The end Netflix Prize is near: BPC has reached 10%.
Now, how do I apply what I've learning during the participation of this contest ? Maybe I should get into trying to predict the stock market or commodity prices, or something. People are already doing that on the stock market, it's called algorithmic trading.
Of course recommender systems and data-mining can be applied to all kinds of products and services, not just movies. For example truck dealers can try to find people who are most likely to buy trucks and send them truck advertisement; online matchmaking services can match people based on user data and history of matches.
Of course, for most products and service there are obvious "indicators" that we all know. For example, males in construction industry are probably far more likely to buy trucks than people in general. But still, there might be some subtle indicators and trends that can only be discovered by a computer algorithm, which might improve the accuracy of recommendations by 10%, as measured by RMSE. And as BellKor's research shows, even a few percentages of RMSE improvement can translate into huge increase in the quality of the recommendations you get. You'll actually like the products recommended to you, or the people.
How Can It Work For Matchmaking ?
Well, now I think about it, it probably won't work for matchmaking services. Why ? Because for matchmaking services, it will take a long time to truly know the quality of the recommendations.
Sure, you can find out about bad ones pretty soon: people go on one date and can't stand each other. But how do you know which recommendations are really good, and which ones only look good now but will end up in messy divorce 10 year later ? I think for matchmaking services, you'll have to wait for a generation to tell which recommendations are truly good. But by that time, society and people in general would have chance a lot, so whatever worked 20 years ago probably won't work so well today.
So for matchmaking services, the value of recommender systems is probably to filter out potential incompatible dates.
Wednesday, July 1, 2009
Thursday, June 11, 2009
Calculating 316 million movie correlations in 2 minutes (down from 2.5 hours)
KNN is one of the most popular CF algorithms. Central to KNN is how to define the neighborhood relationship, or the distance between two objects. For this purpose, Pearson Correlation is a pretty good measure and is used frequently (see my previous post for my basic KNN approach). In the Netflix dataset, there are 17770 movies, so we need to calculate 17770*17770, or about 316 million Pearson correlations. Not a trivial task.
In this post, I'll describe the optimizations I used for the calculation of Pearson Correlation, which cut my running time from 2.5 hours to less than 2 minutes. It won't help your RMSE directly, but it may help indirectly by allowing you to explore the KNN parameter space faster. And although I used Pearson Correlation, the methods described in this post can be applied to many other neighborhood measures too.
Since this is about optimization, system spec is important. The numbers reported here were generated on a PC with Intel Core2 Quad CPU (4 cores) @ 2.4G Hz, 2 GB of RAM, running Window XP. Most code was written in C++, built by Visual Studio 2008. (TODO: get memory spec)
The Matrix: Halved
Before discussing any optimization, I'll discuss an optimization that's NOT taken. Look at Figure 1. It shows the correlation matrix of 5 movies. AB is the correlation of movie A and B, AC is the correlation of movie A and C, and so on.
Now, if you assume movie correlations are symmetrical, then AB=BA, AC=CA, BC=CB, and so on, so only less than half of the matrix needs to be calculated. Now let’s say I calculated the blue-shaded part of the matrix, and saved the data to a big file. To make predictions, I need to read the data back:
For movie A, I need to read AB to AE, and that’s 1 file-seek and 1 file-read operation.
For movie B, I need to read AB, and BC to BE, that’s 2 file-seek and 2 file-read operations.
For movie C, I need to read AC, BC, and CD and CE, that’s 3 file-seek and 3 file-read operations.
And so on…
In the Netflix data set, there are M=17770 movies, so by the time I get to the last movie, I’d need 17769 file seek and read operations per movie. All that file seeking will kill my performance and my hard disk.
The simple solution is to have enough memory to hold half the matrix, so I can load all the data from disk in one shot. But for each matrix entry, I need not only the correlation value, but also two average values, and a common viewer count, making it 16 bytes per entry. So for M=17770 movies, I need about 2.35 GB, but I needed this thing to run on my laptop, which had only 1.25 G when I started coding this. Even after I compacted it down to 5 bytes per entry, memory situation was still tight.
(Actually, I think you only need enough memory to hold one quarter of the matrix to eliminate the huge number of file seeks, but the in-memory indexing scheme gets pretty complicated. There might be clever ways to do this, but I'm not sure yet.)
And all that applies only if you assume movie correlations are symmetrical, and there're ways to calculate correlations where they're not symmetrical. So out of concerns for memory usage and simplicity, I decided to ignore this half-matrix optimization.
Now let's get back to the calculation of Pearson's r.
The Brute-Force Approach
To calculate Pearson's r of two movies X and Y, we need to find viewers who rated both X and Y, and calculate some intermediate values:
For Netflix data, float can be replaced by "unsigned int" when processing raw, integer ratings, which can only be 1, 2, 3, 4, or 5.
As mentioned on my previous blog, I have the rating data sorted by movie index followed by viewer index, so finding common viewers of two movies is simply finding common values in two sorted arrays. Very easy, and very slow. The inner loop of this approach looks this:
I think the Big O Notation of this approach is roughly O(M*M*T/M)=O(M*T), where M is the total number of movies, and T is the total number of ratings. Basically, for each movie, you have to compare against all other movies (M*M), and the average length of comparisions is the average number of ratings per movie (T/M).
My implementation of this approach took 4.5 hours to run on my little laptop, and 2.5 hours on the faster desktop. And I suffered thru this 4.5 hours a few times, due to the inevitable bugs in the initial implementation.
You can definitely optimize the code somewhat, but once I got KNN working correctly, I moved onto other algorithms.
A Break-thru
So my KNN algorithm just sat there, for months. During this time, I worked on other CF algorithms, which is probably why one day, I was working on something totally unrelated to KNN, and I suddenly had one of those "Oh my God I'm an idiot !" moment.
I realized there's a much bette way of calculating correlations. You see, for each movie X, instead of comparing it to every other movie directly, I can go thru all the viewers V who rated movie X; and for each viewer V, go thru all the movies Y rated by that viewer; and for each movie Y, update the PearsonIntermediate of movie pair X and Y. At the end, I'll have PearsonIntermediate of movie X and all the movies in the dataset (including X itself).
In pseudo-code, to calculate Pearson correlations of movie X and all movies in the dataset, you declare:
The Big O Notation for this approach is O(M*(T/M)*(T/V))=O(T*T/V). The brute-force approach is O(M*T). Since T/V<<M, I should get a significant speed-up. Furthermore, the inner loop does not have the unpredictable branchings of the brute-force approach.
This is just the motivation I needed to implement KNN on residues of other algorithms. If the original algorithm is slow with raw ratings (1 to 5 integer), it will be even slower on residues (floating point). After I was done with residues, I retrofitted this approach to my raw-rating KNN, and the improvement was huge: calculation time dropped to about about 15 minutes, a 10x speed-up !
That's pretty good, but wait, there is more ...
Parallelization
At this point I implemented multi-threading to take advantage of multiple CPU cores. Movie correlation calculation is trivially parallelizable. The result is about 3.4x speed up on 4 cores, bringing the running time down to 265 seconds.
But wait, there is more ...
Simplifying the Inner Loop
I thought since raw ratings can be only 1, 2, 3, 4, or 5, I should be able to speed things up significantly by replacing the PearsonIntermediate structure with:
I quickly implemented this "clever" idea, however, the result was disappointing: the speed-up was a few percent in some quick tests. Hmm, what could be wrong ?
After some head-scratching, I guessed the problem was the size of memory used. Each PearsonIntermediate structure takes only 24 bytes, but "unsigned int32 vCount[5][5]" takes 100 bytes. So as the code does the calculations, more memory had to be swapped in and out of cache and this seriously harmed the performance.
Fortunately we can use the nature of the data itself to work around this problem. Of the 17770 movies in the data set, 33% have less than 256 ratings, 65% have between 256 and 65535 ratings, and only 2% have more than 65536 ratings. Of course, if a movie has less than 256 ratings, there is no way the it has 256 or more common viewers with another movie, so the common viewer counts of this movie can be stored using "unsigned char" instead of "unsigned int", or 1 byte instead of 4 bytes. For movies that have between 256 and 65535 ratings, we can used "unsigned short" (2 bytes).
This means I need 3 versions of the correlation calculation function, one each for unsigned int/short/char (or int32/int16/int8). Here C++'s template function feature comes to the rescue (in spite of the awkward syntax). I wrote one function that uses a type T to store the common viewer counts:
With this change, the running time improved to 156 seconds.
But wait, there is more ...
Complicating the Inner Loop
The previous chanage simplified the inner loop, but most gains seem to come from reducing the demand on memory subsytem and cache miss. Can we do better ?
Consider a movie with 3000 ratings, the type of its vCount[][][] array is "unsigned short", or 2 bytes per element. But it's unlikely that the final value of any array element would be 256 or bigger, given 3000/25=120. Even for movies with 6400 or more ratings (6400/25=256), because ratings are not evenly distributed, most vCount[][][] elements are still less than 255.
So for most movies, one byte is enough for most vCount elements. But how do we use the 1-byte (unsigned char) version of vCount, and still be able to handle the occasional elements that go above 255 ? Well, use another array to count the overflows:
At first I was very skeptical of this change, because it adds a branching to the inner-most loop. How could it ever improve performance with that ? However that branching is highly predictable, and everything I read says with modern CPU's branch prediction logic, the cost of predictable branches is almost nothing.
So with great skepticism, I tried the code above. To my relief it did run faster, bringing the running time down to 143 seconds.
But wait, there is more ....
Reordering the Movies
For any viewer V, the IDs of movies V has rated are pretty much randomly distributed. Imaging the array of movies, with red dots indicating the movies rated by viewer V:
and as we go thru the movies rated by viewer V, most red dots incur cache misses.
However, if we sort the movie IDs in increasing order of rating count of movies, then the distribution of red dots changes:
statistically, we have the same or even more cache misses at the beginning, but as we go into higher movie indices, the chances that the viewer rated closely spaced movies increases, thus reducing the cache misses as vCount[][][] is updated.
Notice the "sorting" of the movie IDs by movie rating count is not an O(N*log(N)) quicksort for 100 million ratings, but just an O(N) remapping.
This change reduced the running time to 116 seconds, less than 2 minutes.
Summary
Well, that's it. I've picked all the low-hanging fruits, at least those that I could see.
Here's the list of performance gains from my initial implementation:
*Algorithm change: 10x
*Parallelization: 3.4x (4 cores vs 1 core)
*Low-level tweaks: 2.3x
In total about 78x performance gain, without writing a single line of assembly language.
Of the 3 items above, the biggest performance gain vs ease of implementation is parallization, because KNN calculation can be parallized very easily. I believe more cores will help the performance more, but with diminishing returns.
The Memory Bandwidth Bottleneck
Memory bandwidth seems to be the performance bottleneck of KNN algorithms.
When I was coding the parallization step, I installed some software to monitor MSRs, or model-specific registers. MSRs are very useful for performance monitoring, but unfortunately they're very sparsely documented. The "Intel 64 and IA-32 Architectures Software Developer's Manual, Volume 3A: System Programming Guide" has some details, but nowhere as detailed as the individual CPU instructions.
One particular MSR I've looked at is DCU_MISS_OUTSTANDING. This webpage has the following to say about it:
Anyway, when I first implemented multithreading, the code took 265 seconds to run, and I recorded about 400 billion CPU cycles spent on DCU_MISS_OUTSTANDING events. The final version took 116 seconds to run and I recorded about 100 billion CPU cycles. The difference is 149 seconds and 300 billion cycles. 300 billion cycles divided by 2.4G Hz (CPU clock speed) gives 125 seconds. So it looks like a lot of gains came from reducing the time spent waiting for data to be loaded into the cache.
I won't pretend I understand DCU_MISS_OUTSTANDING or many other MSRs completely, I just hope intel provide better documentation.
In this post, I'll describe the optimizations I used for the calculation of Pearson Correlation, which cut my running time from 2.5 hours to less than 2 minutes. It won't help your RMSE directly, but it may help indirectly by allowing you to explore the KNN parameter space faster. And although I used Pearson Correlation, the methods described in this post can be applied to many other neighborhood measures too.
Since this is about optimization, system spec is important. The numbers reported here were generated on a PC with Intel Core2 Quad CPU (4 cores) @ 2.4G Hz, 2 GB of RAM, running Window XP. Most code was written in C++, built by Visual Studio 2008. (TODO: get memory spec)
The Matrix: Halved
Before discussing any optimization, I'll discuss an optimization that's NOT taken. Look at Figure 1. It shows the correlation matrix of 5 movies. AB is the correlation of movie A and B, AC is the correlation of movie A and C, and so on.
Now, if you assume movie correlations are symmetrical, then AB=BA, AC=CA, BC=CB, and so on, so only less than half of the matrix needs to be calculated. Now let’s say I calculated the blue-shaded part of the matrix, and saved the data to a big file. To make predictions, I need to read the data back:
For movie A, I need to read AB to AE, and that’s 1 file-seek and 1 file-read operation.
For movie B, I need to read AB, and BC to BE, that’s 2 file-seek and 2 file-read operations.
For movie C, I need to read AC, BC, and CD and CE, that’s 3 file-seek and 3 file-read operations.
And so on…
In the Netflix data set, there are M=17770 movies, so by the time I get to the last movie, I’d need 17769 file seek and read operations per movie. All that file seeking will kill my performance and my hard disk.
The simple solution is to have enough memory to hold half the matrix, so I can load all the data from disk in one shot. But for each matrix entry, I need not only the correlation value, but also two average values, and a common viewer count, making it 16 bytes per entry. So for M=17770 movies, I need about 2.35 GB, but I needed this thing to run on my laptop, which had only 1.25 G when I started coding this. Even after I compacted it down to 5 bytes per entry, memory situation was still tight.
(Actually, I think you only need enough memory to hold one quarter of the matrix to eliminate the huge number of file seeks, but the in-memory indexing scheme gets pretty complicated. There might be clever ways to do this, but I'm not sure yet.)
And all that applies only if you assume movie correlations are symmetrical, and there're ways to calculate correlations where they're not symmetrical. So out of concerns for memory usage and simplicity, I decided to ignore this half-matrix optimization.
Now let's get back to the calculation of Pearson's r.
The Brute-Force Approach
To calculate Pearson's r of two movies X and Y, we need to find viewers who rated both X and Y, and calculate some intermediate values:
struct PearsonIntermediate {
//from all viewers who rated both movie X and Y.
float x; //sum of ratings for movie X
float y; //sum of ratings for movie Y
float xy; //sum of product of ratings for movies X and Y
float xx; //sum of square of ratings for movie X
float yy; //sum of square of ratings for movie Y
unsigned int cnt; //number of viewers who rated both movies
};
and you can easily calculate Pearson's r from these intermediate values.For Netflix data, float can be replaced by "unsigned int" when processing raw, integer ratings, which can only be 1, 2, 3, 4, or 5.
As mentioned on my previous blog, I have the rating data sorted by movie index followed by viewer index, so finding common viewers of two movies is simply finding common values in two sorted arrays. Very easy, and very slow. The inner loop of this approach looks this:
while (not at the end of pViewerIndex0 and pViewerIndex1) {
if ( *pViewerIndex0 < *pViewerIndex1) {
pViewerIndex0++;
} else if ( *pViewerIndex0 > *pViewerIndex1) {
pViewerIndex1++;
} else {
//a viewer who rated both movies
//update PearsonIntermediate
}
}
The inner loop is conditional branch-heavy, and even worse, given that the viewer indices are randomly spaced, they are very unpredictable branches, probably completely defeating the branch prediction logic of the CPU.I think the Big O Notation of this approach is roughly O(M*M*T/M)=O(M*T), where M is the total number of movies, and T is the total number of ratings. Basically, for each movie, you have to compare against all other movies (M*M), and the average length of comparisions is the average number of ratings per movie (T/M).
My implementation of this approach took 4.5 hours to run on my little laptop, and 2.5 hours on the faster desktop. And I suffered thru this 4.5 hours a few times, due to the inevitable bugs in the initial implementation.
You can definitely optimize the code somewhat, but once I got KNN working correctly, I moved onto other algorithms.
A Break-thru
So my KNN algorithm just sat there, for months. During this time, I worked on other CF algorithms, which is probably why one day, I was working on something totally unrelated to KNN, and I suddenly had one of those "Oh my God I'm an idiot !" moment.
I realized there's a much bette way of calculating correlations. You see, for each movie X, instead of comparing it to every other movie directly, I can go thru all the viewers V who rated movie X; and for each viewer V, go thru all the movies Y rated by that viewer; and for each movie Y, update the PearsonIntermediate of movie pair X and Y. At the end, I'll have PearsonIntermediate of movie X and all the movies in the dataset (including X itself).
In pseudo-code, to calculate Pearson correlations of movie X and all movies in the dataset, you declare:
PearsonIntermediate array[MOVIE_CNT]; //MOVIE_CNT is 17770
initialize array[] to zeros
for each viewer v who rated movie X {
for each movie Y rated by viewer v {
update values of array[Y]
}
}
go thru array[] to calculate all correlations for movie X
As a sanity check, you can verify the correlation from array[X] is 1 (or very close to 1, due to floating point inaccuracy), which is the correlation of movie X and itself.The Big O Notation for this approach is O(M*(T/M)*(T/V))=O(T*T/V). The brute-force approach is O(M*T). Since T/V<<M, I should get a significant speed-up. Furthermore, the inner loop does not have the unpredictable branchings of the brute-force approach.
This is just the motivation I needed to implement KNN on residues of other algorithms. If the original algorithm is slow with raw ratings (1 to 5 integer), it will be even slower on residues (floating point). After I was done with residues, I retrofitted this approach to my raw-rating KNN, and the improvement was huge: calculation time dropped to about about 15 minutes, a 10x speed-up !
That's pretty good, but wait, there is more ...
Parallelization
At this point I implemented multi-threading to take advantage of multiple CPU cores. Movie correlation calculation is trivially parallelizable. The result is about 3.4x speed up on 4 cores, bringing the running time down to 265 seconds.
But wait, there is more ...
Simplifying the Inner Loop
I thought since raw ratings can be only 1, 2, 3, 4, or 5, I should be able to speed things up significantly by replacing the PearsonIntermediate structure with:
unsigned int32 vCount[5][5];where vCount[m][n] is the number of viewers who gave movie X a score of m+1 and movie Y a score of n+1. This simplifies the inner-loop to just incrementing an integer !
I quickly implemented this "clever" idea, however, the result was disappointing: the speed-up was a few percent in some quick tests. Hmm, what could be wrong ?
After some head-scratching, I guessed the problem was the size of memory used. Each PearsonIntermediate structure takes only 24 bytes, but "unsigned int32 vCount[5][5]" takes 100 bytes. So as the code does the calculations, more memory had to be swapped in and out of cache and this seriously harmed the performance.
Fortunately we can use the nature of the data itself to work around this problem. Of the 17770 movies in the data set, 33% have less than 256 ratings, 65% have between 256 and 65535 ratings, and only 2% have more than 65536 ratings. Of course, if a movie has less than 256 ratings, there is no way the it has 256 or more common viewers with another movie, so the common viewer counts of this movie can be stored using "unsigned char" instead of "unsigned int", or 1 byte instead of 4 bytes. For movies that have between 256 and 65535 ratings, we can used "unsigned short" (2 bytes).
This means I need 3 versions of the correlation calculation function, one each for unsigned int/short/char (or int32/int16/int8). Here C++'s template function feature comes to the rescue (in spite of the awkward syntax). I wrote one function that uses a type T to store the common viewer counts:
T vCount[5][MOVIE_CNT][5]; //initialize to zero
for each viewer V who rated movie X {
m=the rating viewer V gave to movie X
for each movie Y rated by viewer V {
n=the rating viewer V gave to movie Y
vCount[m][Y][n]++;
}
}
and I had 3 "instantiations" (those jargons are as bad as the awkward syntax) of this function where T is unsigned int/short/char. Notice the vCount array is arranged as [5][MOVIE_CNT][5] instead of the more "straight-forward" form of [MOVIE_CNT][5][5]. My tests showed [5][MOVIE_CNT][5] gave the best performance, no doubt because it minimized cache misses.With this change, the running time improved to 156 seconds.
But wait, there is more ...
Complicating the Inner Loop
The previous chanage simplified the inner loop, but most gains seem to come from reducing the demand on memory subsytem and cache miss. Can we do better ?
Consider a movie with 3000 ratings, the type of its vCount[][][] array is "unsigned short", or 2 bytes per element. But it's unlikely that the final value of any array element would be 256 or bigger, given 3000/25=120. Even for movies with 6400 or more ratings (6400/25=256), because ratings are not evenly distributed, most vCount[][][] elements are still less than 255.
So for most movies, one byte is enough for most vCount elements. But how do we use the 1-byte (unsigned char) version of vCount, and still be able to handle the occasional elements that go above 255 ? Well, use another array to count the overflows:
unsigned char vCount[5][MOVIE_CNT][5]; //initialize to zero
unsigned short overflows[5][MOVIE_CNT][5]; //initialize to zero
for each viewer V who rated movie X {
m=the rating viewer V gave to movie X
for each movie Y rated by viewer V {
n=the rating viewer V gave to movie Y
vCount[m][Y][n]++;
if (0==vCount[m][Y][n]) overflows[m][Y][n]++;
}
}
The code above takes advantage of the fact that in 2's complement system, when an integer value overflows, it wraps around to zero.At first I was very skeptical of this change, because it adds a branching to the inner-most loop. How could it ever improve performance with that ? However that branching is highly predictable, and everything I read says with modern CPU's branch prediction logic, the cost of predictable branches is almost nothing.
So with great skepticism, I tried the code above. To my relief it did run faster, bringing the running time down to 143 seconds.
But wait, there is more ....
Reordering the Movies
For any viewer V, the IDs of movies V has rated are pretty much randomly distributed. Imaging the array of movies, with red dots indicating the movies rated by viewer V:
and as we go thru the movies rated by viewer V, most red dots incur cache misses.However, if we sort the movie IDs in increasing order of rating count of movies, then the distribution of red dots changes:
statistically, we have the same or even more cache misses at the beginning, but as we go into higher movie indices, the chances that the viewer rated closely spaced movies increases, thus reducing the cache misses as vCount[][][] is updated.Notice the "sorting" of the movie IDs by movie rating count is not an O(N*log(N)) quicksort for 100 million ratings, but just an O(N) remapping.
This change reduced the running time to 116 seconds, less than 2 minutes.
Summary
Well, that's it. I've picked all the low-hanging fruits, at least those that I could see.
Here's the list of performance gains from my initial implementation:
*Algorithm change: 10x
*Parallelization: 3.4x (4 cores vs 1 core)
*Low-level tweaks: 2.3x
In total about 78x performance gain, without writing a single line of assembly language.
Of the 3 items above, the biggest performance gain vs ease of implementation is parallization, because KNN calculation can be parallized very easily. I believe more cores will help the performance more, but with diminishing returns.
The Memory Bandwidth Bottleneck
Memory bandwidth seems to be the performance bottleneck of KNN algorithms.
When I was coding the parallization step, I installed some software to monitor MSRs, or model-specific registers. MSRs are very useful for performance monitoring, but unfortunately they're very sparsely documented. The "Intel 64 and IA-32 Architectures Software Developer's Manual, Volume 3A: System Programming Guide" has some details, but nowhere as detailed as the individual CPU instructions.
One particular MSR I've looked at is DCU_MISS_OUTSTANDING. This webpage has the following to say about it:
DCU_MISS_OUTSTANDING gives a weighted measure of the cycles spent waiting for a cache line to be brought in. Each cycle spent waiting is weighted by the number of outstanding cache misses ... and has some caveats, but can be considered a rough estimate of the time spent waiting for a cache line to be brought in.
Anyway, when I first implemented multithreading, the code took 265 seconds to run, and I recorded about 400 billion CPU cycles spent on DCU_MISS_OUTSTANDING events. The final version took 116 seconds to run and I recorded about 100 billion CPU cycles. The difference is 149 seconds and 300 billion cycles. 300 billion cycles divided by 2.4G Hz (CPU clock speed) gives 125 seconds. So it looks like a lot of gains came from reducing the time spent waiting for data to be loaded into the cache.
I won't pretend I understand DCU_MISS_OUTSTANDING or many other MSRs completely, I just hope intel provide better documentation.
Monday, February 23, 2009
Almost A Year Since Last Update
Wow, it's almost a year since last update. What have I done in the last year ?
*SVD++. This is similar to the SVD++ described by BellKor, or the transdusive MF described by Gravity.
*Expanded KNN, combinations of movie/viewer and raw/residue/binary, and much faster too (raw movie correlation calcuation takes less than 2 minutes now). For all my KNNs, I no longer save huge correlation files, instead the program just calculates correlations and predictions in one pass and saves predictions only.
*Recently got basic RBM working. This is the most confusing model for me. Basically I got it working thru trial and error. I still don't really understand it, and I have yet to make it work on residues of other predictors.
*Read a lot of research papers. I actually spent more time reading papers than writing code.
*SVD++. This is similar to the SVD++ described by BellKor, or the transdusive MF described by Gravity.
*Expanded KNN, combinations of movie/viewer and raw/residue/binary, and much faster too (raw movie correlation calcuation takes less than 2 minutes now). For all my KNNs, I no longer save huge correlation files, instead the program just calculates correlations and predictions in one pass and saves predictions only.
*Recently got basic RBM working. This is the most confusing model for me. Basically I got it working thru trial and error. I still don't really understand it, and I have yet to make it work on residues of other predictors.
*Read a lot of research papers. I actually spent more time reading papers than writing code.
Monday, February 25, 2008
Residues and Threads
I'm slowly changing all my algorithms to work on either raw data or residue data. Because raw data are integer values in the range of 1 to 5, and thus enabling some performance optimizations and shortcuts, but residue data are essentially floats, all my algorithms took a performance hit.
At the same time, I'm adding multi-threading support to speed things up. The improvement is significant, but I'm already seeing diminishing returns: on a quad-core systems, matrix factorization using 4 cores runs at only 3.27x the speed of using one core.
At the same time, I'm adding multi-threading support to speed things up. The improvement is significant, but I'm already seeing diminishing returns: on a quad-core systems, matrix factorization using 4 cores runs at only 3.27x the speed of using one core.
Monday, January 28, 2008
Back to Basics
New year, now I'm playing with the most basic stuff: global effects. This time I'll try to be more patient and take a more systematic approach, instead of the "hack anything that looks promising" approach I've been taking.
Right now I've a nice number for my qualifying RMSE: 0.8888. Hey, it takes real skill to hit such a nice number by chance ! I'll try to resist the urge to submit better results until I'm sure I've made a big improvement.
Right now I've a nice number for my qualifying RMSE: 0.8888. Hey, it takes real skill to hit such a nice number by chance ! I'll try to resist the urge to submit better results until I'm sure I've made a big improvement.
Friday, December 14, 2007
Monday, September 10, 2007
The Greater Collaborative Filtering Groupthink: KNN
(Note: this post assumes you're familiar with The Netflix Prize.)
In this post I'll describe my implementation of the KNN algorithm, which stands for K Nearest Newbies (with me being the biggest newbie around). A more descriptive name for my implementation is "movie-to-movie Pearson's r-based KNN with slope-one correction and other tweaks" - that's how I see it anyway. I was able to get a 0.9174 qualifying score using this method.
In a nutshell, the KNN algorithm is this: to predict viewer v's rating for movie m, you first get the list of all the movies previously rated by viewer v. Among those movies, you find the K closest "neighbors" to movie m. Then you calculate the weighted-average of viewer v's ratings for those "neighbor" movies, and that’s your final prediction.
Making KNN Predictions
Conceptually, to predict the rating for movie m by viewer v, I do the following:
First get a list of all movies rated by viewer v, let's call this list L0.
Pick a value for parameter MinCV, which is the minimum number of common viewers required. By common viewers I mean viewers who rated both movies. Remove all movies in L0 that have less than MinCV common viewers with movie m. From trial runs, it seems 16 is a good number for MinCV.
For each movie n left in the list L0, calculate the following data structure:
In the structure above, commonViewers is the number of viewers who rated both movie m and n, mAvg and nAvg are the average ratings of movie m and n by common viewers, respectively. And nRating is viewer v's rating for movie n.
The rRaw is Pearson's product moment coefficient, aka Pearson's r, for the movie pair in question. And rLower is the lower bound of the movie pair's Pearson's r at 95% confidence interval. This is calculated using rRaw, commonViewers, and Fisher Transform/Fisher Inverse. If rRaw has a small absolute value, rLower may end up having a different sign than rRaw, in this case, I chose to set rLower to 0.
Once rLower is calculated, I calculate the weight this way:
This is just one of the many ways to calculate weight. For example, you can change the confidence interval, you can use a power function on commonViewers, or a sigmoid function, or just linear with a max limit. I tried many different ways of calculating the weight and found the best ones gave very similar end results.
Put all MovieNeighbor structures in a new list, let's call this list L1. Each item in list L1 represents a neighbor for movie m. And we don't need list L0 any more.
Create a new MovieNeighbor structure, and set the following fields:
add this new structure to list L1. This creates a dummy neighbor that predicts movie m's average rating. The effect of this dummy neighbor depends on the movie m's real neighbors. If movie m has lots of weak neighbors (low correlation, low number of common viewers), then the dummy neighbor will pull the prediction toward movie m's average. On the other hand, if movie m has many good-quality neighbors (high correlation, high number of common viewers), then the dummy neighbor will have no effect on the final outcome.
Pick a value for parameter MaxN, which is the maximum number of neighbors used for prediction. Sort list L1 by weight and keep the MaxN items that have the greatest weights (implementation hint: don't call sort or qsort, use nth_element).
For each item remaining in L1, calculate its prediction:
The final prediction is the weighted-average of predictions of all remaining items in list L1. That's it !
Implementation Notes
A straight-forward implementation of the above will work, but it will be slow and make it almost impractical to explore large number of parameter combinations. As you can see there're a number of parameters you can tune:
To try large combinations of parameter values efficiently, you want to pre-calculate and share the computational overhead as much as possible.
On the top of the list is the following for each movie pair in the data set:
There're 17770 movies in the data set, so there're 17770x17770 (of which half are unique) MovieCorrelation structures we can calculate and save to a large file. This sounds like a major task, but it only needs to be done once (once your code is bug-free).
And thanks to the raw speed of modern computers, it does not take as long as it looks. I managed to get it done in less than 20 minutes on my laptop, and my implementation is by no means optimized to the max. If you have a fast PC, it should take less than 15 minutes; and if you have enough memory to hold 17770x17770/2 (packed) records in memory, it will probably take less than 10 minutes.
Once you have all that MovieCorrelation calculated and saved in a file, arrange your prediction code like this:
The 1st parameter might be confidence interval, the 2nd parameter might be maximum number of neighbors, etc. After I wrote my prediction code like above, it took about one minute to generate one probe prediction, but to generate 32 probe predictions, it only took 3 minutes !
In this post I'll describe my implementation of the KNN algorithm, which stands for K Nearest Newbies (with me being the biggest newbie around). A more descriptive name for my implementation is "movie-to-movie Pearson's r-based KNN with slope-one correction and other tweaks" - that's how I see it anyway. I was able to get a 0.9174 qualifying score using this method.
In a nutshell, the KNN algorithm is this: to predict viewer v's rating for movie m, you first get the list of all the movies previously rated by viewer v. Among those movies, you find the K closest "neighbors" to movie m. Then you calculate the weighted-average of viewer v's ratings for those "neighbor" movies, and that’s your final prediction.
Making KNN Predictions
Conceptually, to predict the rating for movie m by viewer v, I do the following:
First get a list of all movies rated by viewer v, let's call this list L0.
Pick a value for parameter MinCV, which is the minimum number of common viewers required. By common viewers I mean viewers who rated both movies. Remove all movies in L0 that have less than MinCV common viewers with movie m. From trial runs, it seems 16 is a good number for MinCV.
For each movie n left in the list L0, calculate the following data structure:
struct MovieNeighbor { //movie n as neighbor of movie m
unsigned int commonViewers;
float mAvg; //average rating of movie m
float nAvg; //average rating of movie n
float nRating; //current viewer's rating for movie n
float rRaw; //raw Pearson's r value
float rLower;
float weight;
};
In the structure above, commonViewers is the number of viewers who rated both movie m and n, mAvg and nAvg are the average ratings of movie m and n by common viewers, respectively. And nRating is viewer v's rating for movie n.
The rRaw is Pearson's product moment coefficient, aka Pearson's r, for the movie pair in question. And rLower is the lower bound of the movie pair's Pearson's r at 95% confidence interval. This is calculated using rRaw, commonViewers, and Fisher Transform/Fisher Inverse. If rRaw has a small absolute value, rLower may end up having a different sign than rRaw, in this case, I chose to set rLower to 0.
Once rLower is calculated, I calculate the weight this way:
weight = rLower*rLower*log(commonViewers);
This is just one of the many ways to calculate weight. For example, you can change the confidence interval, you can use a power function on commonViewers, or a sigmoid function, or just linear with a max limit. I tried many different ways of calculating the weight and found the best ones gave very similar end results.
Put all MovieNeighbor structures in a new list, let's call this list L1. Each item in list L1 represents a neighbor for movie m. And we don't need list L0 any more.
Create a new MovieNeighbor structure, and set the following fields:
mAvg=average rating of movie m from all viewers
nAvg=ratingn=0;
weight=log(MinCV);
add this new structure to list L1. This creates a dummy neighbor that predicts movie m's average rating. The effect of this dummy neighbor depends on the movie m's real neighbors. If movie m has lots of weak neighbors (low correlation, low number of common viewers), then the dummy neighbor will pull the prediction toward movie m's average. On the other hand, if movie m has many good-quality neighbors (high correlation, high number of common viewers), then the dummy neighbor will have no effect on the final outcome.
Pick a value for parameter MaxN, which is the maximum number of neighbors used for prediction. Sort list L1 by weight and keep the MaxN items that have the greatest weights (implementation hint: don't call sort or qsort, use nth_element).
For each item remaining in L1, calculate its prediction:
dif = ratingn - nAvg;
if (rRaw<0 dif="-dif;<br"> prediction = mAvg + dif;
The final prediction is the weighted-average of predictions of all remaining items in list L1. That's it !
Implementation Notes
A straight-forward implementation of the above will work, but it will be slow and make it almost impractical to explore large number of parameter combinations. As you can see there're a number of parameters you can tune:
- confidence interval
- power rLower
- function of common viewers (log, power, sigmoid, etc)
- minimum number of common viewers required
- max number of neighbors
- etc ...
To try large combinations of parameter values efficiently, you want to pre-calculate and share the computational overhead as much as possible.
On the top of the list is the following for each movie pair in the data set:
struct MovieCorrelation { //for movie pair m-n
unsigned int commonViewers;
float mAvg; //average rating of movie m
float nAvg; //average rating of movie n
float rRaw; //raw Pearson's r value
};
There're 17770 movies in the data set, so there're 17770x17770 (of which half are unique) MovieCorrelation structures we can calculate and save to a large file. This sounds like a major task, but it only needs to be done once (once your code is bug-free).
And thanks to the raw speed of modern computers, it does not take as long as it looks. I managed to get it done in less than 20 minutes on my laptop, and my implementation is by no means optimized to the max. If you have a fast PC, it should take less than 15 minutes; and if you have enough memory to hold 17770x17770/2 (packed) records in memory, it will probably take less than 10 minutes.
Once you have all that MovieCorrelation calculated and saved in a file, arrange your prediction code like this:
for each movie m to predict {
array0=read all MovieCorrelation's of movie m from file
for each value of 1st parameter {
array1=some_operation(array0);
for each value of 2nd parameter {
array2=some_operation(array1);
for each value of 3rd parameter {
array3=some_operation(array2);
do predictions using array3;
}
}
}
}
The 1st parameter might be confidence interval, the 2nd parameter might be maximum number of neighbors, etc. After I wrote my prediction code like above, it took about one minute to generate one probe prediction, but to generate 32 probe predictions, it only took 3 minutes !
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