Reactive Actors

I've been meaning to revisit reactive programming and the actor model for a while now. I first learned about them in the Principles of Reactive Programming Coursera class and then actors came up again in the Seven Concurrency Models in Seven Weeks book. The Scala I picked up is quickly being forgotten and I haven't done a post with code in a while, so here I'll get back into that and create a simple application using Akka and RxScala.

Actor model

The developerWorks article JVM Concurrency: Acting asynchronously with Akka gives a good introduction to the actor model:

The actor model for concurrent computations builds up systems based on primitives called actors. Actors take actions in response to inputs called messages. Actions can include changing the actor's own internal state as well as sending off other messages and even creating other actors. All messages are delivered asynchronously, thereby decoupling message senders from receivers. Because of this decoupling, actor systems are inherently concurrent: Any actors that have input messages available can be executed in parallel, without restriction.

Then JVM Concurrency: Building actor applications with Akka goes on to explain the advantages of this approach:

If you compose your actors and messages correctly, you end up with a system in which most things happen asynchronously. Asynchronous operation is harder to understand than a linear approach, but it pays off in scalability. Highly asynchronous programs are better able to use increased system resources (for example, memory and processors) either to accomplish a particular task more quickly or to handle more instances of the task in parallel. With Akka, you can even extend this scalability across multiple systems, by using remoting to work with distributed actors.

At first the actor model may sound the same as what I described in my Communicating Sequential Processes post because both involve message passing, but the two concurrency models have several differences:

• Actors have identities while CSPs are anonymous
• Actors transmit messages to named actors while CSPs transmit messages using channels
• Actors transmit messages asynchronously while CSPs can't transmit a message until the sender is ready to receive it

My impression, and I could be wrong, is that actors more naturally extend beyond a single machine to a distributed system since the sending and receiving of messages is decoupled. A quick search does turn up distributed channels in pycsp, though, so it seems that both can be distributed.

Reactive applications

The Reactive Manifesto details four qualities of reactive applications:

• responsive - the system responds in a timely manner if at all possible
• resilient - the system stays responsive in the face of failure
• elastic - the system stays responsive under varying workloads
• message driven - the system relies on asynchronous message passing between components

Where Akka describes itself as a toolkit and runtime, RxScala only claims to be a library for composing asynchronous and event-based programs using observable sequences. To me it's not clear how it helps us achieve all four qualities (or if it even intends to).  Nevertheless, the ReactiveX introduction explains their advantages:

The ReactiveX Observable model allows you to treat streams of asynchronous events with the same sort of simple, composable operations that you use for collections of data items like arrays. It frees you from tangled webs of callbacks, and thereby makes your code more readable and less prone to bugs.

This means is that the methods returning Observables can be implemented using thread pools, non-blocking I/O, actors, or anything else. This is how ReactiveX and Akka will be used together: Actors are the concurrency implementation for services communicating with asynchronous messages.

Combining Akka and RxScala

I came up with the following short example. First I wrote a couple of methods returning an Observable to get a feel for it, then added the stockQuote() method which also uses an actor in it's implementation:


Running it produces the expected output, something like:

6

8

broken service

GOOG: 253.22

I can really see the potential in the Observable model, especially after reading more about it at The Netflix Tech Blog. If you were already using actors maybe combining them like this could make sense. I also need to checkout Akka Streams which seems like a similar idea.

UPDATE

Ray is a framework for parallelizing ML workloads. They use the actor model as a way of coordinating work and maintaining state.

Eigenvectors and PageRank

Eigenvectors are another topic that I used in a math class or two but never really understood what they were for. As I mentioned in my last post Bipartite Graphs and Google Adwords, I like how the professors of the Coursera class Mining Massive Datasets gave great examples of how to apply some academic concepts to real world problems. Their PageRank material is no different as it explains how eigenvectors are used in the algorithm. Here is my summary of the class' link analysis chapter which I'm sure is greatly simplified from what Google really does, but I think it's interesting nonetheless.

Early search engines used an inverted index which I covered in my tf-idf post. That approach is vulnerable to term spam. People would put invisible text on their pages (same color as the background) to trick web crawlers into thinking that the page was about a topic it wasn't. The PageRank algorithm solves that problem by giving more importance to pages that have many in-links, and especially important in-links. Even if someone creates millions of fake pages that link to their real page, it's importance will not be that high because the fake pages will have a low importance as no one is linking to them.

The web can be thought of as a directed graph where web pages are the vertexes and links are the arcs. The basic idea of PageRank is that you simulate web surfers starting at random spots in the web graph and then following random out-links. The most important pages, the pages with the most in-links, will end up with the most surfers.

To implement this simulation, consider a transition matrix M which has n rows and columns where n is the number of pages crawled. M_ij has the value 1/k if page j has k arcs out and one of them is to page i, otherwise it has the value 0. Start with a vector v₀ which has all elements set to 1/n. The first step is to multiply v₀ by M, the second step is to multiple by M², and so on (this is an example of a Markov process which means you can make predictions on its future state based only on the present state and do not need the history).

If the web graph is one strongly connected component and doesn't have any dead ends, then v = Mv is the limiting distribution. The limit would be reached when multiplying the distribution by M another time doesn't change the distribution. Vector v is an eigenvector of M because v = λMv. Vector v is also the principal eigenvector because M is a stochastic matrix and the eigenvalue associated with the principal eigenvector is 1. In practice it takes 50-75 iterations to reach this limit.

M is too big for us to use Gaussian elimination. M is also very sparse, so it makes sense to store only the non-zero elements. An entire column can be represented by the out-degree of a page and the row number of the non-zero elements (since those element's values will be 1 divided by the out-degree). That means we only need to store a little over 4 bytes for each non-zero element. Then we can solve for v using MapReduce and v tells us the "page rank" of each page.

In reality the web is not strongly connected. There are dead ends and there are spider traps, or cycles, in the graph. This problem is solved by the concept of taxation. Each surfer has a small probability of teleporting to a random page instead of following an out-link. There are also other algorithm variations for dealing with link spam. For example, besides page content, you also consider the link text or words near the link, so you are getting other people's take on what the page is about instead of relying solely on the page owner.

UPDATE:

This same idea can be applied to sports ratings where links are replaced by something like goals scored. My Machine Learning for NCAA Basketball Prediction - Performance Edition post has more details and some code.

Resources:
http://infolab.stanford.edu/~ullman/mmds/ch5.pdf

Bipartite Graphs and Google Adwords

Continuing my series of posts relating to the Coursera Mining Massive Datasets class, this post summarizes the chapter on Google AdWords. I liked how the professors showed it to be related to matching in a bipartite graph--something that I learned about in a university graph theory class but without such a practical context as web advertising. I think that too often happens, at least in my experience. I've taken a lot of math classes (and done well in them), but I still don't feel like I had much practice applying what I learned to real-world problems.

To be clear, AdWords is for advertisers and AdSense is for website owners. Google provides an explanation in The difference between AdWords and AdSense. Today I'm focusing on how AdWords works.

The "adwords" model matches web searches with advertisements. Advertisers bid to be shown in response to certain search queries. If a user clicks on one of the ads shown to them then the advertiser will pay. Of course, the most relevant ads are clicked more often than less attractive ads, so the challenge is displaying the ads with the highest bids that are most likely to be clicked, while staying within an advertisers budget. This is a problem that traditional newspapers and magazines don't really have. They can only target specific niches of people (like people who buy Golf Digest), but web advertisers can target individuals (like people searching for a specific brand of golf clubs).

Google knows the click-through rate, or the percentage of the time an ad is clicked when it is displayed. It knows how much of an advertiser's budget has been spent. Google also knows what people have searched for in the past, but it doesn't know for sure what people are going to search for in the future. The "adwords" problem therefore needs a greedy algorithm because all you can do is make the best choice for each search and hope it results in the best overall outcome.

A simplified version of this "adwords" problem can be modeled as maximal matching in bipartite graphs. A matching is a subset of edges where no vertex is an end of two or more edges. A perfect matching contains every vertex and a maximal matching is the largest possible matching for the graph. In this case one side of the graph is search queries, the other side is ads, and the edges are who the ads could be shown to. The maximal matching is the best way to display the ads. The simplifying assumptions are that one ad is shown, all advertisers have the same budget, click-through rates are the same, and bids are either 0 or 1.

The obvious greedy algorithm for matching will consider edges in the order they are given. An edge is part of the matching if neither end is connected to an edge already added to the matching. The competitive ratio is defined as the ratio between the worst online algorithm and the best offline algorithm. The offline solution is the optimal solution because all information about the problem is known in advance. For our bipartite matching solution the ratio is only 0.5, that is it will always find at least half as many matches as what is optimal.

It is possible to do better. The more realistic BALANCE algorithm considers the highest bidder and the highest remaining budget. By doing so it results in a competitive ratio of 0.63 which is the highest possible for an online algorithm.

UPDATE:

Bipartite matching implementations can be found in NetworkX and SciPy.

Resources
http://infolab.stanford.edu/~ullman/mmds/ch8.pdf

Locality Sensitive Hashing

In my previous post on tf-idf, I summarized my notes on how to look for similar documents in the sense that the documents are about the same topic. However, there is a slightly different problem of finding similar documents that are really the same document. They could be, for example, plagiarized, a mirror website, or a news story from the same source carried by multiple news outlets. When there are too many pairs of documents to efficiently compare them all, we can use a technique called locality sensitive hashing or LSH. Today I'll summarize my LSH notes.

In order to find these lexically similar documents, you need to represent each document as a set of sentences or phrases. One technique for this is called shingling and a k-shingle is any substring with length k in the document. The value of k needs to be large enough that the probability of a shingle appearing in any given document is low. To represent shingles more concisely, use a hash function to hash the substrings to a number. Then, for example, each shingle is represented as four bytes instead of a k byte substring.

Even with four byte shingles the set for a document still takes up more space than the document itself. We need to replace these large sets with smaller representations called signatures. A signature is the result of several hundred minhash calculations. Consider a matrix where there is a row for each shingle and a column for every document sets. The matrix has a 1 for a particular row and column if that shingle appears in the set. Pick a permutation of rows and then the minhash value for a column is the number of the first row (in permuted order) with a 1 in that column.

In practice, storing the sparse matrix and permuting and sorting millions of rows is too time-consuming, but it can be simulated by random hash functions that map row numbers to as many buckets as there are rows (and ignoring the small number of collisions). Instead of picking n random permutations you pick n randomly chosen hash functions. If a column has a 1 for row r, then the corresponding signature matrix column is updated with the bucket numbers as long as they are less than the current values.

From the signature matrix we can estimate the Jaccard similarities of the underlying sets. Mind = blown.

It still might not be possible to efficiently check the similarity of all pairs of documents because there are just too many pairs. To find the most similar pairs or all pairs above a certain similarity threshold, we need to focus only on pairs that are likely to be similar. The LSH approach here is to hash items several times and hope that similar pairs will end up in the same bucket at least once. Any pair that hashed to the same bucket in any of the hashings is a candidate pair (even though some of them are false positives).

Divide the signature matrix into b bands where each band consists of r rows. Hash each vector (the portion of each column in a band) to a large number of buckets. Each band has its own buckets. The similarity threshold is an S-curve and is approximately:

For a Jaccard similarity s, the probability of a false negative, i.e. missing a candidate pair, is given by:

Finally, examine the candidate pairs to determine if they are above the previously determined similarity threshold.

Besides the minhash family of functions, there are also other function families that apply to other distance measures like Hamming distance or edit distance. LSH works well for fingerprint matching and entity-resolution.

UPDATE:

Spark implements several LSH operations and has more info on how they can be used.


Resources:
http://infolab.stanford.edu/~ullman/mmds/ch3.pdf

Bloom Filters

A bloom filter is a data structure that can be used for checking whether an element is in a set when the set is too large to use conventional hashing techniques. It is probabilistic in the sense that false positives are possible, i.e. it might say that an element is in the set when it is really not. There are, however, never false negatives. You can not remove elements from a basic bloom filter so it is well-suited to data streaming problems where the number of elements only increases.

An example use case would be a web crawler that needs to determine whether is has already visited a URL. A small percentage of false positives will be acceptable because no significant portion of the web has only one link (read one incorrect bloom filter lookup) pointing to it.

The bloom filter is implemented as a large array of bits and several independent hash functions. Initially all bits are set to 0. An element to be added has all the hash functions applied to it. The bit corresponding to the result of each hash function is set to 1. To check whether an element has been seen before the same hashing process is used on the element. If all resulting bit positions are 1 then it has probably been seen before.

If m is the number of bits in the array and k is the number of hash functions and n is the number of elements, then the probability of a false positive is:

As an example, consider the case where m = 1 billion, n = 1 million, and k = 5. The probability of a false positive is about 0.94%. Note how each element requires more than one bit. m has to be bigger than n or else all bits would eventually be set to 1 and every lookup would be true, so the bloom filter only makes sense where the number of possible elements is large and a traditional bit array would not work.

Optimal number of hash functions also shows how to calculate m and k when n and the desired false positive rate are known.

UPDATE:

In an Ethereum JavaScript API dependency I found an implementation and they have a real life usage example.

tf-idf

Term frequency-inverse document frequency (tf-idf) is a measure of word importance in a document within a corpus. Words with the highest tf-idf often best characterize the topic of the document.

Words appearing most frequently in the corpus are not the most important words as might be expected. They are common words like stop-words. Rare words are actually the best indicators of importance, especially if they appear multiple times.

tf-idf can be represented by the following equation (thanks Online LaTeX Equation Editor):

Term frequency is calculated as the frequency of word i in document j is divided (normalized) by the maximum frequency of any word k in document j. Inverse document frequency, which accounts for words that are just more common, is calculated as the number of documents N divided by the number of those documents n the word i appears in, and then scaling that by taking the logarithm (the base of the log function does not matter).

This topic has come up in a couple of Coursera classes I have looked at--Web Intelligence and Big Data and Mining Massive Datasets--in the context of a search engine. Basically, you view each document and query (short document) as a vector of tf-idf scores, then you can find the most similar ones using cosine similarity as a way to rank the search results. Inverted indexes allow us to pre-compute much of the tf-idf score.

UPDATE:

scikit-learn has an tf-idf usage example at Clustering text documents using k-means.