Quantiles
Hacking Algorithms into H2O: Quantiles
This is a presentation of hacking a simple algorithm into the new dev-friendly branch of H2O, h2o-dev.
This is one of three "Hacking Algorithms into H2O" tutorials. All three tutorials start out the same: getting the h2o-dev code and building it. They are the same until the section titled Building Our Algorithm: Copying from the Example, and then the content is customized for each algorithm. This tutorial describes computing Quantiles.
What is H2O-dev ?
As I mentioned, H2O-dev is a dev-friendly version of H2O, and is soon to be our only version. What does "dev-friendly" mean? It means:
- No classloader: The classloader made H2O very hard to embed in other projects. No more! Witness H2O's embedding in Spark.
- Fully integrated into IdeaJ: You can right-click debug-as-junit any of the junit tests and they will Do The Right Thing in your IDE.
- Fully gradle-ized and maven-ized: Running
gradlew buildwill download all dependencies, build the project, and run the tests.
These are all external points. However, the code has undergone a major revision internally as well. What worked well was left alone, but what was... gruesome... has been rewritten. In particular, it's now much easier to write the "glue" wrappers around algorithms and get the full GUI, R, REST & JSON support on your new algorithm. You still have to write the math, of course, but there's not nearly so much pain on the top-level integration.
At some point, we'll flip the usual H2O github repo to have h2o-dev as our main repo, but at the moment, h2o-dev does not contain all the functionality in H2O, so it is in its own repo.
Building H2O-dev
I assume you are familiar with basic Java development and how github repo's work - so we'll start with a clean github repo of h2o-dev:
C:\Users\cliffc\Desktop> mkdir my-h2o
C:\Users\cliffc\Desktop> cd my-h2o
C:\Users\cliffc\Desktop\my-h2o> git clone https://github.com/0xdata/h2o-dev
This will download the h2o-dev source base; took about 30secs for me from home onto my old-school Windows 7 box. Then do an initial build:
C:\Users\cliffc\Desktop\my-h2o> cd h2o-dev
C:\Users\cliffc\Desktop\my-h2o\h2o-dev> .\gradlew build
...
:h2o-web:test UP-TO-DATE
:h2o-web:check UP-TO-DATE
:h2o-web:build
BUILD SUCCESSFUL
Total time: 11 mins 41.138 secs
C:\Users\cliffc\Desktop\my-h2o\h2o-dev>
The first build took about 12mins, including all the test runs. Incremental gradle-based builds are somewhat faster:
C:\Users\cliffc\Desktop\my-h2o\h2o-dev> gradle --daemon build -x test
...
:h2o-web:signArchives SKIPPED
:h2o-web:assemble
:h2o-web:check
:h2o-web:build
BUILD SUCCESSFUL
Total time: 1 mins 44.645 secs
C:\Users\cliffc\Desktop\my-h2o\h2o-dev>
But faster yet will be IDE-based builds. There's also a functioning Makefile
setup for old-schoolers like me; it's a lot faster than gradle for incremental
builds.
While that build is going, let's look at what we got. There are 4 top-level directories of interest here:
h2o-core: The core H2O system - including clustering, clouding, distributed execution, distributed Key-Value store, the web, REST and JSON interfaces. We'll be looking at the code and javadocs in here - there are a lot of useful utilities - but not changing it.h2o-algos: Where most of the algorithms lie, including GLM and Deep Learning. We'll be copying theExamplealgorithm and turning it into a quantiles algorithm.h2o-web: The web interface and JavaScript. We will use jar files from here in our project, but probably not need to look at the code.h2o-app: A tiny sample Application which drives h2o-core and h2o-algos, including the one we hack in. We'll add one line here to teach H2O about our new algorithm.
Within each top-level directory, there is a fairly straightforward maven'ized directory structure:
src/main/java - Java source code
src/test/java - Java test code
In the Java directories, we further use water directories to hold core H2O
functionality and hex directories to hold algorithms and math:
src/main/java/water - Java core source code
src/main/java/hex - Java math source code
Ok, let's setup our IDE. For me, since I'm using the default IntelliJ IDEA setup:
C:\Users\cliffc\Desktop\my-h2o\h2o-dev> gradlew idea
...
:h2o-test-integ:idea
:h2o-web:ideaModule
:h2o-web:idea
BUILD SUCCESSFUL
Total time: 38.378 secs
C:\Users\cliffc\Desktop\my-h2o\h2o-dev>
Running H2O-dev Tests in an IDE
Then I switched to IDEAJ from my command window. I launched IDEAJ, selected
"Open Project", navigated to the h2o-dev/ directory and clicked Open. After IDEAJ opened, I clicked the Make project button (or Build/Make Project or ctrl-F9) and
after a few seconds, IDEAJ reports the project is built (with a few dozen
warnings).
Let's use IDEAJ to run the JUnit test for the Example algorithm I mentioned
above. Navigate to the ExampleTest.java file. I used a quick double-press of
Shift to bring the generic project search, then typed some of
ExampleTest.java and selected it from the picker. Inside the one obvious
testIris() function, right-click and select Debug testIris(). The testIris code
should run, pass pretty quickly, and generate some output:
"C:\Program Files\Java\jdk1.7.0_67\bin\java" -agentlib:jdwp=transport=dt_socket....
Connected to the target VM, address: '127.0.0.1:51321', transport: 'socket'
11-08 13:17:07.536 192.168.1.2:54321 4576 main INFO: ----- H2O started -----
11-08 13:17:07.642 192.168.1.2:54321 4576 main INFO: Build git branch: master
11-08 13:17:07.642 192.168.1.2:54321 4576 main INFO: Build git hash: cdfb4a0f400edc46e00c2b53332c312a96566cf0
11-08 13:17:07.643 192.168.1.2:54321 4576 main INFO: Build git describe: RELEASE-0.1.10-7-gcdfb4a0
11-08 13:17:07.643 192.168.1.2:54321 4576 main INFO: Build project version: 0.1.11-SNAPSHOT
11-08 13:17:07.644 192.168.1.2:54321 4576 main INFO: Built by: 'cliffc'
11-08 13:17:07.644 192.168.1.2:54321 4576 main INFO: Built on: '2014-11-08 13:06:53'
11-08 13:17:07.644 192.168.1.2:54321 4576 main INFO: Java availableProcessors: 4
11-08 13:17:07.645 192.168.1.2:54321 4576 main INFO: Java heap totalMemory: 183.5 MB
11-08 13:17:07.645 192.168.1.2:54321 4576 main INFO: Java heap maxMemory: 2.66 GB
11-08 13:17:07.646 192.168.1.2:54321 4576 main INFO: Java version: Java 1.7.0_67 (from Oracle Corporation)
11-08 13:17:07.646 192.168.1.2:54321 4576 main INFO: OS version: Windows 7 6.1 (amd64)
11-08 13:17:07.646 192.168.1.2:54321 4576 main INFO: Possible IP Address: lo (Software Loopback Interface 1), 127.0.0.1
11-08 13:17:07.647 192.168.1.2:54321 4576 main INFO: Possible IP Address: lo (Software Loopback Interface 1), 0:0:0:0:0:0:0:1
11-08 13:17:07.647 192.168.1.2:54321 4576 main INFO: Possible IP Address: eth3 (Realtek PCIe GBE Family Controller), 192.168.1.2
11-08 13:17:07.648 192.168.1.2:54321 4576 main INFO: Possible IP Address: eth3 (Realtek PCIe GBE Family Controller), fe80:0:0:0:4d5c:8410:671f:dec5%11
11-08 13:17:07.648 192.168.1.2:54321 4576 main INFO: Internal communication uses port: 54322
11-08 13:17:07.648 192.168.1.2:54321 4576 main INFO: Listening for HTTP and REST traffic on http://192.168.1.2:54321/
11-08 13:17:07.649 192.168.1.2:54321 4576 main INFO: H2O cloud name: 'cliffc' on /192.168.1.2:54321, discovery address /227.18.246.131:58130
11-08 13:17:07.650 192.168.1.2:54321 4576 main INFO: If you have trouble connecting, try SSH tunneling from your local machine (e.g., via port 55555):
11-08 13:17:07.650 192.168.1.2:54321 4576 main INFO: 1. Open a terminal and run 'ssh -L 55555:localhost:54321 [email protected]'
11-08 13:17:07.650 192.168.1.2:54321 4576 main INFO: 2. Point your browser to http://localhost:55555
11-08 13:17:07.652 192.168.1.2:54321 4576 main INFO: Log dir: '\tmp\h2o-cliffc\h2ologs'
11-08 13:17:07.719 192.168.1.2:54321 4576 main INFO: Cloud of size 1 formed [/192.168.1.2:54321]
11-08 13:17:07.722 192.168.1.2:54321 4576 main INFO: ###########################################################
11-08 13:17:07.723 192.168.1.2:54321 4576 main INFO: * Test class name: hex.example.ExampleTest
11-08 13:17:07.723 192.168.1.2:54321 4576 main INFO: * Test method name: testIris
11-08 13:17:07.724 192.168.1.2:54321 4576 main INFO: ###########################################################
Start Parse
11-08 13:17:08.198 192.168.1.2:54321 4576 FJ-0-7 INFO: Parse result for _85a160bc2419316580eeaab88602418e (150 rows):
11-08 13:17:08.204 192.168.1.2:54321 4576 FJ-0-7 INFO: Col type min max NAs constant numLevels
11-08 13:17:08.205 192.168.1.2:54321 4576 FJ-0-7 INFO: sepal_len: numeric 4.30000 7.90000
11-08 13:17:08.206 192.168.1.2:54321 4576 FJ-0-7 INFO: sepal_wid: numeric 2.00000 4.40000
11-08 13:17:08.207 192.168.1.2:54321 4576 FJ-0-7 INFO: petal_len: numeric 1.00000 6.90000
11-08 13:17:08.208 192.168.1.2:54321 4576 FJ-0-7 INFO: petal_wid: numeric 0.100000 2.50000
11-08 13:17:08.209 192.168.1.2:54321 4576 FJ-0-7 INFO: class: categorical 0.00000 2.00000 3
11-08 13:17:08.212 192.168.1.2:54321 4576 FJ-0-7 INFO: Internal FluidVec compression/distribution summary:
11-08 13:17:08.212 192.168.1.2:54321 4576 FJ-0-7 INFO: Chunk type count fraction size rel. size
11-08 13:17:08.212 192.168.1.2:54321 4576 FJ-0-7 INFO: C1 1 20.000 % 218 B 19.156 %
11-08 13:17:08.212 192.168.1.2:54321 4576 FJ-0-7 INFO: C1S 4 80.000 % 920 B 80.844 %
11-08 13:17:08.212 192.168.1.2:54321 4576 FJ-0-7 INFO: Total memory usage : 1.1 KB
Done Parse: 488
11-08 13:17:08.304 192.168.1.2:54321 4576 FJ-0-7 INFO: Example: iter: 0
11-08 13:17:08.304 192.168.1.2:54321 4576 FJ-0-7 INFO: Example: iter: 1
11-08 13:17:08.305 192.168.1.2:54321 4576 FJ-0-7 INFO: Example: iter: 2
11-08 13:17:08.306 192.168.1.2:54321 4576 FJ-0-7 INFO: Example: iter: 3
11-08 13:17:08.307 192.168.1.2:54321 4576 FJ-0-7 INFO: Example: iter: 4
11-08 13:17:08.308 192.168.1.2:54321 4576 FJ-0-7 INFO: Example: iter: 5
11-08 13:17:08.309 192.168.1.2:54321 4576 FJ-0-7 INFO: Example: iter: 6
11-08 13:17:08.309 192.168.1.2:54321 4576 FJ-0-7 INFO: Example: iter: 7
11-08 13:17:08.310 192.168.1.2:54321 4576 FJ-0-7 INFO: Example: iter: 8
11-08 13:17:08.311 192.168.1.2:54321 4576 FJ-0-7 INFO: Example: iter: 9
11-08 13:17:08.315 192.168.1.2:54321 4576 main INFO: #### TEST hex.example.ExampleTest#testIris EXECUTION TIME: 00:00:00.586 (Wall: 08-Nov 13:17:08.313)
Disconnected from the target VM, address: '127.0.0.1:51321', transport: 'socket'
Process finished with exit code 0
Ok, that's a pretty big pile of output - but buried it in is some cool stuff we'll need to be able to pick out later, so let's break it down a little.
The yellow stuff is H2O booting up a
cluster of 1 JVM. H2O dumps out a bunch of stuff to diagnose initial cluster
setup problems, including the git build version info, memory assigned to the
JVM, and the network ports found and selected for cluster communication. This
section ends with the line:
11-08 13:17:07.719 192.168.1.2:54321 4576 main INFO: Cloud of size 1 formed [/192.168.1.2:54321]
This tells us we formed a Cloud of size 1: one JVM will be running our program, and its IP address is given.
The lightblue stuff is our
ExampleTest JUnit test starting up and loading some test data (the venerable
iris dataset with headers, stored in the H2O-dev repo's
smalldata/iris/ directory). The printout includes some basic stats about the
loaded data (column header names, min/max values, compression ratios).
Included in this output are the lines Start Parse and Done Parse. These
come directly from the System.out.println("Start Parse") lines we can see in
the ExampleTest.java code.
Finally, the green stuff is our Example
algorithm running on the test data. It is a very simple algorithm (finds the
max per column, and does it again and again, once per requested _max_iters).
Building Our Algorithm: Copying from the Example
Now let's get our own algorithm framework to start playing with in place.
We want to compute Quantiles, so let's call the code quantile. I cloned the
main code and model from the h2o-algos/src/main/java/hex/example/ directory
into h2o-algos/src/main/java/hex/quantile/, and the test from
h2o-algos/src/test/java/hex/example/ directory into
h2o-algos/src/test/java/hex/quantile/.
Then I copied the three GUI/REST files in h2o-algos/src/main/java/hex/schemas
with Example in the name (ExampleHandler.java, ExampleModelV2.java,
ExampleV2) to their Quantile* variants.
I also copied the h2o-algos/src/main/java/hex/api/ExampleBuilderHandler.java
file to its Quantile variant. Finally, I renamed the files and file contents
from Example to Quantile.
I also dove into h2o-app/src/main/java/water/H2OApp.java and copied the two
Example lines and made their Quantile variants. Because I'm old-school,
I did this with a combination of shell hacking and Emacs; about 5 minutes all
told.
At this point, back in IDEAJ, I nagivated to QuantileTest.java, right-clicked
debug-test testIris again - and was rewarded with my Quantile clone running
a basic test. Not a very good Quantile, but definitely a start.
Whats in All Those Files?
What's in all those files? Mainly there is a Model and a ModelBuilder, and
then some support files.
A model is a mathematical representation of the world, an effort to approximate some interesting fact with numbers. It is a static concrete unchanging thing, completely defined by the rules (algorithm) and data used to make it.
A model-builder builds a model; it is transient and active. It exists as long as we are actively trying to make a model, and is thrown away once we have the model in-hand.
In our case, we want quantiles as a result - a mathematical representation of
the world - so that belongs in the QuantileModel.java file. The algorithm
to compute quantiles belongs in the QuantileModelBuilder.java file.
We also split Schemas from Models to isolate slow-moving external APIs from rapidly-moving internal APIs. As a Java dev, you can hack the guts of Quantile to your heart's content, including the inputs and outputs, as long as the externally facing V2 schemas do not change. If you want to report new stuff or take new parameters, you can make a new V3 schema (which is not compatible with V2) for the new stuff. Old external V2 users will not be affected by your changes - you'll still have to make the correct mappings in the V2 schema code from your V3 algorithm.
One other important hack: quantiles is an unsupervised algorithm - no training
data (no "response") tells it what the results "should" be. So we need to hack
the word Supervised out of all the various class names it appears in. After
this is done, your QuantileTest probably fails to compile, because it is trying
to set the response column name in the test, and unsupervised models do not
get a response to train with. Just delete the line for now:
parms._response_column = "class";
At this point, we can run our test code again (still finding the max-per-column).
The Quantile Model
The Quantile model, in the file QuantileModel.java, should contain what we
expect out of quantiles: the quantile value - a single double. Usually people
request a set of Q probabilities (e.g., the 1%, 5%, 25% and 50% probabilities),
so we'll report a double[/*Q*/] of quantiles. Also, we'll report them for
all N columns in the dataset, so it will really be a double[/*N*/][/*Q*/]. For our Q
probabilities, this will be:
public double _quantiles[/*N*/][/*Q*/]; // Our N columns, Q quantiles reported
Inside the QuantileModel class, there is a class for the model's output:
class QuantileOutput. We'll put our quantiles there. The various support
classes and files will make sure our model's output appears in the correct REST &
JSON responses, and gets pretty-printed in the GUI. There is also the
left-over _maxs array from the old Example code; we can delete that now.
And finally a quick report on the effort used to build the model: the number of iterations we actually took. Our quantiles will run in iterations, improving with each iteration until we get the exact answer. I'll describe the algorithm in detail below, but it's basically an aggressive radix sort.
My final QuantileOutput class looks like:
public static class QuantileOutput extends Model.Output {
public int _iters; // Iterations executed
public double _quantiles[/*N*/][/*Q*/]; // Our N columns, Q quantiles reported
public QuantileOutput( Quantile b ) { super(b); }
@Override public ModelCategory getModelCategory() { return Model.ModelCategory.Unknown; }
}
Now, let's turn to the input for our model-building process. These are stored
in the class QuantileModel.QuantileParameters. We already inherit an input
training dataset (returned with train()), and some other helpers (e.g. which
columns to ignore if we do an expensive scoring step with each iteration). For
now, we can ignore everything except the input dataset from train().
However, we want some more parameters for quantiles: the set of probabilities.
Let's also put in some default probabilities (the GUI & REST/JSON layer will let
these be overridden, but it's nice to have some sane defaults). Define it next
to the left-over _max_iters from the old Example code (which we might as well
also nuke):
// Set of probabilities to compute
public double _probs[/*Q*/] = new double[]{0.01,0.05,0.10,0.25,0.333,0.50,0.667,0.90,0.95,0.99};
My final QuantileParameters class looks like:
public static class QuantileParameters extends Model.Parameters {
// Set of probabilities to compute
public double _probs[/*Q*/] = new double[]{0.01,0.05,0.10,0.25,0.333,0.50,0.667,0.90,0.95,0.99};
}
A bit on field naming: I always use a leading underscore _ before all
internal field names - it lets me know at a glance whether I'm looking at a
field name (stateful, can changed by other threads) or a function parameter
(functional, private). The distinction becomes interesting when you are
sorting through large piles of code. There's no other fundamental reason to
use (or not use) the underscores. External APIs, on the other hand, generally do
not care for leading underscores. Our JSON output and REST URLs will strip the
underscores from these fields.
To make the GUI functional, I need to add my new probabilities field to the external schema
in h2o-algos/src/main/java/hex/schemas/QuantileV2.java:
public static final class QuantileParametersV2 extends ModelParametersSchema<QuantileModel.QuantileParameters, QuantileParametersV2> {
static public String[] own_fields = new String[] {"probs"};
// Input fields
@API(help="Probabilities for quantiles") public double probs[];
}
And I need to add my result fields to the external output schema in
h2o-algos/src/main/java/hex/schemas/QuantileModelV2.java:
public static final class QuantileModelOutputV2 extends ModelOutputSchema<QuantileModel.QuantileOutput, QuantileModelOutputV2> {
// Output fields
@API(help="Iterations executed") public int iters;
@API(help="Quantiles per column") public double quantiles[/*Q*/][/*N*/];
The Quantile Model Builder
Let's turn to the Quantile model builder, which includes some boilerplate we
inherited from the old Example code, and a place to put our real algorithm.
There is a basic Quantile constructor which calls init:
public Quantile( ... ) { super("Quantile",parms); init(false); }
In this case, init(false) means "only do cheap stuff in init". Init is
defined a little ways down and does basic (cheap) argument checking.
init(false) is called every time the mouse clicks in the GUI and is used to
let the front-end sanity parameters function as people type. In this case
"only do cheap stuff" really means "only do stuff you don't mind waiting on
while clicking in the browser". No computing quantiles in the init() call!
Speaking of the init() call, the one we got from the old Example code limits
sanity checks the now-deleted _max_iters. Let's add some lines to check that
our _probs are sane:
for( double p : _parms._probs )
if( p < 0.0 || p > 1.0 )
error("probs","Probabilities must be between 0 and 1 ");
In the Quantile.java file there is a trainModel call that is used when you
really want to start running quantiles (as opposed to just checking arguments).
In our case, the old boilerplate starts a QuantileDriver in a background
thread. Not required, but for any long-running algorithm, it is nice to have it
run in the background. We'll get progress reports from the GUI (and from
REST/JSON) with the option to cancel the job, or inspect partial results as the
model builds.
The class QuantileDriver holds the algorithmic meat. The compute2() call
will be called by a background Fork/Join worker thread to drive all the hard
work. Again, there is some brief boilerplate we need to go over.
First up: we need to record Keys stored in H2O's DKV: Distributed
Key/Value store, so a later cleanup, Scope.exit();, will wipe out any temp
keys. When working with Big Data, we have to be careful to clean up after
ourselves - or we can swamp memory with Big Temps.
Scope.enter();
Next, we need to prevent the input datasets from being manipulated by other threads during the model-build process:
_parms.lock_frames(Quantile.this);
Locking prevents situations like accidentally deleting or loading a new dataset
with the same name while quantiles is running. Like the Scope.exit() above,
we will unlock in the finally block. While it might be nice to use Java locking,
or even JDK 5.0 locks, we need a distributed lock, which is not provided by
JDK 5.0. Note that H2O locks are strictly cooperative - we cannot
enforce locking at the JVM level like the JVM does.
Next, we make an instance of our model object (with no clusters yet) and place it in the DKV, locked (e.g., to prevent another user from overwriting our model-in-progress with an unrelated model).
model = new QuantileModel(dest(), _parms, new QuantileModel.QuantileOutput(Quantile.this));
model.delete_and_lock(_key);
Also, near the file bottom is a leftover class Max from the old Example code.
Might as well nuke it now.
The Quantile Main Algorithm
Finally we get to where the Math is!
Quantiles can be computed in a variety of ways, but the most common starting point is to sort the elements - then, finding the Nth element is easy. Alas, sorting large datasets is expensive. Also, sorting provides a total order on the data, which is overkill for what we need. Instead, we'll do a couple of rounds of a radix-sort, refining our solution with each iteration until we get the exact answer. The key metric in a radix sort is the number of bins used; we want all our bins to fit in cache, probably even L1 cache, so we'll start with limiting ourselves to 1024 bins.
Given a histogram / radix-sort of 1024 bins on a billion rows, each bin will represent approximately 1 million rows. We then pick the bin holding the Nth element, and re-bin / re-histogram / re-radix-sort another 1024 bins refining the bin holding the Nth element. We expect this bin to hold about 1000 elements. Finding the Nth element in a billion rows then probably takes 3 or 4 passes (worst case is sorta bad: 100 passes), and each pass will be really fast.
Basically, we're gonna fake a sort - we'll end up with the elements for the exact row numbers we want for each probability (and not all the other row numbers). E.g., if the probability of 0.4567 on a billion row dataset needs the element (in sorted order) for row 456,700,000 - we'll get it (this element is not related to the unsorted row number 456,700,000).
Besides a classic histogram (element counts per bucket), we will also need an actual value - if there is a unique one. This is always true if we only have one element in a bucket, but it might also be true if we have a long run of the same value (e.g., lots of the year 2014's in our dataset).
My Quantile now has a leftover loop from the old Example code running up to
some max iteration count. Let's nuke it and just run a loop over all the
dataset columns.
// Run the main Quantile Loop
Vec vecs[] = train().vecs();
for( int n=0; n<vecs.length; n++ ) {
...
}
Let's also stop if somebody clicks the "cancel" button in the GUI:
for( int n=0; n<vecs.length; n++ ) {
if( !isRunning() ) return; // Stopped/cancelled
...
}
I removed the "compute Max" code from the old Example code in the loop body. Next up, I see code to record any new model (e.g. quantiles), and save the results back into the DKV, bump the progress bar, and log a little bit of progress:
// Update the model
model._output._quantiles = ????? // we need to figure these out
model._output._iters++; // One iter per-prob-per-column
model.update(_key); // Update model in K/V store
update(1); // One unit of work in the GUI progress bar
StringBuilder sb = new StringBuilder();
sb.append("Quantile: iter: ").append(model._output._iters);
Log.info(sb);
The Quantile Main Loop
And now we need to figure what do in our main loop. Somewhere between the
loop-top-isRunning check and the model.update() call, we need to compute
something to update our model with! This is the meat of Quantile - for each
point, bin it - build a histogram. With the histogram in hand, we see if we
can compute the exact quantile, if we cannot and then build a new refined
histogram from the bounds computed in the prior histogram.
Anything that starts out with the words "for each point" when you have a billion points needs to run in-parallel and scale-out to have a chance of completing fast - and this is exactly H2O is built for! So let's write code that runs scale-out for-each-point... and the easiest way to do that is with an H2O Map/Reduce job - an instance of MRTask. For Quantile, this is an instance of a histogram. We'll call it from the main-loop like this, and define it below (extra lines included so you can see how it fits):
if( !isRunning() ) return; // Stopped/cancelled
Vec vec = vecs[n];
// Compute top-level histogram
Histo h1 = new Histo(/*bounds for the full Vec*/).doAll(vec); // Need to figure this out???
// For each probability, see if we have it exactly - or else run
// passes until we do.
for( int p = 0; p < _parms._probs.length; p++ ) {
double prob = _parms._probs[p];
Histo h = h1; // Start from the first global histogram
while( h.not_have_exact_quantile??? ) // Need to figure this out???
h = h.refine_histogram???? // Need to figure this out???
// Update the model
model._output._iters++; // One iter per-prob-per-column
model.update(_key); // Update model in K/V store
update(1); // One unit of work
}
StringBuilder sb = new StringBuilder();
sb.append("Quantile: iter: ").append(model._output._iters).append(" Qs=").append(Arrays.toString(model._output._quantiles[n]));
Log.info(sb);
Basically, we just called some not-yet-defined Histo code on the entire Vec
(column) of data, then looped over each probability. If we can compute the
quantile exactly for the probability, great! If not, we build and run another
histogram over a refined range, repeating until we can compute the exact
histogram. I also printed out the quantiles per Vec, so we can watch the
progress over time. Now class Histo can be coded as an inner class to the
QuantileDriver class:
class Histo extends MRTask<Histo> {
private static final int NBINS=1024; // Default bin count
private final int _nbins; // Actual bin count
private final double _lb; // Lower bound of bin[0]
private final double _step; // Step-size per-bin
private final long _start_row; // Starting row number for this lower-bound
private final long _nrows; // Total datasets rows
// Big Data output result
long _bins [/*nbins*/]; // Rows in each bin
double _elems[/*nbins*/]; // Unique element, or NaN if not unique
private Histo( double lb, double ub, long start_row, long nrows ) {
_nbins = NBINS;
_lb = lb;
_step = (ub-lb)/_nbins;
_start_row = start_row;
_nrows = nrows;
}
@Override public void map( Chunk chk ) {
...
}
@Override public void reduce( Histo h ) {
...
}
}
A Quick H2O Map/Reduce Diversion
This isn't your Hadoop-Daddy's Map/Reduce. This is an in-memory super-fast map-reduce... where "super-fast" generally means "memory bandwidth limited", often 1000x faster than the usual hadoop-variant - MRTasks can often touch a gigabyte of data in a millisecond, or a terabyte in a second (depending on how much hardware is in your cluster - more hardware is faster for the same amount of data!)
The map() call takes data in Chunks - where each Chunk is basically a
small array-like slice of the Big Data. Data in Chunks is accessed with basic
at0 and set0 calls (vs accessing data in Vecs with at and set). The
output of a map() is stored in the Histo object itself, as a Plain Old
Java Object (POJO). Each map() call has private access to its own fields and
Chunks, which implies there are lots of instances of Histo objects
scattered all over the cluster (one such instance per Chunk of data...
well, actually one instance per call to map(), but each map call is handed an
aligned set of Chunks, one per feature or column in the dataset).
Since there are lots of little Histos running about, their results need to be
combined. That's what reduce does - combine two Histos into one.
Typically, you can do this by adding similar fields together - often array
elements are added side-by-side, similar to a saxpy operation.
All code here is written in a single-threaded style, even as it runs in parallel and distributed. H2O handles all the synchronization issues.
Histogram
Back to class Histo, we create arrays to hold our results and loop over the
data:
@Override public void map( Chunk[] chks ) {
long bins [] = _bins =new long [_nbins];
double elems[] = _elems=new double[_nbins];
for( int row=0; row<chk._len; row++ ) {
double d = chk.at0(row);
....
}
}
Then we need to find the correct bin (simple linear interpolation). If the bin
is in-range, increment the bin count. Note that if a value is missing, it
will represented as a NaN, then the computation of idx will be a NaN and the
range test will fail and no bin will be incremented.
for( int row=0; row<chk._len; row++ ) {
double d = chk.at0(row);
double idx = (d-_lb)/_step;
if( !(0.0 <= idx && idx < bins.length) ) continue;
int i = (int)idx;
...
bins[i]++;
}
Also gather a unique element in the bin, if there is a unique element (otherwise, use NaN).
int i = (int)idx;
if( bins[i]==0 ) elems[i] = d; // Capture unique value
else if( !Double.isNaN(elems[i]) && elems[i]!=d )
elems[i] = Double.NaN; // Not unique
bins[i]++; // Bump row counts
And that ends the map call and the Histo main work loop. To recap, here it
is all at once:
@Override public void map( Chunk chk ) {
long bins [] = _bins =new long [_nbins];
double elems[] = _elems=new double[_nbins];
for( int row=0; row<chk._len; row++ ) {
double d = chk.at0(row);
double idx = (d-_lb)/_step;
if( !(0.0 <= idx && idx < bins.length) ) continue;
int i = (int)idx;
if( bins[i]==0 ) elems[i] = d; // Capture unique value
else if( !Double.isNaN(elems[i]) && elems[i]!=d )
elems[i] = Double.NaN; // Not unique
bins[i]++; // Bump row counts
}
}
The reduce needs to fold together the returned results; the _elems and the
_bins. It's a bit tricky for the unique elements - either the left Histo or
the right Histo might have zero, one, or many elements - and if they both have a
unique element, it might not be the same unique element.
@Override public void reduce( Histo h ) {
for( int i=0; i<_nbins; i++ ) // Keep unique elements
if( _bins[i]== 0 ) _elems[i] = h._elems[i]; // Left had none, so keep right unique
else if( h._bins[i] > 0 && _elems[i] != h._elems[i] )
_elems[i] = Double.NaN; // Left & right both had elements, but not equal
ArrayUtils.add(_bins,h._bins);
}
And that completes the Big Data portion of Quantile.
From Histogram to Quantile
Now we get to the nit-picky part of Quantiles. This isn't Big Data, but it is Math. Gotta get it right! We have this code we need to figure out:
while( h.not_have_exact_quantile??? ) // Need to figure this out???
h = h.refine_histogram???? // Need to figure this out???
Let's make a function to return the exact Quantile (if we can compute it, or use NaN otherwise). We'll call it like this, capturing the quantile in the model output as we test for NaN:
while( Double.isNaN(model._output._quantiles[n][p] = h.findQuantile(prob)) )
h = h.refine_histogram???? // Need to figure this out???
For the findQuantile function we will:
- Find the fractional (sorted) row number for the given probability.
- Find the lower actual integral row number. If the probability evenly divides the dataset, then we want the element for the sorted row. If not, we need both elements on either side of the fractional row number.
- Find the histogram bin for the lower row.
- Find the unique element for this bin, or return a NaN (need another refinement pass)
- Repeat for the higher row number: find the bin, find the unique element or return NaN.
- With the two values spanning the desired probability in-hand, compute the quantile. We'll use R's Type-7 linear interpolation, but we could just as well use any of the other types.
Here's my findQuantile code:
double findQuantile( double prob ) {
double p2 = prob*(_nrows-1); // Desired fractional row number for this probability
long r2 = (long)p2; // Lower integral row number
int loidx = findBin(r2); // Find bin holding low value
double lo = (loidx == _nbins) ? binEdge(_nbins) : _elems[loidx];
if( Double.isNaN(lo) ) return Double.NaN; // Needs another pass to refine lo
if( r2==p2 ) return lo; // Exact row number? Then quantile is exact
long r3 = r2+1; // Upper integral row number
int hiidx = findBin(r3); // Find bin holding high value
double hi = (hiidx == _nbins) ? binEdge(_nbins) : _elems[hiidx];
if( Double.isNaN(hi) ) return Double.NaN; // Needs another pass to refine hi
return computeQuantile(lo,hi,r2,_nrows,prob);
}
And a couple of simple helper functions:
private double binEdge( int idx ) { return _lb+_step*idx; }
// bin for row; can be _nbins if just off the end (normally expect 0 to nbins-1)
private int findBin( long row ) {
long sum = _start_row;
for( int i=0; i<_nbins; i++ )
if( row < (sum += _bins[i]) )
return i;
return _nbins;
}
Finally the computeQuantiles call:
static double computeQuantile( double lo, double hi, long row, long nrows, double prob ) {
if( lo==hi ) return lo; // Equal; pick either
// Unequal, linear interpolation
double plo = (double)(row+0)/(nrows-1); // Note that row numbers are inclusive on the end point, means we need a -1
double phi = (double)(row+1)/(nrows-1); // Passed in the row number for the low value, high is the next row, so +1
assert plo <= prob && prob < phi;
return lo + (hi-lo)*(prob-plo)/(phi-plo); // Classic linear interpolation
}
Refining a Histogram
Now we need to define how to refine a Histogram. Let's just admit it needs a
Big Data pass up front and call the doAll() in the driver loop directly:
while( Double.isNaN(model._output._quantiles[n][p] = h.findQuantile(prob)) )
h = h.refinePass(prob).doAll(vec); // Full pass at higher resolution
Then the refinePass call needs to make a new Histogram from the old one, with
refined endpoints. The original Histogram had endpoints of vec.min() and
vec.max(), but here we'll use endpoints from the same bins that
findQuantiles uses.
Histo refinePass( double prob ) {
double prow = prob*(_nrows-1); // Desired fractional row number for this probability
long lorow = (long)prow; // Lower integral row number
int loidx = findBin(lorow); // Find bin holding low value
// If loidx is the last bin, then high must be also the last bin - and we
// have an exact quantile (equal to the high bin) and we didn't need
// another refinement pass
assert loidx < _nbins;
double lo = binEdge(loidx); // Lower end of range to explore
// If probability does not hit an exact row, we need the elements on
// either side - so the next row up from the low row
long hirow = lorow==prow ? lorow : lorow+1;
int hiidx = findBin(hirow); // Find bin holding high value
// Upper end of range to explore - except at the very high end cap
double hi = hiidx==_nbins ? binEdge(_nbins) : binEdge(hiidx+1);
long sum = _start_row; // Compute adjusted starting row for this histogram
for( int i=0; i<loidx; i++ )
sum += _bins[i];
return new Histo(lo,hi,sum,_nrows);
}
Running Quantile
Running the QuantileTest returns:
11-13 12:19:04.179 172.16.2.47:54321 1940 FJ-0-7 INFO: Quantile: iter: 11 Qs=[4.4, 4.6000000000000005, 4.800000000000001, 5.1000000000000005, 5.4, 5.800000000000001, 6.300000000000001, 6.4, 6.9, 7.255000000000001, 7.7]
11-13 12:19:04.183 172.16.2.47:54321 1940 FJ-0-7 INFO: Quantile: iter: 22 Qs=[2.2, 2.345, 2.5, 2.8000000000000003, 2.9000000000000004, 3.0, 3.2, 3.3000000000000003, 3.61, 3.8000000000000003, 4.151]
11-13 12:19:04.187 172.16.2.47:54321 1940 FJ-0-7 INFO: Quantile: iter: 33 Qs=[1.1490000000000002, 1.3, 1.4000000000000001, 1.6, 2.5787, 4.35, 4.9, 5.1000000000000005, 5.800000000000001, 6.1000000000000005, 6.7]
11-13 12:19:04.191 172.16.2.47:54321 1940 FJ-0-7 INFO: Quantile: iter: 44 Qs=[0.1, 0.2, 0.2, 0.30000000000000004, 0.8468, 1.3, 1.6, 1.8, 2.2, 2.3000000000000003, 2.5]
11-13 12:19:04.194 172.16.2.47:54321 1940 FJ-0-7 INFO: Quantile: iter: 55 Qs=[0.0, 0.0, 0.0, 0.0, 0.6169999999999999, 1.0, 1.383, 2.0, 2.0, 2.0, 2.0]
You can see the Quantiles-per-column (and there are 11 of them by default, so
the iteration printout bumps by 11's). I checked these numbers vs R's
default Type 7 quantiles for the iris dataset and got the same numbers.
Iris is too small to trigger the refinement pass, so I also tested on covtype.data - an old forest covertype dataset with about a half-million rows. Took about 0.7 seconds on my laptop to compute 11 quantiles exactly, on 55 columns and 581000 rows.
Good luck with your own H2O algorithm,
Cliff