In part 1 and part 2 of this series, we set both the theoretical and practical foundation of logistic regression and saw how a state of the art implementation can all be implemented in roughly 30 lines of code. In this third (and last) post of this series, we’ll demonstrate the use of a very effective and powerful library to build logistic regression models in practice: Vowpal Wabbit.

What is Vowpal Wabbit

Vowpal Wabbit (VW) is a general purpose machine learning library which is implementing, among other things, logistic regression with the same ideas we presented in our previous post like the hashing trick and per-coordinate adaptive learning rates  (in fact, the hashing trick was made popular by that library). A big advantage  of Vowpal Wabbit is that it is blazing fast. Not only because its underlying implementation is in C++, but also because it is using the L-BFGS optimization method. L-BGFS  stand for  “Limited-memory Broyden–Fletcher–Goldfarb–Shanno” and basically approximates the Broyden–Fletcher–Goldfarb–Shanno (BFGS) method using a limited amount of memory.  This method is much more complex to implement than Stochastic Gradient descent (which can be implemented in few lines of code as we saw in our previous post), but is supposedly converging faster (in less iterations). If you want to read more about L-BFGS and/or understand its difference with other optimisation methods, you can check this  (doc from Vowpal Wabbit) or this (nice blog post). Note that L-BFGS was empirically observed to be superior to SGD in many cases, in particular in deep learning settings (check out that paper on that topic).

Input format, Namespaces and more

Many times, i’ve heard people giving up on Vowpal Wabbit because of its input format, even after going quickly over its documentation . So let’s try to present it through a toy (yet real) example that will be used throughout this post to illustrate the main concepts of Vowpal Wabbit. On top of it, i’ll provide an helper tool (in next section) allowing to transform your tabular dataset into the VW input format easily.

So, the dataset we’ll use can be found here and represents the attempt of a bank trying to predict if a marketing phone call will end up in a bank term deposit by the customer, based on a bunch of signals like socio-economic factors of the customer like “does he have a loan?”, etc..

The traditional way to represent such datasets is to have a tsv or csv file, with the header being the name of the signals and each line representing the value of the training example on each signal. Each line of the training set has thus a fixed size, and missing values are just a blank cell or some specific value to indicate that it’s missing. Typically, for that dataset, the header looks like that:

age;job;marital;education;default;balance;housing;loan;contact;day;month;duration;campaign;pdays;previous;poutcome;y

With y being the actual supervision (i.e. did the call ended up in bank term deposit). And a typical training example looks like that:

58;management;married;tertiary;no;2143;yes;no;unknown;5;may;261;1;-1;0;unknown;no

In Vowpal Wabbit, there is no header, and each signal name is embedded in the training example itself. For example, the training example above can look like that in Vowpal Wabbit format:

-1 |i age:58 balance:2143 duration:261 campaign:1 pdays:-1 previous:0 |c job=management marital=married education=tertiary default=no housing=yes contact=unknown day=5 month=may poutcome=unknown

Let’s discuss multiple important things there:

• -1 says that this was a negative example.
• The |i and |c  are here to specify that the following features are part of a same feature namespace.  Being part of a namespace simply means that all the features in the namespace will be hashed together in a same feature space (this relates to the hashing trick, c.f. the previous post of that series).
• Here, i artificially created two namespaces: one for numerical features and another one for categorical ones. But that was just to illustrate the idea of namespace .
• In practice, namespaces can be used for different reasons (check the doc here) but one that is particularly useful  is that it allows you to do feature interactions:
• For instance, in the command line, using --quadratic ic would combine all the features of the namespaces i and c in our example above to create on the fly 2-way interacting features .  For instance the value of age and job together would be a new signal (maybe if you are a certain age in a certain profession, you’re more or less likely to do a bank term deposit).
• Note as well that for the numerical features, i used the colon ‘:‘ and for categorical ones i used ‘=‘ .
• Only the  ‘:‘ will be interpreted by Vowpal Wabbit. Both in training and when applying the model, the weight of the corresponding numerical feature (let’s say age) will be multiplied by the actual numerical value in the weighted linear product of the logistic hypothesis (more on that later).
• The  ‘=‘ is just cosmetic and for clarity. Technically, writing  married instead of marital=married makes absolutely no difference for the training, except if the value  married could show up in different contexts. E.g. if there were another signal childMarital indicating the marital status of customer’s children,  then you’d have to differentiate if the value married refers to the customer or his children, in which case the feature name would be necessary. Note that if you’d put two such features in different namespaces then they could not be mixed together and the prefix would be again not necessary.
• Note that for each signal, i’ve used the full name of the signal as a prefix (e.g. age or marital). First, we just saw that for categorical feature, this is not necessarily  required. For numerical signal though, it is (i.e. you cannot just throw a number without context). Now, for huge training sets, you don’t necessarily want to have a long string repeated millions (or more) of times. A good compromise is to have a mapping between signal names and very short string (like e.g. F1, F2, F3 ….). In the following section, i provide some code that allows to generate such training set with signal names mapping.
• There is a nice answer on Quora here exposing a short cheat-sheet  to remind those and how to encode boolean, categorical, ordinal+monotonic or numerical variables in VW.
• Last but not least, one thing i love about this format, is that it is very adapted to sparse data. Think that you have thousands of features or maybe just a list of words, then you don’t care about the order of the features or the missing values, you just  throw the features with the right prefix and/or in the right namespace and you’re done. VW will then hash them in their proper bucket in their proper hashing namespace.

How to transform your TSV/CSV datasets into VW format

Most often, classification or regression training sets are coming in the form of TSV or CSV files as mentioned previously.  Transforming them into the VW input format is not difficult, but it does require a minimum of attention. Indeed, depending on the training set, the target variable (or label) might be a word like “yes/no” or a number like “1/0” , while VW requires it to be -1/1 . Also, if a signal is numerical or categorical, in VW you need to transform it into different things (using e.g ‘:’ for numerical features, c.f remarks in previous section).

I wrote a very simple java (8) class that does this, find it here and feel free to use it.  You’ll just need to create (or edit an existing) method there to set up the characteristics of your data set. It doesn’t use any external library (other than pure java 8 libraries) so you if don’t have a Java IDE already installed, you can easily edit it from a text editor and compile/run it from command line.

Then you simply specify:

• the separator (e.g. ‘\t’ or ‘,’ or ‘;’)
• the name of the target variable (as it appears in the header)
• the value of the positive target variable in the dataset (e.g. ‘yes’ or ‘1’ or ‘click’)
• two separate lists: the list of names of numerical variables and the list of names of categorical variables (the names must be in the header as well).

All those are specified as parameters inside a side method that you just invoke in the main (which is then invoking the core method of that class calledtabular2VWGenerator). You have two examples of such side methods in the code:  generateBankTraningSet representing our dataset discussed above   and generateDonationTrainingSet representing another more complex dataset with a lot a sparse features (check the full list here).  You  invoke the appropriate method from the main.

The program then parses each line of the original training set, and, based on the list of numerical/categorical variables names, will generates two files:

• the corresponding appropriate training examples in VW format (which also take care of missing values, that are assumed to be empty string, even though you can change that in the code). Feature names from the header are transformed into short names: F1, F2, … This is to make the training set file weight lower (it does makes a difference for huge datasets)
• A small “.txt” file, mapping the short signal names with the original signal name from the header (e.g. F0 corresponds to age).

Important note: as in the example in previous section, the program is separating the numerical and categorical features into two namespaces (respectively named i and c ).  You can also decide to put all the features in the same namespaces (c.f. last parameter of the tabular2VWGenerator method). For our previous example, a training example will look like this:

-1 |i F0:58 F5:2143 F11:261 F12:1 F13:-1 F14:0 |c F1=management F2=married F3=tertiary F4=no F6=yes F8=unknown F9=5 F10=may F15=unknown

Note that the program can easily be enhanced to e.g. support as input multiple lists, each one would represent a namespace, and in the list, you could represent the feature type as a character, e.g. one of the list could look like {“age:n”, “balance:n”, “education:c”} and the program would parse this and know that age is numerical and education is categorical and encode them accordingly. Feel free to modify it!

The VW command line and its powerful options

Once you have your training set in the VW input format, you can start playing around with building some models from the command line. To illustrate it, we’ll take the small dataset we mentioned before about predicting bank term deposit. You can find here the training set in the VW input format and its short name signal mapping  (which were created using the tool described in previous section).

Let’s start by a first command to train a logistic regression model:

vw train.vw -f model.vw --loss_function logistic


It is pretty much self explained (-f is to specify the filename of the output mode and – –loss_function specifies which loss function to use, logistic in our case).

The output will show you some useful information on the progress of the training, along with the final obtained loss (average loss = 0.253874 in that case).

Then, to actually use the model on a separate test set (more later on how to easily create one), you simply do:

vw test.vw -t -i model.vw -p preds.txt --link logistic


The -t option specifies that you’re in test mode and VW will thus ignore the label of the training examples. -i specifies the model to use (typically the one that was created by the previous training command). --link logistic  says that the logistic regression is applied on top of the linear combinations. Without it, the file preds.txt will contains only the result of  \( \theta^Tx \) and not the sigmoid function applied on top of it.

Some options i found useful and interesting for the training part:

• • -c --passes N .  This specifies to do N passes on the training set while learning the optimal weights. In deep learning, the term epoch is often used instead of pass, and basically represents a full pass over the whole training set to update the weights. Doing several passes often leads to stronger models but the ideal number of passes can be tuned as an hyper parameter.  Note that the  -c option specifying to use caching is necessary when doing multiple passes because from the second pass, VW is using pre-compiled information that it prepared/cached during the first pass.
  • -b N  . The -b option allows you to control the number of bits in the hashing namespace (c.f part 2 of this series to understand what is the hashing trick ) and set it to \(2^N\) . The default value for N is 18, which might be more than ok (e.g. for the toy bank dataset) or not enough depending on the cardinality of your features values. If you need to encode  features having an high cardinality, i.e. a lot of different values like e.g. a product id in a catalog of millions of product, or, more frequently, if you need to create interactions of features (i.e. the cartesian product of two features values) which is also often leading to an high cardinality features, then you’ll probably need to increase N. Obviously the higher it is, the less collisions you’ll have in your namespace, but the more memory you’ll need.
  • --interactions arg . This is a very powerful one. Basically  arg is a list of letters, and each letter represents a namespace (assuming you organised your features around namespaces, like e.g. in our example in previous section). Applying that option means that it will automatically create interactions between all features in the corresponding namespaces. For instance, in our example above, adding e.g.  --interactions ic   will instantly create a whole bunch of new features in the model: all the interactions pairs between features in the namespace i and in the namespace c . Note that in this case the option is equivalent to --quadratic ic but the --interactions option is more general as it allows to create not only quadratic interactions but even more (triplets, quadruplets etc…). Such a feature somehow allows you to get closer to factorization machine models.

So here is an example of training command using those parameters:

vw train.vw -c --passes 4 -f model.vw --loss_function logistic --interactions ci -b 26

Using VW from a python Jupyter Notebook

A lot of ML engineers/data scientists nowadays (including myself) are using jupyter notebooks to explore/play with/compare various models interactively right from the notebook, thanks to the huge ML ecosystem we have in python (scikit-learn, keras, etc…) . While VW command line is nice, i still wanted to be able to play with it from a notebook, to easily control the train/test split, graph the results, switch between datasets, compare to other algos/libraries etc…

There are some python wrappers for VW (e.g. here) but they are either painful to install or slower. So i used a  less clean yet very practical solution: calling VW as an external command from the notebook and loading the results of the training via the output file. See a full example below. Feel free to run it in your own notebook, you’ll only need to specify the right path and have a training set in the VW format. Here again i used the banking training set in VW format (re-sharing the link here) that was generated by the tool i presented previously.

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Once you’re in the python ecosystem, you can feel at home, use any of the libraries you’re familiar with, e.g. calculating the AUC as we did above, or e.g plotting the ROC curve (as a continuity of previous notebook, see below). Bottom line: the sky (or maybe your python skills) is the limit :).

Note that an AUC of 0.91 on the test set is very respectable. Well, we used a rather simple dataset here, mainly to illustrate the concepts more easily, but i played with much less trivial datasets as well, having hundreds of sparse features and hundreds of gigabytes of data, and VW in most cases eats them for breakfast and gives very strong results.

Auditing the weights of your model

When you want to debug your model to check if something is wrong, VW proposes a very nice auditing option :  -a . It also allows to explore how VW is representing the core info of your model behind the scene. Let’s use that option with the following command:

vw -d train.vw -f model.vw --loss_function logistic --link=logistic -p probs.txt -a > weights_details.txt

Note that the -a option is not working if you also have the --interactions option in the same command. If you open the weights_details.txt file, a typical line will look like this:

-2.546350

       i^F11:137987:380:-0.000686432@20293.9   c^F15=unknown:86264:1:-0.241524@0.414363        Constant:116060:1:-0.241524@0.414363    i^F12:217054:1:-0.241524@0.414363       i^F13:200603:-1:0.241524@0.414363       c^F10=may:104323:1:-0.241524@0.414363   c^F9=5:218926:1:-0.241524@0.414363      c^F8=unknown:86079:1:-0.241524@0.414363 c^F6=yes:6939:1:-0.220727@0.39844       i^F0:48942:42:-0.00340787@1112.67       c^F3=tertiary:235513:1:-0.121834@0.26117        c^F1=entrepreneur:69649:1:-0.10903@0.03325   i^F5:165402:2:-5.23114e-05@1.19111e+06  c^F4=yes:211075:1:0@0   c^F2=divorced:209622:1:0@0

In the auditing section of that page, you have the details of each piece of this format, but   let’s analyze one piece of it together, e.g. c^F1=entrepreneur:69649:1:-0.10903@0.03325 :

• c^ means that the signal is part of the c namespace (this is the categorical namespace, c.f. previous section)
• F1=entrepreneur .  This is the actual feature value in the format we built, with F1 being the name of the feature (which corresponds to Job in our dataset, c.f. previous section)
• 69649 is the actual index in the namespace c , i.e. after applying the hash function on the string “F1=entrepreneur” . Note we didn’t use the -b option, thus the default size of each namespace is 2^18 , which is 262144 , and thus the weight of the feature F1=entrepreneur is stored in that namespace at index 69649 .
• 1 is the value of the feature. For a numerical feature it will be a number, but for categorical values (like here) it is 1 by default.
• -0.10903 is the actual weight of the feature
• 0.03325 is the is the sum of gradients squared for that feature. This is used for per coordinate adaptive learning rate (see part 2 of that series for the intuition behind it).

You now might ask, from where comes the number -2.546350 at the beginning of the line? This actually represents the linear sum of the weights for that example, i.e. \(\theta^Tx \) (c.f. part 1 of this series) . A bit tedious, but to convince yourself, you can observe the actual calculation from the above example :

380*-0.000686432 -0.241524 -0.241524 -0.241524 -1*0.241524 -0.241524 -0.241524 -0.241524 -0.220727 +42*-0.00340787 -0.121834 -0.10903 + 2*-0.0000523114

This gives the output -2.546 . Now, this is not the actual final prediction. To get it, you just need to pass it through the logistic function, i.e. \( \frac{1}{1+e^{-\theta^Tx}} \)  (again, c.f. part 1 of this series) , and you obtain \( \frac{1}{1+e^{- (-2.546350)}} = 0.072672 \) . You can find this number in the corresponding line in the probs.txt file (c.f. the -p option in the command line above). Btw, deciding if 0.072672 should end up in a “yes” or “no” prediction depends on the threshold you picked (the optimal threshold could be picked using the ROC curve above, c.f. this post i wrote some time ago for more details about the intuition behind this). 

Explore the weights of your (hashed) signals

One of the first thing i like to check after building a logistic regression model is the weights that each of the signals received. For a categorical feature, each value of the category is getting its own weight. Note that with the hashing trick, this corresponds to the weights stored in an entry of the hash space. But knowing that e.g. entry 3235 of the hash space got a weight of 0.34 is not very useful. What would be useful is to be able to map this hashed entry to an actual real feature value of your dataset. Happily, VW makes that easy for you, via another command line tool called vw-varinfo . Let’s use it on the dataset of the previous section (putting again the link of the VW version of it here). So you can run for instance this command line:

vw-varinfo --loss_function logistic --link=logistic train.vw > weights_details.txt

This will output a file  weights_details.txt for which the first lines look like that:

FeatureName HashVal MinVal MaxVal Weight RelScore
c^F15=success 182344 0.00 1.00 +1.3503 100.00%
c^F8=cellular 52869 0.00 1.00 +0.1913 14.16%
c^F6=no 182486 0.00 1.00 +0.1777 13.16%
c^F4=no 88500 0.00 1.00 +0.1759 13.03%

This represents the weights of each feature from the highest to the lowest. For instance, the feature that got the highest weight is c^F15=success with a weight of ~1.35 . F15 is the short name given by the dataset creator tool presented in a previous section above. To know to which feature it corresponds to, you can open the feature name mapping also created by the same tool  (see file featuresIndexes.txt in the zip file provided above). There you’ll see that F15 corresponds to the feature poutcome .  And as per the dataset description, poutcome corresponds to “outcome of the previous marketing campaign (categorical: ‘failure’,’nonexistent’,’success’)”.  So, that makes sense that it would get some high weight. The second one is c^F8=cellular . Using the same process you can see that F8 corresponds to contact , which is described as “contact communication type (categorical: ‘cellular’,’telephone’) “. Obviously, having the cellular phone number of the customer rather than his landline significantly increases the chances for the bank to contact him at all, so it make sense as well that such a feature would get an higher weight.

A very nice aspect of the vw-varinfo command is that it supports advanced options like e.g.  --interactions   . I.e you can run this for example:

vw-varinfo -c --passes 4 --interactions ci -b 26 --loss_function logistic --link=logistic train.vw > weights_details.txt

In this case, you’ll be able to observe the weights of feature interactions, e.g.  c^F9=28*i^F14 .

Btw, to be able to give an intuitive interpretation of the weights created by the model, check again part 1 of this series ;-).

Conclusion

By now, if you made it through all the posts of this series,  hopefully logistic regression don’t have much secrets to you anymore.

We’ve described the core theoretical foundation of the model and how to interprets the learned weights  (in part 1),  described  techniques that makes it work at scale in practice like the hashing trick and per coordinate learning rate and how it can be all implemented in 30 lines of code  (in part 2) and, in this post, how to use a very powerful general purpose machine learning library (Vowpal Wabbit) to build state of the art logistic regression models. We also introduced a simple helper tool to transform your standard tabular binary classification datasets into the Vowpal Wabbit format to be able to use this powerful librairy even more easily.

I hope you’re now convinced how simple yet powerful is logistic regression and thus why it is so important to master it as part of the standard set of tools of the modern data scientist/machine learning practitioner. See you in future posts!

In the first post of this series, we set the theoretical foundation of logistic regression. In this post we’ll explore the different techniques that make it work at scale and we’ll even go over a remarkably simple yet massively effective and scalable implementation (in python) connecting all the theoretical and practical concepts in only 30 lines of code.

Signals representation: the Hashing Trick

In the previous post, we presented a canonical way of representing  the signals (a.k.a. features) of a given training example   \(  (x^{(i)} , y_i)\) . We said that  \(x^{(i)} ∈ \{0, 1\}^d\) is a sparse binary feature vector in a d-dimensional space. How does it connect to reality?

Let’s say that you are trying to predict if a person will be willing to make a donation based on a list of previous donators (c.f. this dataset with which we’ll play in the next part of this series). You have a lot of signals around the previous donators like e.g. their location (State/Zipcode),  their title (Mr. , Dr., Professor, etc), their socio-economic status  and hundreds of others (c.f. the full list here).

Each signal has its own cardinality, e.g. the socio-economic status can take only three values (High/Average/Low), while the Zipcode can take up to ~43000 values (according to that page). The standard way to represent such categorical features for classification/regression problems is  called one-hot encoding.  For example, for the socio-economic status , you’d have a vector of three bits, one for each possible value. For “High” the encoding would give (1,0,0) , for “Average” (0,1,0) and for “Low” (0,0,1). For the  US state you’d have 50 bits, and for the title you’d  have 100 bits (their are 100 different titles in the fields description page).

For Zip Code? Hmm it could start to be overkilled/cumbersome to have a vector of ~43000 bits with only one 1 in it. What about a feature that is unbounded like e.g. an email? And when you connect all those “one-hot encoded” features together, you’re left with a pretty huge sparse binary vector of signals, potentially with much more entries (and thus weights to learn) than the number of training examples in your training set, which would be doomed to overfitting.

Meet the hashing trick.

The hashing trick was first introduced in that paper (by the creators of Vowpal Wabbit that we’ll cover extensively in our next blog post). Its principle is pretty simple: instead of having a huge sparse binary vector (of potentially unbounded size) for representing the signals, project each signal values into a much smaller fixed size binary vector. How? by hashing each signal values into that smaller vector. Let’s illustrate this on a simple toy example.

Suppose your training example has 3 features: Socio-economic level, US state, and  Title, and that we’re trying to encode  “social=High, State=WI (Wisconsin) and title=Mayor ”  . With one hot encoding, it would look like this:

With the hashing trick, let’s suppose we pick a fixed size of 50 for the underlying vector, the idea would be to hash each feature and apply a modulo of the size of the underlying vector (here 50), and send it to the corresponding cell of the vector. On our example, the encoding of the signals would look like this:

Few remarks around this example and the hashing trick in general:

• In our example, we reduced the size of the vector by about a factor of 3 (from 153 to 50), but in real life examples  it can go much beyond that and with much higher numbers (say from hundreds of of millions to tens of millions).
• This is mainly happening when you are doing feature interactions, i.e. when you take two features of very high cardinality and combine them together to have a new feature.
• Obviously, when reducing that much the feature space, collisions might (and will) happen (i.e. two different values hashing to the same bucket). Surprisingly, when the size of the feature space is still big enough, the impact of those collisions is negligible at best and acceptable at worst (some more details about it in that paper section 4.4, or in that very nice blog post).
• One could think there are obviously much smarter ways to reduce feature space or reducing/mitigating collisions, and there are, but most of them are involving keeping data structures and mappings in-memory, defeating the biggest added value of feature hashing which allows to process huge dataset with a tiny memory footprint (which is more or less the size of the hashed feature vector).
• You can do use some tricks to try minimizing collisions while still keeping the feature hashing trick, like e.g hashing features in different feature space (to e.g. make sure that two important values like for example two countries getting hashed to the same place and thus getting the same weight).
• Note that in Vowpal Wabbit, you can easily use one different hashed feature space (called namespace) for different subsets of the signals. Each namespace has its own hash index, and each signal belonging to it get hashed with a different hash function.
• Ideally, always treat the hashed feature space size as an hyper-parameter and tune/cross validate it by observing its impact on your loss.

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Per-coordinate Adaptive Learning Rate

In the first post of this series, we mentioned the notion of learning rate \( \alpha \) in the gradient descent step. This is controlling the rate at which we’re updating the weights of each parameter in the direction of the gradient. The lower is that rate, the slower will be the gradient descent (as it will only do baby steps in weights updates). The higher it is, the faster will be the gradient descent (as it will do big jumps in the direction of the gradient) but with a risk of missing the target and diverging.

One idea would be to adapt the learning rate along the way. Intuitively, we could have some high learning rate at the beginning of the gradient descend, and reduce it along the way while it is getting closer to its target (to not miss it). Even more than that, we could adapt the learning rate specifically for each specific parameter (weight) separately ! This is what per-parameter (or per-coordinate) adaptive learning rate is all about.

A lot of research was (and is still) done around that topic, specifically in deep learning, around different variation that would optimally adapt the learning rate per-coordinate during gradient descent (first popular proposed method is AdaGrad and some  more recent tunings like RMSProp and Adam, more on them here or here  ).

So, intuitively, the idea would be to make the learning rate smaller as a function of how much data was observed for the specific corresponding parameter. A simple example of such an idea (inspired from the code we’ll present in the next section and from that paper from google) would be to set the learning rate \( \alpha_c\)  of coordinate \(c\) :

$$ \alpha_c = \frac{\alpha}{\sqrt{n_c}+1} $$

where \( \alpha \) is a general learning rate and \(n_c\) is simply a count of the number of times the coordinate \( c\) was observed in the data.

Mapping that to our example above, let’s assume a global learning rate  \(\alpha = 0.1\) and that after 100k examples, the value “Title=Mayor” was observed only twice, while the value “Title=Mr” was obviously observed many more times (say 50k). Then the learning rate of the coordinate “Title=Mayor” will be \( 0.1/(\sqrt(2)+1) \approx 0.041 \) while the learning rate of coordinate “Title=Mr”  will be \( 0.1/(\sqrt(50000)+1) \approx0.00045 \)  .

observing a training example with the value “Title=Mayor” will obviously be much more rarely observed (than e.g. training examples with the value “Title=Mr”) and thus each time you observe it you want to do big jumps in the update of its weights (and for “Title=Mr”, it would be observed so often than the learning rate would decrease quickly). This is much more powerful than having a global learning rate for all learned parameters.

Connecting it all in 30 lines of python

Probably the most popular optimization method in machine learning is (Stochastic) Gradient Descent (SGD). But we won’t cover the intuition/details behind it, there are already (too?) many ressources to learn that 🙂 (if you’re interested by the basics, maybe start here and follow some links there for more details).

SGD basically boils down to a simple formula updating weights, so it is not surprising that its implementation is compact. But here i want to illustrate how beautifully simple yet powerful it is to connect all the dots and obtain an highly scalable implementation of Stochastic Gradient Descent for Logistic regression able to learn millions of parameters  using the hashing trick and per-coordinate adaptive learning rate with a tiny memory footprint, all that in a bunch of lines of code. To do so, i’ll use the very clear and elegant implementation by @tinrtgu that he shared as part of a kaggle competition. Here we expose  only the core code that is useful for our purpose (and not the code that read/writes the training set and produce a prediction file etc..). For the full version of his code (which is not much longer and include more comments ), check it here.

### This code is from https://kaggle2.blob.core.windows.net/forum-message-attachments/53646/1539/fast_solution.py
from math import exp, log, sqrt

D = 2 ** 20 # number of weights use for learning
alpha = .1 # learning rate for sgd optimization

def logloss(p, y):
    p = max(min(p, 1. - 10e-12), 10e-12)
    
    return -log(p) if y == 1. else -log(1. - p)

def get_x(csv_row, D):
    x = [0] # 0 is the index of the bias term
    for key, value in csv_row.items():
        index = int(value + key[1:], 16) % D # weakest hash ever ;)
        x.append(index)
   
    return x

def get_p(x, w):
    wTx = 0.
    for i in x: # do wTx
        wTx += w[i] * 1. # w[i] * x[i], but if i in x we got x[i] = 1.
    return 1. / (1. + exp(-max(min(wTx, 20.), -20.))) # bounded sigmoid

def update_w(w, n, x, p, y):
    for i in x:
        w[i] -= (p - y) * alpha / (sqrt(n[i]) + 1.)
        n[i] += 1.
    return w, n

Let’s go over it piece by piece.

D = 2 ** 20 # number of weights use for learning
alpha = .1 # learning rate for sgd optimization

The variable D is the size of the feature vector we discussed above. Each cell of that (very sparse) vector will be either 0 or 1 for each training example (c.f. example in the first section). Note that \( 2^{20}  \) is about 1 million, but the same code could easily work with say \( 2^{25}  \) and learn 33+ millions of parameters. The only limit is your memory size and the size of your training set (you need to have much more training examples than parameters to learn otherwise you’re doomed to overfit ). The variable alpha is the global learning rate that we mentioned  above.

def logloss(p, y):
    p = max(min(p, 1. - 10e-12), 10e-12)
    
    return -log(p) if y == 1. else -log(1. - p)

This is the log loss function (c.f. the beginning of Part 1  of this series) where p is the predicted value and y is the actual real value (in our case either 0 or 1) . Note the clipping of the predicted value p into the range \( [\epsilon , 1-\epsilon] \) with \( \epsilon = 10^{-12}\) . This is to avoid undefined values in case the prediction p would be either 0 or 1 (as it could lead to a loss of log(0) ).

def get_x(csv_row, D):
    x = [0] # 0 is the index of the bias term
    for key, value in csv_row.items():
        index = int(value + key[1:], 16) % D # weakest hash ever ;)
        x.append(index)
   
    return x

This function is the core of the hashing trick. Indeed, it transforms the original training example (variable csv_row) into its hashed version. The training example here is assumed to be a dictionary of key-value pairs. If we take our previous training example “social=High, State=WI and title=Mayor ” , then  we would have csv_row = {‘social’: ‘High’, ‘State’: ‘WI’, ‘title’: ‘Mayor’} . Each key-value pair is then hashed. The hash function used here is very weak as @tinrtgu writes in his comment and is subject to an high collision rate, but it is easy to replace it by some stronger (collisions resistant) hash functions like e.g. murmurhash  ( @tinrtgu  goal here  was to have a code not using even one external library).

Note that the function does not return a big vector of size D, but just the list of the indexes  of the 1s in the vector (in our example at most 3 if no collisions). The is an efficient way to represent an highly sparse vector full of million(s) of 0s and having only a bunch of 1s.

def get_p(x, w):
    wTx = 0.
    for i in x: # do wTx
        wTx += w[i] * 1. # w[i] * x[i], but if i in x we got x[i] = 1.
    return 1. / (1. + exp(-max(min(wTx, 20.), -20.))) # bounded sigmoid

Given a training example x (built with the previous function) and the current vector of weights w (of size D, see next function to see how it is updated), this function compute the prediction for this training example x.

If you remember from my previous post, the prediction in logistic regression is:

$$h_{\theta}(x) =  \frac{1}{1+e^{-\theta^Tx}} $$

\( \theta \) corresponds to the weight vector w and \( \theta^Tx \) is the linear combination calculated in the for loop. Note that it is of course not necessary to go compute the linear combination over the whole weights vector w but just for the weights of the corresponding features that are present in the training example. Note as well that in the feature hashing representation, the presence of a signal is simply represented by a 1 , and thus, the linear combination is nothing else than the sum of the weights (this was done to make the code simple to read, but technically, numerical features can also be represented and in which case it won’t a just a 0 or 1, but the actual numerical feature). In our running example (“social=High, State=WI and title=Mayor “) , the linear combination would simply correspond to the sum of the 3 weights associated with those 3 signal values.

The actual prediction then simply  corresponds to plug that linear combination into the sigmoid function (see above formula). Note that in the code, the linear combination is clipped into the range [-20,20] to avoid numerical overflows (in computing \( e^{-\theta^Tx} \) ) and because sigmoid(-20) and sigmoid(20) are respectively so close from 0 and 1 that it doesn’t have any impact anyway.

def update_w(w, n, x, p, y):
    for i in x:
        w[i] -= (p - y) * alpha / (sqrt(n[i]) + 1.)
        n[i] += 1.
    return w, n

Alright, now we reach the heart of the whole thing: the actual update of the weight vector w using the current training example x, the prediction on it (p) and the actual value y (the supervision in our training set, typically 0 or 1).

Again, if you remember from my previous post, the update rule for a given weight \( \theta_j  \) is:

$$ \theta_j = \theta_j \thinspace  – \alpha  \sum\limits_{i=1}^{N} ( h_{\theta}(x^{(i)}) – y_i )x_j^{(i)}  $$

But this rule is defined for the whole training set in one shot (N is the number of training examples). In SGD, the weights gets updated for a bunch of examples rather than for the whole training set, and specifically in the implementation here, they get updated one  training example at a time. In other words, given a training example x the update of weight  \( \theta_j  \) more corresponds to:

$$ \theta_j = \theta_j \thinspace  – \alpha \thinspace (h_{\theta}(x ) – y_i ) x_j   $$

Note that in our case, \(x_j  \)   is always 0 or 1 (depending if the corresponding signal is present or not in the training example), so when we update \(\theta_j \) , \(x_j  \)   will always be 1 and can thus be removed in the equation above.

Now, regarding the learning rate \( \alpha \) , we mentioned above that instead of a general global learning rate, we want a per-coordinate adaptive one, \( \alpha_c\)  , for the coordinate (weight/signal) \(c\) . But replacing \( \alpha \) by the formula we mentioned  previously, we obtain:

$$\theta_j = \theta_j  – \alpha  \frac{ h_{\theta}(x ) – y_i  }{\sqrt{n_{\theta_j}}+1} $$

In our case, \( n_{\theta_j} \) is simply the number of times that this weight was updated so far (equivalently how many time its corresponding signal was observed in the training set) and this number is maintained in a simple array (the n variable in the code) of size D, with n[i] being that counter.  And \( h_{\theta}(x ) \) is nothing else than the prediction (produced by the get_p function). So yes, the equation above is exactly the line of code performing the gradient update step:

w[i] -= (p - y) * alpha / (sqrt(n[i]) + 1.)

This line of code is obviously the most important one, where all the magic happens. Beautiful isn’t it? 😀

Note that this implementation is what is commonly called “out-of-core”, meaning that it does not require to load the whole training set in memory, making it thus massively scalable.

I hope that at this stage you have a deep understanding of the theoretical foundation and practical aspects around logistic regression.

In the next post (yet to be written 🙂 ), we’ll go over a very popular and highly scalable library implementing all the tricks we saw above and more: Vowpal Wabbit.  Stay tuned.

Logistic regression is arguably the most widely used machine learning algorithm in production systems when it comes to classify or predict the likelihood of some events to happen, often  in the context of modelling online users behaviour like e.g. the likelihood of a user clicking (a.k.a CTR estimation) or buying something (well, factorization machines are getting some serious momentum as well, to be discussed in some future posts). There is a reason for that: logistic regression is incredibly powerful, scalable, simple to implement and blazing fast to apply online once the model was trained offline.

In this post, we’ll deep dive into the theory behind logistic regression, giving the intuition behind its core concepts and its multiple faces across various fields of statistics and computer science. This will involve some maths, but nothing too deep assuming you have some notions of calculus and core statistics.

In the second part of this series, we’ll be much more concrete and deep dive into the implementation details of logistic regression, and go over some tricks like the hashing trick and the per coordinate adaptive learning rate which are making logistic regression works very well in practice on real (big) data sets. In that second post we’ll also go over a beautifully simple and elegant implementation of online logistic regression including all those tricks. In the third part of this series we’ll demonstrate the usage of a very powerful and popular library implementing logistic regression (and more) at scale: Vowpal Wabbit.

For now, let’s start with the theory 🙂

A classical derivation of logistic regression

We’ll start by introducing a standalone description of logistic regression, similar to what you can find in any classical introduction to machine learning course (e.g. that one to cite the most popular of them all).

So you have a training set of N examples  \(\{ (x^{(1)} , y_1) , …, (x^{(N)} , y_N) \}\) where \(x^{(i)} ∈ \{0, 1\}^d\) is a sparse binary feature vector in a d-dimensional space, a.k.a. the signals or features of the \(i\)th training example (more on that signals representation later, especially in part 2 of that series), and \(y_i \in \{0,1\}\) is the label associated to that example (which could represent a click/non click, spam/not spam, malignant/benign, …):

$$\left\{
\begin{array}{ll}
x_1^{(1)},…,x_d^{(1)} & y_1 \\
… \\
x_1^{(N)},…,x_d^{(N)} & y_N
\end{array}
\right. $$

To make a prediction for a given signal vector \(x ∈ \{0, 1\}^d\) ,  the logistic regression model proposes to take a linear combination \( \theta^Tx \)  where \(\theta\) is a vector of parameters (weights) \(\theta_1, … ,  \theta_n\) , and to project it into the \([0..1]\)  range by applying the logistic (or sigmoid) function directly to that linear product,  giving the following model representation:

$$h_{\theta}(x) = logistic( \theta^Tx ) = \frac{1}{1+e^{-\theta^Tx}} $$

The usual interpretation of \(h_{\theta}(x) \)  is that it represents the estimated probability that \(y=1\)  on input \(x\), in other words:  \(h_{\theta}(x) = p(y=1|x)\) .  Then, if you have to use that number to predict weather \(y = 1\) or \(y =0\), some threshold is picked , either simply 0.5 (i.e. predicting \(y = 1\) when  \(h_{\theta}(x) >= 0.5\) and 0 otherwise) or any other threshold empirically chosen using the classifier’s ROC curve (c.f. my older other post for more details on that).

Note that logistic regression is a linear classifier given that its decision boundary is a linear combination of the input. Indeed, if your threshold is e.g. 0.5, then you have \(y = 1\) when  \(h_{\theta}(x) >= 0.5\) . If you draw the sigmoid function \(g(z)\), you can see that it is >= 0.5 when \(z>=0\) . Thus \(h_{\theta}(x) >= 0.5\)   when \(\theta^Tx >= 0 \) , which is a linear decision boundary.

Let’s now talk about the cost function which is the most important part when building a model given that it is what need to be minimised on the training data to learn the optimal weight vector\( \theta\). Given the model representation, we cannot take a standard cost function based on MSE because it would make it non convex. All the power  of logistic regression is in its cost function which looks as follow:

$$
Cost(h_{\theta}(x),y) = \left\{
\begin{array}{ll}
-log(h_{\theta}(x)) & \textrm{if} \quad y =1 \\
-log(1-h_{\theta}(x)) & \textrm{if} \quad y =0
\end{array}
\right.
$$

The beauty behind that cost function is first that it is very intuitive, because when you predict 0 instead of 1 (or 1 instead of 0), then your cost tends to infinity (and thus you penalize the learning algorithm by a very large cost), but most importantly, this cost function is convex (check here for a proof), thus allowing to use any standard gradient descent based optimization algorithm .

Note that this function can be written \(-[y\thinspace log(h_{\theta}(x))  +(1-y) log(1-h_{\theta}(x))] \)  (just replace \(  y\)  by 0 or 1 to be convinced). We’ll denote \(Cost(\theta)\) the average cost on the whole training set\(\{ (x^{(1)} , y_1) , …, (x^{(N)} , y_N) \}\) , which is defined as:

$$Cost(\theta) = -\frac{1}{N}\sum\limits_{i=1}^{N} [y_i\thinspace log(h_{\theta}(x^{(i)}))  +(1-y_i) log(1-h_{\theta}(x^{(i)}))] $$

This is also sometimes called the logarithmic loss. You can define a multi-class version of it (when your output can take more than 2 values) , see e.g. here or here for some intuitive explanations.

So, bottom line, we need to find the optimal weight vector \( \theta \) by solving \(\underset{\theta}{min} \thinspace Cost(\theta) \) . To do so, gradient descent is the natural tool. We simply need to compute the partial derivative of \( Cost(\theta) \) according to each weight \( \theta_j \) of  \( \theta \),  i.e. \( \frac{\partial }{\partial \theta_j} Cost(\theta) \) . We won’t go into the details of the actual derivative calculation (you can find it e.g. here ) but just remember the notations:  the \(i\) training example \( x^{(i)} \)  is  a vector \((x_1^{(i)},…,x_d^{(i)}) \) , and  \(\theta^Tx^{(i)} =  \theta_0 + \theta_1 x_1^{(i)} + … + x_d^{(i)}\) and thus, for instance, \( \frac{\partial }{\partial \theta_j} \theta^Tx^{(i)} = x_j^{(i)}  \) . The result of the calculation of the partial derivative gives:

$$ \frac{\partial }{\partial \theta_j} Cost(\theta)  = \sum\limits_{i=1}^{N} ( h_{\theta}(x^{(i)}) – y_i )x_j^{(i)}  $$

This concludes all what is needed to solve \(\underset{\theta}{min} \thinspace Cost(\theta) \) to find  the optimal weight vector \( \theta \) from our training data. Indeed,  assuming some learning rate \( \alpha \) , we simply have to iterate enough times over updating all the weights \( \theta_j \) of \( \theta \)  using the gradient step below, until we observe that the cost is not reducing anymore :

$$ \theta_j = \theta_j \thinspace  – \alpha  \sum\limits_{i=1}^{N} ( h_{\theta}(x^{(i)}) – y_i )x_j^{(i)}  $$

How to interpret the learned weights?

At the end of your learning procedure via gradient descent as described above, you end with an “optimal”  weight vector \( \theta = (\theta_0,…, \theta_d) \) , with \( \theta_j \)   the weight associated with the input signal \( x_j \) . In a simple linear regression model, the interpretation for that weight would be that if the corresponding signal  \( x_j \) increases by one unit, then the predicted output increases by \( \theta_j \)  units. In logistic regression it cannot be really interpreted that way given that we’re dealing with the sigmoid function and probabilities.

To understand how to interpret the learned weights in logistic regression, we first need to define and understand the notion of odds ratio. Let’s say that the probability of some event to happen (e.g. a basketball team winning a game) is \(p=0.8 \) . The probability of them loosing is \(1-p = 0.2 \) . The odds ratio is simply defined as the ratio between probability of success  and probability of failure,  \( \frac{p}{1-p} \) i.e. 0.8 / 0.2 = 4 in our example. The interpretation is that the odds for the basketball team to win are 4 to 1.

How does that relate to logistic regression? To answer, you just need to know that the inverse function of the logistic function is the logit function . We thus have:

When it comes to glass you have to consider what tadalafil 10mg uk type you should use. Stay careful not to intake more than one published here viagra from canada dosage in 24 hours as this may create very unfavorable critical health conditions. The activity of spermatozoids of men-volunteers was analyzed before and after taking http://amerikabulteni.com/2011/10/19/cumhuriyetci-adaylarin-las-vegas-bulusmasi-secim-yarisinda-tansiyonu-yukseltti/ buy viagra italy. This amerikabulteni.com cialis online canada implies a remedy from the erectile dysfunction as well as those with an enlarged prostate. $$ logit(h_{\theta}(x)) =  logit(logistic(\theta^Tx)) = \theta^Tx $$

Let’s remind that  \(h_{\theta}(x) \) represents the probability of the outcome being 1 (given a signal vector  \(x \) ). Let’s denote that probability p. We thus have:

$$ logit(p) = \theta^Tx $$

Now the interesting part is that \(logit(p) = log(\frac{p}{1-p}) \) . Noticed \(\frac{p}{1-p}\)? Yep, that’s the odds ratio defined above 🙂 . In other words, logistic regression is a model relating the log odds probability of the outcome as a linear combination of the input signals:

$$log(\frac{p}{1-p}) = \theta_0 + \theta_1x_1 + … + \theta_dx_d  $$

We can now interpret the meaning of a weight \( \theta_j \) : if the signal \( x_j \) increases by one unit (or if it is present in case it is a boolean signal), then it increases by \( \theta_j \) the log odds of the outcome. Even more interpretable, if you take the exponent of both sides in the expression above you get:

$$\frac{p}{1-p} = e^{ \theta^T x} = \prod\limits_{j=0}^{d}e^{ \theta_j x_j}  $$

which gives a direct relation with the odds and thus an even more simple interpretation of the weight \( \theta_j \)  : the value  \( e^{\theta_j} \) directly gives you the increase in the odds of the outcome if the value signal \( x_j \) increases by one unit (or if it is present in case it is a boolean signal) . Example: if one of your signal is a boolean “already won NBA finals” for your predicting probability of a basketball team to win, and that it gets a weight of say \(1.2\) , the interpretation would be: if the team already won an NBA finals, then it increases its odds of winning by \( e^{1.2}  \approx 3.32 \)  , meaning an increase of 232% (i.e. \((3.32-1)*100\) ) in the odds of winning.

Bottom line: If a signal \( x_j \) ends up with  a weight \( \theta_j \) in logistic regression, it means that if the signal increases by one unit (or just if it is equal to 1 in case of boolean signal), then it increases the odds of the outcome to be 1 (e.g. a click happening) by \( (e^{\theta_j} -1)*100 \)%.

Log Loss vs. Cross Entropy vs. Negative Log Likelihood??

The concept behind logistic regression is so remarkable and efficient that it arose from  various different fields, including different branches of computer science and statistics, and often, you stumble upon different ways of deriving it, including various different names for the cost function or what needs to be maximised or minimised etc.., which might make the whole thing quiet confusing. For instance, in NLP, logistic regression (more precisely the multi-class version of it) is often called Maximum Entropy (or MaxEnt), first defined in that paper .  In this section, i’ll just recall the probabilistic view of logistic regression and connect the dots between cross-entropy, MLE, negative log likelihood, and logLoss .

First, entropy is a powerful concept invented by Claude Shanon who basically set the ground for information theory (if you want to get the gist of it from scratch, check this very nice vulgarization video). Cross-entropy is often used as a way to measure the difference between two probability vectors in the context of multinomial classification (a generalisation of the binary classification problem we’re interested in ), c.f. e.g that short video .  The “binary” version of cross entropy (i.e. its particular case when you have only two output classes like in our setting) is defined over the two vectors \( p = (y, 1-y)\) and \(q =  (\hat{y} , 1-\hat{y}) \) where \( y\) is the observed true value and \( \hat{y} \)  is the prediction:

$$  H(p,q) = -\sum_{i=1} p_i log q_i \\ = -ylog(\hat{y}) – (1-y)log(1-\hat{y})$$

This gives you a measure of “disorder” between the two vectors (the true one and the predicted one). In our case, \(\hat{y} = h_{\theta}(x) \) , so the average cross entropy on the whole training set is:

$$ -\frac{1}{N}\sum\limits_{i=1}^{N} [y_i\thinspace log(h_{\theta}(x^{(i)}))  +(1-y_i) log(1-h_{\theta}(x^{(i)}))]  $$

Wait, did you notice? This is exactly the log loss cost function we had in the first section!!

And there is more.

Let’s move to another very popular concept in machine learning called Maximum Likelihood Estimation (MLE) . MLE is a simple yet very powerful tool to estimate a (set of) parameter(s) based on observed data (if you have never heard about it and need an explanation “for dummies” then you can check this video for the high level idea and that one for a specific example). When you want to use MLE, the first step is to write down the probability of observing the data (in our case the \( y_i, …, y_N \) ) given the input signals \( x^{(1)}, …, x^{(N)} \) and the vector of parameters \( \theta \) :

$$ Pr(y_1, …, y_N  |x^{(1)}, …, x^{(N)}  , \theta) =  \prod\limits_{i=1}^{N}Pr(y_i| x^{(i)}, \theta)  $$

Given that in our case \( y_i \) is either 0 or 1, a common trick is to write that:

$$Pr(y_i| x^{(i)}, \theta)  = \\  Pr(y_i=1 |x^{(i)}, \theta)^{y_i} \thinspace Pr(y_i=0 |x^{(i)}, \theta)^{1-y_i} $$

The actual likelihood function always inverse the parameters in the notation to make clear that we are looking for an optimal \( \theta \) given the fixed observations of the training set:

$$ L(\theta , x^{(1)}, …, x^{(N)} | y_1, …, y_N ) = \\ \prod\limits_{i=1}^{N} Pr(y_i=1 |x^{(i)}, \theta)^{y_i} \thinspace Pr(y_i=0 |x^{(i)}, \theta)^{1-y_i}  $$

Note that the same form could have been obtained without the need for the previous trick by simply noticing that in the case of binary classification, the proper likelihood function is Bernoulli .   Now, we denote \(Pr(y_i=1 |x^{(i)}, \theta) \)  as \(h_{\theta}(x)\) (exact same notation as in the first section). We’ll also denote \( L(\theta)  \) the likelihood function for convenience. MLE thus suggest we find the \( \theta \)  maximizing that likelihood function (hence the name maximum likelihood), in other words:

$$ \underset{\theta}{\arg\max}   L(\theta) = \underset{\theta}{\arg\max} \prod\limits_{i=1}^{N}h_{\theta}(x^{(i)}) ^{y_i} \thinspace (1-h_{\theta}(x^{(i)}))^{1-y_i} $$

Since the next step is always to find a derivative of the likelihood, you almost always take the log of the likelihood  since it transforms the product into a sum (on which it is much easier to apply derivatives), and that the logarithm function is monotonic (strictly increasing), and thus maximizing the log likelihood is equivalent to maximizing the likelihood, as well as minimizing the negative log likelihood. So applying a log on the above product gives:

$$ \underset{\theta}{\arg\max}  \thinspace  log \thinspace  L(\theta) = \\ \underset{\theta}{\arg\max}   \sum\limits_{i=1}^{N}  y_i log(h_{\theta}(x^{(i)})) (1-y_i)log(1-h_{\theta}(x^{(i)}))   $$

Instead of looking for the maximum of the log likelihood, you can equivalently look for the minimum of the negative log likelihood. If you take the average negative log likelihood on the training set, what do you obtain? you guessed it, once again, the exact same log loss cost function we found both in the first section and also via cross entropy!!!

As a final link between logistic regression and other well known concepts in ML or statistics, logistic regression is often compared with Naive Bayes, see here (wikipedia), here (more detailed book chapter) and here (high level Quora answer). But the point is that naive bayes can be seen as a generative version of logistic regression (which is a discriminative model, here is a nice Quora discussion if you want to understand the difference between generative and discriminative models ).

Bottom line: in the context of logistic regression, when you’ll hear about log loss or cross entropy or negative log likelihood, you’ll now know why and how they are so closely related.

I hope you enjoyed  that post. If you want to get to the details allowing to make this work at scale and actually see an implementation connecting it all in 30 lines of python, continue to part 2 of this series 🙂 .