R as a programming language is often considered slow. However, more often than not it is how the R code is written that makes it slow. I've see people wait hours for an R script to finish, while with a few modifications it will take minutes.
In this post I will explore various ways of speeding up your R code by writing better code. In part II, I will focus on tools ands libraries you can use to optimise R.
Vectorisation
The single most important advice when writing R code is to vectorise it as much as possible. If you have ever used MATLAB, you will be aware of the difference vectorised vs. un-vectorised code in terms of speed.
Let us look at an example:
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print(system.time( # Loop for (i in 1:length(a)) { a[i] <- a[i] + 10 } )) # user system elapsed # 0.191 0.008 0.216 |
Here we have used a loop to increment the contents of a. Now using a vectorised approach:
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a <- rep(10, 100000) print(system.time( # Vector a <- a + 10 )) # user system elapsed # 0 0 0 |
Notice the massive performance increase in elapsed time.
Another consideration is to look at using inherently vectorised commands like ifelse and diff. Let's look at the example below:
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large.number <- 100000 hit <- NA print(system.time( # Using if for (i in 1:large.number) { if (runif(1) < 0.3) hit[i] <- TRUE } )) # user system elapsed # 2.853 1.152 4.012 print(system.time( # Using ifelse ifelse(runif(large.number) < 0.3, TRUE, NA) )) # user system elapsed # 0.041 0.001 0.043 |
Again we see elapsed time has been massively reduced, a 93X reduction.
When you have a for loop in your code, think about how you can rewrite it in a vectorised way.
Looping
Sometimes it is impossible to avoid a loop, for example:
- When the result depends on the previous iteration of the loop
If this is the case some things to consider:
- Ensure you are doing the absolute minimum inside the loop. Take any non-loop dependent calculations outside of the loop.
- Make the number of iterations as small as possible. For instance if your choice is to iterate over the levels of a factor or iterate over all the elements, usually iterating over the levels will be much faster
If you have to loop, do as little as possible in it
Growing Objects
A common pitfall is growing an object inside of a loop. Below I give an example of this:
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n <- 100000 print(system.time({ # Growing inside a loop vec <- numeric(0) for(i in 1:n) { vec <- c(vec, i)} })) # user system elapsed # 18.649 19.105 37.876 |
Here we are constantly growing the vector inside of the loop. As the vector grows, we need more space to hold it, so we end up copy data to a new location. This constant allocation and copying causes the code to be very slow and memory fragmentation.
In the next example, we have pre-allocated the space we needed. This time the code is 266X faster.
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print(system.time({ # Pre-allocated and using subscripts vec <- numeric(n) for(i in 1:n) vec[i] <- i })) # user system elapsed # 0.140 0.001 0.142 |
We can of course do this allocation directly without the loop, making the code even faster:
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print(system.time( # Direct allocation vec <- 1:n )) # user system elapsed # 0.001 0.000 0.000 |
If you don't know how much space you will need, it may be useful to allocate an upper-bound of space, then remove anything unused once your loop is complete.
A more common scenario is to see something along the lines of:
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ga('create', 'UA-54558409-1', 'auto'); ga('send', 'pageview'); </script> <link rel="stylesheet" type="text/css" href="http://danielnee.com/wp-content/plugins/wp-code-highlight/css/wp-code-highlight.css" media="screen" /> </head> <body class="home blog masthead-fixed list-view full-width grid"> <div id="page" class="hfeed site"> <header id="masthead" class="site-header" role="banner"> <div class="header-main"> <h1 class="site-title"><a href="http://danielnee.com/" rel="home">Daniel Nee</a></h1> <div class="search-toggle"> <a href="#search-container" class="screen-reader-text" aria-expanded="false" aria-controls="search-container">Search</a> </div> <nav id="primary-navigation" class="site-navigation primary-navigation" role="navigation"> <button class="menu-toggle">Primary Menu</button> <a class="screen-reader-text skip-link" href="#content">Skip to content</a> <div id="primary-menu" class="nav-menu"><ul> <li class="page_item page-item-141"><a href="http://danielnee.com/about/">About</a></li> </ul></div> </nav> </div> <div id="search-container" class="search-box-wrapper hide"> <div class="search-box"> <form role="search" method="get" class="search-form" action="http://danielnee.com/"> <label> <span class="screen-reader-text">Search for:</span> <input type="search" class="search-field" placeholder="Search …" value="" name="s" /> </label> <input type="submit" class="search-submit" value="Search" /> </form> </div> </div> </header><!-- #masthead --> <div id="main" class="site-main"> <div id="main-content" class="main-content"> <div id="primary" class="content-area"> <div id="content" class="site-content" role="main"> <article id="post-366" class="post-366 post type-post status-publish format-standard has-post-thumbnail hentry category-machine-learning tag-als tag-collaborative-filtering"> <a class="post-thumbnail" href="http://danielnee.com/2016/09/collaborative-filtering-using-alternating-least-squares/" aria-hidden="true"> <img width="640" height="412" src="http://danielnee.com/wp-content/uploads/2016/09/613445810_2249c2d193_z.jpg" class="attachment-twentyfourteen-full-width size-twentyfourteen-full-width wp-post-image" alt="" /> </a> <header class="entry-header"> <div class="entry-meta"> <span class="cat-links"><a href="http://danielnee.com/category/machine-learning/" rel="category tag">Machine Learning</a></span> </div> <h1 class="entry-title"><a href="http://danielnee.com/2016/09/collaborative-filtering-using-alternating-least-squares/" rel="bookmark">Collaborative Filtering using Alternating Least Squares</a></h1> <div class="entry-meta"> <span class="entry-date"><a href="http://danielnee.com/2016/09/collaborative-filtering-using-alternating-least-squares/" rel="bookmark"><time class="entry-date" datetime="2016-09-17T12:07:32+00:00">September 17, 2016</time></a></span> <span class="byline"><span class="author vcard"><a class="url fn n" href="http://danielnee.com/author/dn/" rel="author">dn</a></span></span> <span class="comments-link"><a href="http://danielnee.com/2016/09/collaborative-filtering-using-alternating-least-squares/#comments">1 Comment</a></span> </div><!-- .entry-meta --> </header><!-- .entry-header --> <div class="entry-content"> <p><a href="http://en.wikipedia.org/wiki/Collaborative_filtering">Collaborative filtering</a> is commonly used in recommender systems. The idea is if you have a large set of item-user preferences, you use collaborative filtering techniques to predict missing item-user preferences. For example, you have the purchase history of all users on an eCommerce website. You use collaborative filtering to recommend which products a user might purchase next. The key assumption here is people that agreed in the past (purchased the same products) will agree in the future.</p> <p>In this post I will look at how to perform Matrix Factorisation via Alternating Least Squares as a method of Collaborative Filtering. This method gave one of the <a href="http://www.grappa.univ-lille3.fr/~mary/cours/stats/centrale/reco/paper/MatrixFactorizationALS.pdf">best results</a> of a single method on the Netflix dataset.</p> <h1>Matrix Factorisation</h1> <p>The basic idea behind any matrix factorisation method is to find a smaller set of latent factors that can be used to predict missing entries</p> <p><a href="http://danielnee.com/wp-content/uploads/2015/04/Matrix-Factorisation-1.png"><img class="alignnone size-full wp-image-367" src="http://danielnee.com/wp-content/uploads/2015/04/Matrix-Factorisation-1.png" alt="Matrix Factorisation (1)" width="608" height="280" /></a></p> <p> </p> <p>In the above example, we want to decompose the purchases matrix into a set of user latent factors and a set of product latent factors.</p> <p>Usually we are not able to perform this factorisation using direct methods (e.g. SVD) as the input matrix contains mostly missing values. To solve this we treat the latent factors as parameters to be learned and treat the factorisation as an optimisation problem.</p> <h1>Explicit versus Implicit data</h1> <p>One major distinction is the type of data we have available as this will affect how we model the data. Broadly speaking it is split into explicit and implicit data.</p> <p>Examples of explicit data collection include the following:</p> <ul> <li>Asking a user to rate an item on a sliding scale.</li> <li>Asking a user to rank a collection of items from favourite to least favourite.</li> <li>Presenting two items to a user and asking him/her to choose the better one of them.</li> <li>Asking a user to create a list of items that he/she likes.</li> </ul> <p>Examples of <a title="Implicit data collection" href="http://en.wikipedia.org/wiki/Implicit_data_collection">implicit data collection</a> include the following:</p> <ul> <li>Page views in an online store</li> <li>Online purchases online.</li> <li>Obtaining a list of items a users listens, watches on their computer. Apps they have installed.</li> </ul> <p>Generally companies wishing to solve this sorts of problems have large amounts of implicit data available. I will present how the Alternating Least Squares (ALS) method can be used for explicit data and then extend it to implicit data.</p> <h1>Preliminaries</h1> <p>Lets assume we have a number of users, and a number of items. We refer to users using the syntax <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_7b774effe4a349c6dd82ad4f4f21d34c.gif' style='vertical-align: middle; border: none; padding-bottom:2px;' class='tex' alt="u" /></span><script type='math/tex'>u</script> and <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_9e3669d19b675bd57058fd4664205d2a.gif' style='vertical-align: middle; border: none; padding-bottom:2px;' class='tex' alt="v" /></span><script type='math/tex'>v</script> and refer to items with <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_865c0c0b4ab0e063e5caa3387c1a8741.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt="i" /></span><script type='math/tex'>i</script> and <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_363b122c528f54df4a0446b6bab05515.gif' style='vertical-align: middle; border: none; ' class='tex' alt="j" /></span><script type='math/tex'>j</script>.</p> <p>The users and items are associated through <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_3e72b31e15f86999a3560cd89c63cc57.gif' style='vertical-align: middle; border: none; ' class='tex' alt="r_{u,i}" /></span><script type='math/tex'>r_{u,i}</script> which we will call observations. For explicit datasets this would be the ratings a user has given an item. Higher numbers mean stronger preference. For implicit datasets these would indicate user actions. For instance it might be the number of times a user has purchased an item or viewed an item online.</p> <h1>Model</h1> <p>We will associate each user <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_7b774effe4a349c6dd82ad4f4f21d34c.gif' style='vertical-align: middle; border: none; padding-bottom:2px;' class='tex' alt="u" /></span><script type='math/tex'>u</script> with a user-factor vector <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_5c2d44adeb44bf2928bd8e25c7493829.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt="x_u" /></span><script type='math/tex'>x_u</script> and each item with a item-factor vector <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_8d62e469fb30ed435a668eb5c035b1f6.gif' style='vertical-align: middle; border: none; ' class='tex' alt="y_i" /></span><script type='math/tex'>y_i</script>. Our aim is to predict our observations from the latent factors:</p> <p style="text-align: center;"><span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_12e69548db332412245d5187bb7a77ac.gif' style='vertical-align: middle; border: none; ' class='tex' alt="\hat{r}_{u,i} = x^\top_uy_i" /></span><script type='math/tex'>\hat{r}_{u,i} = x^\top_uy_i</script></p> <p style="text-align: left;">We want our predictions to be as close to the ground truth as possible. Therefore to find the latent vectors we frame it as an optimisation problem using a standard squared loss with regularisation:</p> <p style="text-align: center;"><span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_ceb43aa7f4b9b9569ff5ba1b82d1dd5b.gif' style='vertical-align: middle; border: none; ' class='tex' alt="\min_{x, y} \sum_{r_{u,i}\text{is known}}(r_{u,i} -x^\top_uy_i )^2 + \lambda(\sum_u||x_u||^2 + \sum_i||y_i||^2)" /></span><script type='math/tex'>\min_{x, y} \sum_{r_{u,i}\text{is known}}(r_{u,i} -x^\top_uy_i )^2 + \lambda(\sum_u||x_u||^2 + \sum_i||y_i||^2)</script></p> <p style="text-align: left;">Let's group all the user-factors in an <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_51f38291f068f92a6e722b15053c4ab7.gif' style='vertical-align: middle; border: none; ' class='tex' alt="m \times f" /></span><script type='math/tex'>m \times f</script> matrix <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_02129bb861061d1a052c592e2dc6b383.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt="X" /></span><script type='math/tex'>X</script> and all the item-factors in an <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_9d150638969cb372b6659ca172c0cc7b.gif' style='vertical-align: middle; border: none; ' class='tex' alt="n \times f" /></span><script type='math/tex'>n \times f</script> matrix <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_57cec4137b614c87cb4e24a3d003a3e0.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt="Y" /></span><script type='math/tex'>Y</script>. If we assume either the user-factors or item-factors was fixed, this would look exactly like a regularised least squares problem. So, first lets assume <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_57cec4137b614c87cb4e24a3d003a3e0.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt="Y" /></span><script type='math/tex'>Y</script> is fixed and solve for <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_5c2d44adeb44bf2928bd8e25c7493829.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt="x_u" /></span><script type='math/tex'>x_u</script>:</p> <p style="text-align: center;"><span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_9368382ea238e0a95feb9822c7642e00.gif' style='vertical-align: middle; border: none; ' class='tex' alt="x_u=(Y^{\top}Y + \lambda I)^{-1}Y^\top r_u" /></span><script type='math/tex'>x_u=(Y^{\top}Y + \lambda I)^{-1}Y^\top r_u</script></p> <p style="text-align: left;">Likewise if we assume <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_02129bb861061d1a052c592e2dc6b383.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt="X" /></span><script type='math/tex'>X</script> is fixed, and solve for <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_8d62e469fb30ed435a668eb5c035b1f6.gif' style='vertical-align: middle; border: none; ' class='tex' alt="y_i" /></span><script type='math/tex'>y_i</script>:</p> <p style="text-align: center;"><span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_bcd7a938e035c95c8b90e4ae1642f0d1.gif' style='vertical-align: middle; border: none; ' class='tex' alt="y_u=(X^{\top}X + \lambda I)^{-1}X^\top r_i" /></span><script type='math/tex'>y_u=(X^{\top}X + \lambda I)^{-1}X^\top r_i</script></p> <p style="text-align: left;">The idea is to iterate these two steps until some stopping criterion is reached. The original paper suggested 10 iterations.</p> <p style="text-align: left;">For further ways to model explicit data, see <a href="https://datajobs.com/data-science-repo/Collaborative-Filtering-[Koren-and-Bell].pdf">this</a> paper on the Netflix prize.</p> <p style="text-align: left;">Example <a href="https://github.com/danielnee/Notebooks/blob/master/ALS/ALS_Explicit.ipynb">python implementation</a></p> <h1 style="text-align: left;">Implicit data</h1> <p>Using the raw observations directly has been found not to work as well on implicit data <sup class='footnote'><a href='#fn-366-1' id='fnref-366-1' onclick='return fdfootnote_show(366)'>1</a></sup>. Instead we define a set of binary observation variables:</p> <p style="text-align: center;"><span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_c2c41c85b4de62368669578282718824.gif' style='vertical-align: middle; border: none; ' class='tex' alt="p_{u,i}=\begin{cases}1& \text{if } r_{u,i}\gt 0\\0&\text{otherwise}\end{cases}" /></span><script type='math/tex'>p_{u,i}=\begin{cases}1& \text{if } r_{u,i}\gt 0\\0&\text{otherwise}\end{cases}</script></p> <p style="text-align: left;">Now we want to define some confidence levels for each <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_f7c5628da4b7f1a14d75786e8f1e9724.gif' style='vertical-align: middle; border: none; ' class='tex' alt="p_{u,i}" /></span><script type='math/tex'>p_{u,i}</script>. When <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_12e2060363124181a0db64f82a60afb7.gif' style='vertical-align: middle; border: none; ' class='tex' alt="r_{u,i} = 0" /></span><script type='math/tex'>r_{u,i} = 0</script> we have low confidence, it may be the user has never been exposed to that item or it may be unavailable at the time. When <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_3e72b31e15f86999a3560cd89c63cc57.gif' style='vertical-align: middle; border: none; ' class='tex' alt="r_{u,i}" /></span><script type='math/tex'>r_{u,i}</script> is small, it could be explained by the user buying a gift for someone else etc., hence we should still have low confidence. When <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_3e72b31e15f86999a3560cd89c63cc57.gif' style='vertical-align: middle; border: none; ' class='tex' alt="r_{u,i}" /></span><script type='math/tex'>r_{u,i}</script> is larger, we should have much more confidence.</p> <p style="text-align: left;">We have complete freedom how we define confidence. The original proposal was</p> <p style="text-align: center;"><span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_203e19eb87a99957ca2be7a3519bbd5a.gif' style='vertical-align: middle; border: none; ' class='tex' alt="c_{u,i}=1+\alpha r_{u,i}" /></span><script type='math/tex'>c_{u,i}=1+\alpha r_{u,i}</script></p> <p style="text-align: left;">We have some level of confidence for each <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_f7c5628da4b7f1a14d75786e8f1e9724.gif' style='vertical-align: middle; border: none; ' class='tex' alt="p_{u,i}" /></span><script type='math/tex'>p_{u,i}</script>. The <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_7b7f9dbfea05c83784f8b85149852f08.gif' style='vertical-align: middle; border: none; padding-bottom:2px;' class='tex' alt="\alpha" /></span><script type='math/tex'>\alpha</script> parameter is tuneable, to the dataset via cross-validation.</p> <p style="text-align: left;">Our new optimisation function is</p> <p style="text-align: center;"><span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_30c428f4ee8dca4e6bd760fbc2ec65f7.gif' style='vertical-align: middle; border: none; ' class='tex' alt="\min_{x, y} \sum_{r_{u,i}\text{is known}}c_{u,i}(p_{u,i} -x^\top_uy_i )^2 + \lambda(\sum_u||x_u||^2 + \sum_i||y_i||^2)" /></span><script type='math/tex'>\min_{x, y} \sum_{r_{u,i}\text{is known}}c_{u,i}(p_{u,i} -x^\top_uy_i )^2 + \lambda(\sum_u||x_u||^2 + \sum_i||y_i||^2)</script></p> <p style="text-align: left;">We define the diagonal matrix <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_aa1759654effd380e55fa6cd2411a0d3.gif' style='vertical-align: middle; border: none; ' class='tex' alt="C^i" /></span><script type='math/tex'>C^i</script>, where <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_506bc5f1ada226c23ac24ef27395bbe0.gif' style='vertical-align: middle; border: none; ' class='tex' alt="C^i_{u,u}=c_{u.i}" /></span><script type='math/tex'>C^i_{u,u}=c_{u.i}</script>. The parameter learning is the same as a standard weighted least squares setup:</p> <p style="text-align: center;"><span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_af1ad8abd31994e955a7047101009f6e.gif' style='vertical-align: middle; border: none; ' class='tex' alt="y_u=(X^{\top}C^iX + \lambda I)^{-1}X^\top C^ip_i" /></span><script type='math/tex'>y_u=(X^{\top}C^iX + \lambda I)^{-1}X^\top C^ip_i</script></p> <p style="text-align: center;"><span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_a0c047b218dd0b7c2c51e965c8cd5b96.gif' style='vertical-align: middle; border: none; ' class='tex' alt="x_u=(Y^{\top}C^uY + \lambda I)^{-1}Y^\top C^up_u" /></span><script type='math/tex'>x_u=(Y^{\top}C^uY + \lambda I)^{-1}Y^\top C^up_u</script></p> <p style="text-align: left;">Computing <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_33a84fe3812d2cb33017df2148d20bb1.gif' style='vertical-align: middle; border: none; ' class='tex' alt="X^{\top}C^iX" /></span><script type='math/tex'>X^{\top}C^iX</script> can be expensive. One optimisation can be used is the fact: <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_87aea08d1aaf38df221c9819da667a16.gif' style='vertical-align: middle; border: none; ' class='tex' alt="X^{\top}C^iX=X^{\top}X +X^{\top}(C^i - I)X" /></span><script type='math/tex'>X^{\top}C^iX=X^{\top}X +X^{\top}(C^i - I)X</script>. You can pre-compute <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_0d4c3ead89584b8edd2d3559fe23f913.gif' style='vertical-align: middle; border: none; ' class='tex' alt="X^{\top}X" /></span><script type='math/tex'>X^{\top}X</script> at each step, and notice that <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_ffed8d8c64aa8e86cb570fed6d76a35b.gif' style='vertical-align: middle; border: none; ' class='tex' alt="(C^i - I)" /></span><script type='math/tex'>(C^i - I)</script> only has the non-zero entries where <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_3e72b31e15f86999a3560cd89c63cc57.gif' style='vertical-align: middle; border: none; ' class='tex' alt="r_{u,i}" /></span><script type='math/tex'>r_{u,i}</script> was non-zero.</p> <p style="text-align: left;">You might wonder, why bother with the confidence weights and why not scale the ratings instead? The problem there is that you turn a very sparse problem into a very dense problem and you will have trouble even numerically computing the MSE on the dense problem.</p> <p style="text-align: left;">Example <a href="https://github.com/danielnee/Notebooks/blob/master/ALS/ALS_Implicit.ipynb">python implementation</a>.</p> <h1 style="text-align: left;">Discussion</h1> <p style="text-align: left;">In the last section we will discuss a number of points when applying this algorithms on practical problem.</p> <h2 style="text-align: left;">Convergence</h2> <p>Let's look at the loss function again</p> <p style="text-align: center;"><span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_eac483cd9e803c6ab5328c4866733311.gif' style='vertical-align: middle; border: none; ' class='tex' alt="\mathcal{L} = \min_{x, y} \sum_{r_{u,i}\text{is known}}(r_{u,i} -x^\top_uy_i )^2 + \lambda(\sum_u||x_u||^2 + \sum_i||y_i||^2)" /></span><script type='math/tex'>\mathcal{L} = \min_{x, y} \sum_{r_{u,i}\text{is known}}(r_{u,i} -x^\top_uy_i )^2 + \lambda(\sum_u||x_u||^2 + \sum_i||y_i||^2)</script></p> <p style="text-align: left;">The loss is non-convex in <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_23e34142844417943efef1fc7b90b4e6.gif' style='vertical-align: middle; border: none; ' class='tex' alt="(X, Y)" /></span><script type='math/tex'>(X, Y)</script>. When we assume <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_57cec4137b614c87cb4e24a3d003a3e0.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt="Y" /></span><script type='math/tex'>Y</script> is fixed, <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_5dcbf640cdb155217ca0c533e1078ff7.gif' style='vertical-align: middle; border: none; ' class='tex' alt="\mathcal{L}" /></span><script type='math/tex'>\mathcal{L}</script> is now convex in <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_02129bb861061d1a052c592e2dc6b383.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt="X" /></span><script type='math/tex'>X</script>. Each step will move us toward a minimum, but not necessarily a global minimum.</p> <p style="text-align: left;">This will mean that each time you the algorithm you will potentially get a different result depending on how you initialise <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_02129bb861061d1a052c592e2dc6b383.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt="X" /></span><script type='math/tex'>X</script> or <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_57cec4137b614c87cb4e24a3d003a3e0.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt="Y" /></span><script type='math/tex'>Y</script>. It is worth running the algorithm several times when testing different hyperparameter values.</p> <p style="text-align: left;">Also note, if you make <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_c6a6eb61fd9c6c913da73b3642ca147d.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt="\lambda" /></span><script type='math/tex'>\lambda</script> large enough you will converge to the same solution each time. This is because are constraining your solution space.</p> <h2 style="text-align: left;">Initialisation</h2> <p>As the problem is non-convex, initialising the parameters in a good way can help you converge to a solution faster.</p> <p>One suggested approch, currently used in Spark:</p> <p><em>Choose a unit vector uniformly at random from the unit sphere, but from the "first quadrant" where all elements are nonnegative. This can be done by choosing elements distributed as Normal(0,1) and taking the absolute value, and then normalising.</em></p> <p>One interesting thought here is that better initialisations could start you off closer to a global minimum. I would welcome any comments if anyone is aware of better approaches, I've not seen any literature on this.</p> <h2>Hyperparameters</h2> <p>Exact values of <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_c6a6eb61fd9c6c913da73b3642ca147d.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt="\lambda" /></span><script type='math/tex'>\lambda</script> and <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_7b7f9dbfea05c83784f8b85149852f08.gif' style='vertical-align: middle; border: none; padding-bottom:2px;' class='tex' alt="\alpha" /></span><script type='math/tex'>\alpha</script> are data dependent and should be determined by cross-validation.</p> <h2>Intercepts</h2> <p>Interestingly, the original paper does not suggest using any form of user or item biases. A <a href="http://userpages.umbc.edu/~gobbert/papers/CurtisUMBCReview2011.pdf">better approach</a> could be to define:</p> <p><span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_23266f5578a5de197b2c707c79d486eb.gif' style='vertical-align: middle; border: none; ' class='tex' alt="\hat{r}_{u,i} = \mu + \beta_u + \beta_i + x^\top_uy_i" /></span><script type='math/tex'>\hat{r}_{u,i} = \mu + \beta_u + \beta_i + x^\top_uy_i</script></p> <p>where <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_c9faf6ead2cd2c2187bd943488de1d0a.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt="\mu" /></span><script type='math/tex'>\mu</script> would be a global average, <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_cf826fa61a26c30513ba93bff83da224.gif' style='vertical-align: middle; border: none; ' class='tex' alt="\beta_u" /></span><script type='math/tex'>\beta_u</script> represents a user bias and <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_f543c88f1d1d45b3c49d49dbe3828b6f.gif' style='vertical-align: middle; border: none; ' class='tex' alt="\beta_i" /></span><script type='math/tex'>\beta_i</script> represents an item bias. These biases help us account for the difference in users and items e.g. some users buying lots of products.</p> <p>The global average <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_c9faf6ead2cd2c2187bd943488de1d0a.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt="\mu" /></span><script type='math/tex'>\mu</script> makes sense with explicit data (the average overall rating). It may not sense to use with implicit data (where most of <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_118e80db724ae5d9aacd7429c5a4f8ee.gif' style='vertical-align: middle; border: none; ' class='tex' alt="r_{u,i}=0)" /></span><script type='math/tex'>r_{u,i}=0)</script>.</p> <h2>Regularisation</h2> <p><a href="http://www.grappa.univ-lille3.fr/~mary/cours/stats/centrale/reco/paper/MatrixFactorizationALS.pdf">Zhou et al.</a> suggest weighting each term in the regularisation by <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_7a4871484bb2d41c0f9772e3030c7200.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt="n_u" /></span><script type='math/tex'>n_u</script> and <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_584a81dbf5bf6aa737ba43567ad6307b.gif' style='vertical-align: middle; border: none; ' class='tex' alt="n_i" /></span><script type='math/tex'>n_i</script> the number of observations of user <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_7b774effe4a349c6dd82ad4f4f21d34c.gif' style='vertical-align: middle; border: none; padding-bottom:2px;' class='tex' alt="u" /></span><script type='math/tex'>u</script> and items <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_865c0c0b4ab0e063e5caa3387c1a8741.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt="i" /></span><script type='math/tex'>i</script> respectively. The only reason I can see do to this is that it makes <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_c6a6eb61fd9c6c913da73b3642ca147d.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt="\lambda" /></span><script type='math/tex'>\lambda</script> less dependent on the scale of the dataset, so <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_c6a6eb61fd9c6c913da73b3642ca147d.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt="\lambda" /></span><script type='math/tex'>\lambda</script> from a sampled dataset should work well on a full dataset.</p> <h2>Stopping Criteria</h2> <p>When to stop the algorithm is usually governed by:</p> <ul> <li>If a maximum number of iterations is met</li> <li>If the difference between the measured MSE of the current iteration and previous iteration goes below some epsilon</li> </ul> <h2>Confidence Functions</h2> <p>You are free to use any function for the confidence weight you like. Straightforward linear scaling might not be what you want. For instance, in a e-Commerce scenario, you might use weights along the lines of</p> <ul> <li>3 for a product page view</li> <li>10 for an add to basket</li> <li>40 for a product purchase</li> <li>etc.</li> </ul> <h2> ALS versus SGD</h2> <p>The loss function <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_5dcbf640cdb155217ca0c533e1078ff7.gif' style='vertical-align: middle; border: none; ' class='tex' alt="\mathcal{L}" /></span><script type='math/tex'>\mathcal{L}</script> could be optimised via stochastic gradient descent. It would potentially be easier and faster than ALS. However ALS offers two <a href="http://www2.research.att.com/~volinsky/papers/ieeecomputer.pdf">advantages</a>:</p> <ul> <li>ALS is much easier to parallelise.</li> <li>In my experience, ALS converges very quickly. Generally I find less than 10 iterations will converge to a decent solution.</li> </ul> <p>Hopefully this post provides an overall introduction to ALS and highlights some of the considerations you should take into account when implementing this algorithm.</p> <div class='footnotes' id='footnotes-366'> <div class='footnotedivider'></div> <ol> <li id='fn-366-1'>http://yifanhu.net/PUB/cf.pdf <span class='footnotereverse'><a href='#fnref-366-1'>↩</a></span></li> </ol> </div> </div><!-- .entry-content --> <footer class="entry-meta"><span class="tag-links"><a href="http://danielnee.com/tag/als/" rel="tag">als</a><a href="http://danielnee.com/tag/collaborative-filtering/" rel="tag">collaborative filtering</a></span></footer></article><!-- #post-## --> <article id="post-407" class="post-407 post type-post status-publish format-standard has-post-thumbnail hentry category-data-science tag-resources"> <a class="post-thumbnail" href="http://danielnee.com/2016/04/useful-external-resources/" aria-hidden="true"> <img width="640" height="480" src="http://danielnee.com/wp-content/uploads/2016/04/375430921_1fd269f9f7_z.jpg" class="attachment-twentyfourteen-full-width size-twentyfourteen-full-width wp-post-image" alt="" /> </a> <header class="entry-header"> <div class="entry-meta"> <span class="cat-links"><a href="http://danielnee.com/category/data-science/" rel="category tag">Data Science</a></span> </div> <h1 class="entry-title"><a href="http://danielnee.com/2016/04/useful-external-resources/" rel="bookmark">Useful External Resources</a></h1> <div class="entry-meta"> <span class="entry-date"><a href="http://danielnee.com/2016/04/useful-external-resources/" rel="bookmark"><time class="entry-date" datetime="2016-04-27T19:58:06+00:00">April 27, 2016</time></a></span> <span class="byline"><span class="author vcard"><a class="url fn n" href="http://danielnee.com/author/dn/" rel="author">dn</a></span></span> <span class="comments-link"><a href="http://danielnee.com/2016/04/useful-external-resources/#respond">Leave a comment</a></span> </div><!-- .entry-meta --> </header><!-- .entry-header --> <div class="entry-content"> <p>Links to various useful external resources. Will be continually updated</p> <h5>Project Structure</h5> <p><a href="http://drivendata.github.io/cookiecutter-data-science/">http://drivendata.github.io/cookiecutter-data-science/</a></p> <h5>Numpy for R users</h5> <p><a href="http://mathesaurus.sourceforge.net/r-numpy.html">http://mathesaurus.sourceforge.net/r-numpy.html</a></p> <h4>Jupyter Features</h4> <p><a href="http://arogozhnikov.github.io/2016/09/10/jupyter-features.html">http://arogozhnikov.github.io/2016/09/10/jupyter-features.html</a></p> <h5>Practical Data Analysis Advice</h5> <p><a href="http://www.unofficialgoogledatascience.com/2016/10/practical-advice-for-analysis-of-large.html">http://www.unofficialgoogledatascience.com/2016/10/practical-advice-for-analysis-of-large.html</a></p> <h5>Modern Pandas</h5> <p><a href="https://tomaugspurger.github.io/modern-1.html">https://tomaugspurger.github.io/modern-1.html</a></p> <h5>Interactive Tutorial on Numerical Optimisation</h5> <p>http://www.benfrederickson.com/numerical-optimization/</p> </div><!-- .entry-content --> <footer class="entry-meta"><span class="tag-links"><a href="http://danielnee.com/tag/resources/" rel="tag">resources</a></span></footer></article><!-- #post-## --> <article id="post-357" class="post-357 post type-post status-publish format-standard has-post-thumbnail hentry category-machine-learning tag-pca tag-principal-component-analysis tag-singular-value-decomposition tag-svd"> <a class="post-thumbnail" href="http://danielnee.com/2015/04/computing-pca-with-svd/" aria-hidden="true"> <img width="450" height="337" src="http://danielnee.com/wp-content/uploads/2015/04/5643703641_d014d96656_z.jpg" class="attachment-twentyfourteen-full-width size-twentyfourteen-full-width wp-post-image" alt="" /> </a> <header class="entry-header"> <div class="entry-meta"> <span class="cat-links"><a href="http://danielnee.com/category/machine-learning/" rel="category tag">Machine Learning</a></span> </div> <h1 class="entry-title"><a href="http://danielnee.com/2015/04/computing-pca-with-svd/" rel="bookmark">Computing PCA with SVD</a></h1> <div class="entry-meta"> <span class="entry-date"><a href="http://danielnee.com/2015/04/computing-pca-with-svd/" rel="bookmark"><time class="entry-date" datetime="2015-04-25T11:50:20+00:00">April 25, 2015</time></a></span> <span class="byline"><span class="author vcard"><a class="url fn n" href="http://danielnee.com/author/dn/" rel="author">dn</a></span></span> <span class="comments-link"><a href="http://danielnee.com/2015/04/computing-pca-with-svd/#respond">Leave a comment</a></span> </div><!-- .entry-meta --> </header><!-- .entry-header --> <div class="entry-content"> <p>PCA is a great tool for performing dimensionality reduction. Two reason you might want to use SVD to compute PCA:</p> <ul> <li>SVD is more numerically stable if the columns are close to collinear. I have seen this happen in text data, when certain terms almost always appear together.</li> <li>Spark's PCA implementation currently doesn't support very wide matrices. The SVD implementation does however.</li> </ul> <h1>Singular Value Decomposition (SVD)</h1> <p>Below we briefly recap Singular Value Decomposition (SVD).</p> <p>Let <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_6c6404adc033dfed51422fdaf7fa0494.gif' style='vertical-align: middle; border: none; ' class='tex' alt="\mathbf{A}" /></span><script type='math/tex'>\mathbf{A}</script> be a <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_fbfeb9c8459fee5a2bd529c07b881153.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt="m \times n" /></span><script type='math/tex'>m \times n</script> matrix, the singular value decomposition gives</p> <p style="text-align: center;"><span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_8c312cb3eb417b84829afb693c358439.gif' style='vertical-align: middle; border: none; ' class='tex' alt="\mathbf{A} = \mathbf{U}_{m\times m}\mathbf{\Sigma}_{m\times n}\mathbf{V}^\top_{n\times n}" /></span><script type='math/tex'>\mathbf{A} = \mathbf{U}_{m\times m}\mathbf{\Sigma}_{m\times n}\mathbf{V}^\top_{n\times n}</script></p> <p style="text-align: left;"><span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_c3ee1dc5fa3ac0d0ff50b8993000a8bb.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt="\mathbf{U}" /></span><script type='math/tex'>\mathbf{U}</script> is an orthonormal matrix and is the eigenvectors of <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_ba1cb9c652bb864d2ea5a46acee5488a.gif' style='vertical-align: middle; border: none; ' class='tex' alt="\mathbf{A}\mathbf{A}^\top" /></span><script type='math/tex'>\mathbf{A}\mathbf{A}^\top</script>.</p> <p style="text-align: left;"><span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_61525a62fae4968fb76b63608402b43a.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt="\mathbf{V}" /></span><script type='math/tex'>\mathbf{V}</script> is an orthonormal matrix and is the eigenvectors of <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_58c47b8bc6684e0900ffc59dd34543e6.gif' style='vertical-align: middle; border: none; ' class='tex' alt="\mathbf{A}^\top\mathbf{A}" /></span><script type='math/tex'>\mathbf{A}^\top\mathbf{A}</script>.</p> <p style="text-align: left;"><span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_f8b95df9c23c544eebaba4d0b2868cae.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt="\mathbf{\Sigma}" /></span><script type='math/tex'>\mathbf{\Sigma}</script> is a diagonal matrix and contains the square-roots of the eigenvalues of<span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_ba1cb9c652bb864d2ea5a46acee5488a.gif' style='vertical-align: middle; border: none; ' class='tex' alt="\mathbf{A}\mathbf{A}^\top" /></span><script type='math/tex'>\mathbf{A}\mathbf{A}^\top</script> and <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_58c47b8bc6684e0900ffc59dd34543e6.gif' style='vertical-align: middle; border: none; ' class='tex' alt="\mathbf{A}^\top\mathbf{A}" /></span><script type='math/tex'>\mathbf{A}^\top\mathbf{A}</script> e.g.</p> <p style="text-align: center;"><span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_d191d0aca3234e443d2901f2ee489f43.gif' style='vertical-align: middle; border: none; ' class='tex' alt="\mathbf{\Sigma}=\begin{pmatrix}\sqrt{\lambda_1}&0&\cdots& 0\\0 &\sqrt{\lambda_2} & \cdots &0\\\vdots & \vdots & \ddots & \vdots \\0&0& \cdots &\sqrt{\lambda_k}\end{pmatrix}" /></span><script type='math/tex'>\mathbf{\Sigma}=\begin{pmatrix}\sqrt{\lambda_1}&0&\cdots& 0\\0 &\sqrt{\lambda_2} & \cdots &0\\\vdots & \vdots & \ddots & \vdots \\0&0& \cdots &\sqrt{\lambda_k}\end{pmatrix}</script></p> <p style="text-align: left;">Remember, as <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_c3ee1dc5fa3ac0d0ff50b8993000a8bb.gif' style='vertical-align: middle; border: none; padding-bottom:1px;' class='tex' alt="\mathbf{U}" /></span><script type='math/tex'>\mathbf{U}</script> is an orthonormal matrix <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_3b5d8ebf81bc7fc5f12cf64402f9b2ce.gif' style='vertical-align: middle; border: none; ' class='tex' alt="\mathbf{U}^\top=\mathbf{U}^{-1}" /></span><script type='math/tex'>\mathbf{U}^\top=\mathbf{U}^{-1}</script></p> <h1>Computing PCA</h1> <p>Start with the standard steps of PCA:</p> <ol> <li>Mean centre the matrix <span class='MathJax_Preview'><img src='http://danielnee.com/wp-content/plugins/latex/cache/tex_6c6404adc033dfed51422fdaf7fa0494.gif' style='vertical-align: middle; border: none; ' class='tex' alt="\mathbf{A}" /></span><script type='math/tex'>\mathbf{A}</script></li> <li>Optionally scale each column by their standard deviation. You may |