robust.
+
+```python
+import dask.array as da
+
+x = da.random.random((10000000, 20))
+x
+```
+
+<table>
+<tr>
+<td>
+<table>  <thead>    <tr><td> </td><th> Array </th><th> Chunk </th></tr>
+  </thead>
+  <tbody>
+    <tr><th> Bytes </th><td> 1.60 GB </td> <td> 100.00 MB </td></tr>
+    <tr><th> Shape </th><td> (10000000, 20) </td> <td> (625000, 20) </td></tr>
+    <tr><th> Count </th><td> 16 Tasks </td><td> 16 Chunks </td></tr>
+    <tr><th> Type </th><td> float64 </td><td> numpy.ndarray </td></tr>
+  </tbody></table>
+</td>
+<td>
+<svg width="75" height="170" style="stroke:rgb(0,0,0);stroke-width:1" >
+
+  <!-- Horizontal lines -->
+  <line x1="0" y1="0" x2="25" y2="0" style="stroke-width:2" />
+  <line x1="0" y1="7" x2="25" y2="7" />
+  <line x1="0" y1="15" x2="25" y2="15" />
+  <line x1="0" y1="22" x2="25" y2="22" />
+  <line x1="0" y1="30" x2="25" y2="30" />
+  <line x1="0" y1="37" x2="25" y2="37" />
+  <line x1="0" y1="45" x2="25" y2="45" />
+  <line x1="0" y1="52" x2="25" y2="52" />
+  <line x1="0" y1="60" x2="25" y2="60" />
+  <line x1="0" y1="67" x2="25" y2="67" />
+  <line x1="0" y1="75" x2="25" y2="75" />
+  <line x1="0" y1="82" x2="25" y2="82" />
+  <line x1="0" y1="90" x2="25" y2="90" />
+  <line x1="0" y1="97" x2="25" y2="97" />
+  <line x1="0" y1="105" x2="25" y2="105" />
+  <line x1="0" y1="112" x2="25" y2="112" />
+  <line x1="0" y1="120" x2="25" y2="120" style="stroke-width:2" />
+
+  <!-- Vertical lines -->
+  <line x1="0" y1="0" x2="0" y2="120" style="stroke-width:2" />
+  <line x1="25" y1="0" x2="25" y2="120" style="stroke-width:2" />
+
+  <!-- Colored Rectangle -->
+  <polygon points="0.000000,0.000000 25.412617,0.000000 25.412617,120.000000 0.000000,120.000000" style="fill:#ECB172A0;stroke-width:0"/>
+
+  <!-- Text -->
+  <text x="12.706308" y="140.000000" font-size="1.0rem" font-weight="100" text-anchor="middle" >20</text>
+  <text x="45.412617" y="60.000000" font-size="1.0rem" font-weight="100" text-anchor="middle" transform="rotate(-90,45.412617,60.000000)">10000000</text>
+</svg>
+</td>
+</tr>
+</table>
+
+
+```python
+u, s, v = da.linalg.svd(x)
+```
+
+
+This works fine in the short and fat case too (when you have far more columns
+than rows) but we're always going to assume that one of your dimensions is
+unchunked, and that the other dimension has chunks that are quite a bit
+longer, otherwise, things might not fit into memory.
+
+
+Approximate SVD
+---------------
+
+If your dataset is large in both dimensions then the algorithm above won't work
+as is.  However, if you don't need exact results, or if you only need a few of
+the components, then there are a number of excellent approximation algorithms.
+
+Dask array has one of these approximation algorithms implemented in the
+[da.linalg.svd_compressed](https://docs.dask.org/en/latest/array-api.html#dask.array.linalg.svd_compressed)
+function.  And with it we can compute the approximate SVD of some very large
+matrices, at least in theory.
+
+I was recently working on a problem (explained below) and found that I was
+still running out of memory when dealing with this algorithm.  There were two
+challenges that I ran into:
+
+1.  The algorithm requires multiple passes over the data, but the Dask task
+    scheduler was keeping the input matrix in memory after it had been loaded once
+    in order to avoid recomputation.
+    Things still worked, but Dask had to move the data to disk and back
+    repeatedly, which reduced performance significantly.
+
+    We resolved this by including explicit persist/wait calls in the algorithm.
+
+2.  Related chunks of data would be loaded at different times, and so would
+    need to stick around longer than necessary to wait for their associated
+    chunks.
+
+    We resolved this by engaging task fusion as an optimization pass.
+
+Before diving further into the technical solution
+I'm going to quickly provide the use case that was motivating this work.
+
+
+Application - Genomics
+----------------------
+
+Many studies are using genome sequencing to study genetic variation
+between different individuals within a species.  These includes
+studies of human populations, but also other species such as mice,
+mosquitoes or disease-causing parasites.  These studies will, in
+general, find a large number of sites in the genome sequence where
+individuals differ from each other.  For example, humans have more
+than 100 million variable sites in the genome, and modern studies
+like the [UK BioBank](https://www.ukbiobank.ac.uk/) are working towards
+sequencing the genomes of 1 million individuals or more.
+
+In diploid species like humans, mice or mosquitoes, each individual
+carries two genome sequences, one inherited from each parent.  At each
+of the 100 million variable genome sites there will be two or more
+"alleles" that a single genome might carry.  One way to think about
+this is via the [Punnett
+square](https://en.wikipedia.org/wiki/Punnett_square), which
+represents the different possible genotypes that one individual might
+carry at one of these variable sites:
+
+<td>
+<img src="https://upload.wikimedia.org/wikipedia/commons/9/93/Punnett_Square_Genetic_Carriers.PNG" alt="punnet square" height="40%" width="40%">
+</td>
+
+In the above there are three possible genotypes: AA, Aa, and aa.  For
+computational genomics, these genotypes can be encoded as 0, 1, or 2.
+In a study of a species with M genetic variants assayed in N
+individual samples, we can represent these genotypes as an (M x N)
+array of integers.  For a modern human genetics study, the scale of
+this array might approach (100 million x 1 million).  (Although in
+practice, the size of the first dimension (number of variants) can be
+reduced somewhat, by at least an order of magnitude, because many
+genetic variants will carry little information and/or be correlated
+with each other.)
+
+These genetic differences are not random, but carry information about
+patterns of genetic similarity and shared ancestry between
+individuals, because of the way they have been inherited through many
+generations.  A common task is to perform a dimensionality reduction
+analysis on these data, such as a [principal components
+analysis](https://journals.plos.org/plosgenetics/article?id=10.1371/journal.pgen.0020190)
+(SVD), to identify genetic structure reflecting these differencies in
+degree of shared ancestry.  This is an essential part of discovering
+genetic variants associated with different diseases, and for learning
+more about the genetic history of populations and species.
+
+Reducing the time taken to compute an analysis such as SVD, like all
+science, allows for exploring larger datasets and testing more
+hypotheses in less time.  Practically, this means not simply a fast
+SVD but an accelerated pipeline end-to-end, from data loading to
+analysis, to understanding.
+
+*We want to run an experiment in less time than it take to make a cup of tea*
+
+Performant SVDs w/ Dask
+-----------------------
+
+To stop Dask from holding onto the data we intentionally trigger computation as
+we build up the graph.  This is a bit atypical in Dask calculations (we prefer
+to have as much of the computation at once before computing) but useful given
+the multiple-pass nature of this problem.  This was a fairly easy change, and
+is available in [dask/dask #5041](https://github.com/dask/dask/pull/5041).
+
+Additionally, we found that it was helpful to turn on moderately wide task
+fusion.
+
+```python
+import dask
+dask.config.set(fuse_ave_width=5)
+```
+
+Then things work fine
+---------------------
+
+I can happily perform an SVD on a 20GB array on my Macbook Pro
+
+```python
+import dask.array as da
+
+x = da.random.random(size=(1_000_000, 20_000), chunks=(20_000, 5_000))
+
+u, s, v = da.linalg.svd_compressed(x, k=10, compute=True)
+v.compute()
+```
+
+This call is no longer entirely lazy, and it recomputes `x` a couple times, but
+it works, and it works using only a few GB of RAM on my consumer laptop.
+
+It takes around 2min 30s time to compute that on my laptop.  That's great!  But we can do better
+
+
+Adding GPUs (a 15 second SVD)
+-----------------------------
+
+We can increase performance considerably by using a multi-GPU machine.
+Here is almost the same code, running in significantly less same time but we make the
+following changes:
+
+1.  We increase the size of the array by a factor of **10x**
+2.  We switch out NumPy for CuPy, a GPU NumPy implementation
+3.  We use a sixteen-GPU DGX-2 machine with NVLink interconnects between GPUs (this will dramatically decrease transfer time between workers)
+
+On A DGX2 we can calculate an SVD on a 200GB Dask array between 10 and15 seconds: [SVD Multi-GPU Notebook](https://gist.github.com/quasiben/98ee254920837313946f621e103d41f4)
+
+To see this run, we recommend viewing
+[the attached screencast](https://youtu.be/4X5yky2lvEw)
+
+
+Read dataset from Disk
+----------------------
+
+While impressive, the computation above is mostly bound by generating random
+data and then performing matrix calculations.  GPUs are good at both of these
+things.
+
+In practice though, our input array won't be randomly generated, it will be
+coming from some dataset stored on disk or increasingly more common, stored in the cloud.
+To make things more realistic we perform a similar calculation with data stored in a [Zarr format](https://zarr.readthedocs.io/en/stable/) (which is what Alistair, our genomics collaborator, uses) in [GCS](https://cloud.google.com/storage)
+
+In this [Zarr SVD example](https://gist.github.com/quasiben/e52bc740ae22ae321f30987c65998078), we load a 25GB GCS backed data set onto a DGX2, run a few processing steps, then perform an SVD.  The combination of preprocessing and SVD calculations ran in 18.7 sec and the data loading took 14.3 seconds.
+
+Again, on a DGX2, from data loading to SVD we are running in time less than it would take to make a cup of tea.  However, the data loading can be accelerated.  From GCS we are reading into host memory, uncompressing the zarr bits, then moving the data from host memory to device memory.  This is be improved if we cut out the host entirely and read directly into the GPU.
+
+And so we come back to a common lesson of high performance computing:
+
+*High performance computing isn't about doing one thing exceedingly well, it's
+about doing nothing poorly*.
+
+In this case GPUs made our computation fast enough that we now need to focus on
+other components of our system, notably disk bandwidth, and a direct reader for
+Zarr data to GPU memory.