What are the advantages of different classification algorithms?

Here are some general guidelines I‘ve found over the years.

How large is your training set?

If your training set is small, high bias/low variance classifiers (e.g., Naive Bayes) have an advantage over low bias/high variance classifiers (e.g., kNN or logistic regression), since the latter will overfit. But low bias/high variance classifiers start to win out as your training set grows (they have lower asymptotic error), since high bias classifiers aren‘t powerful enough to provide accurate models. 

You can also think of this as a generative model vs. discriminative model distinction.

Advantages of some particular algorithms

Advantages of Naive Bayes: Super simple, you‘re just doing a bunch of counts. If the NB conditional independence assumption actually holds, a Naive Bayes classifier will converge quicker than discriminative models like logistic regression, so you need less training data. And even if the NB assumption doesn‘t hold, a NB classifier still often performs surprisingly well in practice. A good bet if you want to do some kind of semi-supervised learning, or want something embarrassingly simple that performs pretty well.

Advantages of Logistic Regression: Lots of ways to regularize your model, and you don‘t have to worry as much about your features being correlated, like you do in Naive Bayes. You also have a nice probabilistic interpretation, unlike decision trees or SVMs, and you can easily update your model to take in new data (using an online gradient descent method), again unlike decision trees or SVMs. Use it if you want a probabilistic framework (e.g., to easily adjust classification thresholds, to say when you‘re unsure, or to get confidence intervals) or if you expect to receive more training data in the future that you want to be able to quickly incorporate into your model.

Advantages of Decision Trees: Easy to interpret and explain (for some people -- I‘m not sure I fall into this camp). Non-parametric, so you don‘t have to worry about outliers or whether the data is linearly separable (e.g., decision trees easily take care of cases where you have class A at the low end of some feature x, class B in the mid-range of feature x, and A again at the high end). Their main disadvantage is that they easily overfit, but that‘s where ensemble methods like random forests (or boosted trees) come in. Plus, random forests are often the winner for lots of problems in classification (usually slightly ahead of SVMs, I believe), they‘re fast and scalable, and you don‘t have to worry about tuning a bunch of parameters like you do with SVMs, so they seem to be quite popular these days.

Advantages of SVMs: High accuracy, nice theoretical guarantees regarding overfitting, and with an appropriate kernel they can work well even if you‘re data isn‘t linearly separable in the base feature space. Especially popular in text classification problems where very high-dimensional spaces are the norm. Memory-intensive and kind of annoying to run and tune, though, so I think random forests are starting to steal the crown.

To go back to the particular question of logistic regression vs. decision trees (which I‘ll assume to be a question of logistic regression vs. random forests) and summarize a bit: both are fast and scalable, random forests tend to beat out logistic regression in terms of accuracy, but logistic regression can be updated online and gives you useful probabilities. And since you‘re at Square (not quite sure what an inference scientist is, other than the embodiment of fun) and possibly working on fraud detection: having probabilities associated to each classification might be useful if you want to quickly adjust thresholds to change false positive/false negative rates, and regardless of the algorithm you choose, if your classes are heavily imbalanced (as often happens with fraud), you should probably resample the classes or adjust your error metrics to make the classes more equal.

But...

Recall, though, that better data often beats better algorithms, and designing good features goes a long way. And if you have a huge dataset, your choice of classification algorithm might not really matter so much in terms of classification performance (so choose your algorithm based on speed or ease of use instead).
 
And if you really care about accuracy, you should definitely try a bunch of different classifiers and select the best one by cross-validation. Or, to take a lesson from the Netflix Prize and Middle Earth, just use an ensemble method to choose them all!

There are a number of dimensions you can look at to give you a sense of what will be a reasonable algorithm to start with, namely:

• Number of training examples
• Dimensionality of the feature space
• Do I expect the problem to be linearly separable?
• Are features independent?
• Are features expected to be in a linear scale?
• Is overfitting expected to be a problem?
• What are the system‘s requirement in terms of speed/performance/memory usage...?
• ...

This list may seem a bit daunting because there are many issues that are not straightforward to answer. The good news though is, that as many problems in life, you can address this question by following the Occam‘s Razor principle: use the least complicated algorithm that can address your needs and only go for something more complicated if strictly necessary. 

Logistic Regression

As a general rule of thumb, I would recommend to start with Logistic Regression. Logistic regression is a pretty well-behaved classification algorithm that can be trained as long as you expect your features to be roughly linear and the problem to be linearly separable. You can do some feature engineering to turn most non-linear features into linear pretty easily. It is also pretty robust to noise and you can avoid overfitting and even do feature selection by using l2 or l1 regularization. Logistic regression can also be used in Big Data scenarios since it is pretty efficient and can be distributed using, for example, ADMM (seelogreg). A final advantage of LR is that the output can be interpreted as a probability. This is something that comes as a nice side effect since you can use it, for example, for ranking instead of classification. 

Even in a case where you would not expect Logistic Regression to work 100%, do yourself a favor and run a simple l2-regularized LR to come up with a baseline before you go into using "fancier" approaches.

Ok, so now that you have set your baseline with Logistic Regression, what should be your next step. I would basically recommend two possible directions: (1) SVM‘s, or (2) Tree Ensembles. If I knew nothing about your problem, I would definitely go for (2), but I will start with describing why SVM‘s might be something worth considering. 

Support Vector Machines

Support Vector Machines (SVMs) use a different loss function (Hinge) from LR. They are also interpreted differently (maximum-margin). However, in practice, an SVM with a linear kernel is not very different from a Logistic Regression (If you are curious, you can see how Andrew Ng derives SVMs from Logistic Regression in his Coursera Machine Learning Course). The main reason you would want to use an SVM instead of a Logistic Regression is because your problem might not be linearly separable. In that case, you will have to use an SVM with a non linear kernel (e.g. RBF). The truth is that a Logistic Regression can also be used with a different kernel, but at that point you might be better off going for SVMs for practical reasons. Another related reason to use SVMs is if you are in a highly dimensional space. For example, SVMs have been reported to work better for text classification. 

Unfortunately, the major downside of SVMs is that they can be painfully inefficient to train. So, I would not recommend them for any problem where you have many training examples. I would actually go even further and say that I would not recommend SVMs for most "industry scale" applications. Anything beyond a toy/lab problem might be better approached with a different algorithm.

Tree Ensembles

This gets me to the third family of algorithms: Tree Ensembles. This basically covers two distinct algorithms: Random Forests and Gradient Boosted Trees. I will talk about the differences later, but for now let me treat them as one for the purpose of comparing them to Logistic Regression.

Tree Ensembles have different advantages over LR. One main advantage is that they do not expect linear features or even features that interact linearly. Something I did not mention in LR is that it can hardly handle categorical (binary) features. Tree Ensembles, because they are nothing more than a bunch of Decision Trees combined, can handle this very well. The other main advantage is that, because of how they are constructed (using bagging or boosting) these algorithms handle very well high dimensional spaces as well as large number of training examples.

As for the difference between Random Forests (RF) and Gradient Boosted Decision Trees (GBDT), I won‘t go into many details, but one easy way to understand it is that GBDTs will usually perform better, but they are harder to get right. More concretely, GBDTs have more hyper-parameters to tune and are also more prone to overfitting. RFs can almost work "out of the box" and that is one reason why they are very popular.

Deep Learning

Last but not least, this answer would not be complete without at least a minor reference to Deep Learning. I would definitely not recommend this approach as a general-purpose technique for classification. But, you might probably have heard how well these methods perform in some cases such as image classification. If you have gone through the previous steps and still feel you can squeeze something out of your problem, you might want to use a Deep Learning approach. The truth is that if you use an open source implementation such as Theano, you can get an idea of how some of these approaches perform in your dataset pretty quickly. 

Summary

So, recapping, start with something simple like Logistic Regression to set a baseline and only make it more complicated if you need to. At that point, tree ensembles, and in particular Random Forests since they are easy to tune, might be the right way to go. If you feel there is still room for improvement, try GBDT or get even fancier and go for Deep Learning.

You can also take a look at the Kaggle Competitions. If you search for the keyword "classification" and select those that are completed, you will get a good sense of what people used to win competitions that might be similar to your problem at hand. At that point you will probably realize that using an ensemble is always likely to make things better. The only problem with ensembles, of course, is that they require to maintain all the independent methods working in parallel. That might be your final step to get as fancy as it gets.

Machine Learning Cheat Sheet (for scikit-learn)

A few important criterions should be addressed: 

Does it require variables to be normally distributed?
Does it suffer multicollinearity issue? 
Dose it do as well with categorical variables as continuous variables?
Does it calculate CI without CV?
Does it conduct variables selection without stepwise?
Does it apply to sparse data?

Here is the comparison:

Logistic regression: no distribution requirement, perform well with few categories categorical variables, compute the logistic distribution, good for few categories variables, easy to interpret, compute CI, suffer multicollinearity

Decision Trees: no distribution requirement, heuristic, good for few categories variables, not suffer multicollinearity (by choosing one of them)

NB: generally no requirements,  good for few categories variables, compute the multiplication of independent distributions, suffer multicollinearity

LDA(Linear discriminant analysis not latent Dirichlet allocation):require normal, not good for few categories variables, compute the addition of Multivariate distribution, compute CI, suffer multicollinearity

SVM:  no distribution requirement, compute hinge loss, flexible selection of kernels for nonlinear correlation, not suffer multicollinearity, hard to interpret

Lasso: no distribution requirement, compute L1 loss, variable selection, suffer multicollinearity

Ridge:  no distribution requirement, compute L2 loss, no variable selection, not suffer multicollinearity

Bagging, boosting, ensemble methods(RF, Ada, etc): generally outperform single algorithm listed above.



Above all, Logistic regression is still the most widely used for its good features, but if the variables are normally distributed and the categorical variables all have 5+ categories, you may be surprised by the performance ofLDA, and if the correlations are mostly nonlinear, you can‘t beat a good SVM,
and if sparsity and multicollinearity are a concern, I would recommendAdaptive Lasso with Ridge(weights) + Lasso, this would suffice for most scenarios without much tuning.

And in the end if you need one fine tuned model, go for ensemble methods.

PS: Just see the sub question, with 10000 instances and more than 100000 features, the quick answer will be Lasso.

It wasn‘t mentioned in the question, but Random Forests should almost always be considered and at least tried as well. Lately, I‘ve seen SVMs and Logistic Regression picked over these by many of my acquaintances involved in machine learning (I‘m guessing this has a lot to do with Random Forests not being brought up in Andrew Ng‘s excellent Machine Learning class on Coursera), but to me this is a mistake.

Random Forests:
1. Almost always have lower classification error and better f-scores than decision trees.
2. Almost always perform as well as or better than SVMs, but are far easier for humans to understand.
3. Deal really well with uneven data sets that have missing variables.
4. Give you a really good idea of which features in your data set are the most important for free.
5. Generally train faster than SVMs (though this obviously depends on your implementation).

More info at Random forests - classification description

There was a European project called StatLog which "compare[d] and evaluated a range of classification techniques, with an assessment of their merits, disadvantages and range of application[s]." In 1994, they published a book about the algorithms they looked at and the results and it‘s now available for free here:

http://www.amsta.leeds.ac.uk/~ch...  

D. Michie, D.J. Spiegelhalter, C.C. Taylor (eds). Machine Learning, Neural and Statistical Classification

• Decision Trees are fast to train and easy to evaluate and interrupt.
• Support vector machine gives good accuracy, power of flexibility from kernels.
• Neural network are slow to converge and hard to set parameters but if done with care it work wells
• Bayesian classifiers  are easy to understand.   

I‘m currently working on these kinds of algorithms for newswire classification into about 10 categories. I‘m comparing kNN, tweaked Naive Bayes and Rocchio‘s algorithm. I wanted very simple algorithms since my dataset is quite unlimited and because SVM, for instance, seems a pain to implement myself. 

- kNN should be avoided in my case since the evaluation is quite heavy if your training dataset contains several thousand elements ; although it gives really good results.

- Naive Bayes is very simple and quicky to evaluate but I had to tweak it to handle unbalanced classes.

- Rocchio seems very naive but it works surprisingly well and is very efficient.

Finally, I use a combination of Naive Bayes and Rocchio to gain accuracy on the same principal than boosting (linear mixing obtained by cross-validation). You can also use EM on NB or Rocchio, since the formulation is very simple in these cases. This could help.

All in all, I‘d say that this is very data-dependent. Best technics depends on the data and what accuracy/efficiency trade off you are expecting.

Decision trees and rule based algorithms are good because you can understand the model that was built for classifying, unlike with neural networks.
Support Vector Machines work very well in many circumstances and perform very good with large amounts of data.
Association ones such as Apriori have an excellent performance, due to how the algorithm is build, and it always reaches the proper solution.
Naive Bayes mechanism is very simple to understand, it has also a high performance and is also easy to implement

The 2 generally accepted high performance methods are SVMs , Random Forests, and Boosted Trees.

see these older references
An Empirical Comparison of Supervised Learning Algorithms
Page on cornell.edu

An Empirical Comparison of Supervised Learning Algorithms Using Different Performance Metrics
Page on niculescu-mizil.org

and the video lecture
Which Supervised Learning Method Works Best for What? An Empirical Comparison of Learning Methods and Metrics

   Advanced methods like Deep Learning are now available but probably pretty hard to use for a novice.  An exception would be SVM based Factorization Machines (see below) 

Many modern methods can be formulated as regularization problems solving a  convex optimizations with some loss function.  So Logistic Regression, SVM, L1-SVM, Boosting, etc, can be viewed as variations of the same method and, these days, are usually implemented with a common solver.

I would hesitate to distinguish between logistic regression and an SVM as different methods for the purposes of classification. Many packages, like LibSVM, provide a variety of methods and can be applied to the same data set for comparing.

MultiClass or MultiLabel:
SVMs are very good at basic and multiclass classification, and Random Forests are probably better for multilabel classification.  

There are specific SVM implementations for Multiclass (Cramer & Singer algo) and Structural (SvmLight) problems, and even MultiLabel SVMs (M3L).  These are a bit esoteric but it is important to be able to solve more than just binary classification problems.  

Still, it is non-trivial to modify an SVM for multilabel problems whereas Random Forests are easily adapted to this.

see:
Multi-Label Learning with Millions of Labels: Recommending Advertiser Bid Phrases for Web Pages
Page on microsoft.com

Instance Weights:
Sometimes we may have more information than just labels; we may have weights on instances.  Here, there is an open source  implementation of LibLinear that provides instance weights.  

If you are using Scikit learn, one can use instance (sample) weights with Random Forests (but not the SVM)


Incomplete Information:
Machine Learning methods do very poorly when we don‘t have enough labels for our data, or the labels are wrong.

SVMs can even be applied in situations when the labels are only partially or weakly known, but we have additional information about the global statistics

Convex Relaxations of Transductive Learning

And this works particularly well for text classification when choosing an extended basis set, such as using a word2vec or glove.

Kernels--and your nature features:
Using an SVM or any Kernel Method requires choosing a regularizer, and maybe a Kernel.  Generally you should not use an RBF Kernel unless the underlying problem is like a signal processing problem:

Kernels Part 1: What is an RBF Kernel?  Really?

That is, you are looking for something like a smooth solution

A recent extension to SVMs Kernels is Factorization Machines (FMs), and while this method are typically thought of as recommender system, it can be used for SVM like classification as well.

FMs would be useful for picking up very weak correlations between sparse, discrete features

Regularization:
Most modern methods include some basic regularizers like L1, L2, or some combination of the 2 (elastic net)

Why would one choose, say, an L1 SVM over an L2 SVM?  The L1 SVM pre-supposed the features are very, very sparse.  For example, I have used an L1 SVM to reduce a model with 50K features to a few hundred.  L2 SVM can capture models that require many tiny features


Numerical performance:
SVMs, like many convex methods, have been optimized for the past 15 years and these days scale very well...I routinely use linear SVMs  to solve classifications problems with tens of millions of instances and say half a million features.  I can do this on my laptop.   I am bit puzzled why others would claim these are hard to train (unless they are trying to use a non-linear / RBF kernel and have a poor implementation)  And in my experience in using them over 15 years in production, they are very easy train at very large scale for most commercial applications.  They are convex methods with only a 1-2 adjustable parameters.  

I have routinely trained linear SVMs with nearly 10M instances and 1M features--on my laptop in near real time.  

For more than say 10M instances and 1M features,  the training can be done in parallel using lock free stochastic coordinate descent, or in online mode using.  Modern open source implementations include libsvm/liblinear, Graphlab/Dato, and Vowpal Wabbit.  There are even GPU accelerated versions like BidMach

Page on gputechconf.com

 and there are distributed memory, parallel implementations running on upto 1000 nodes.

Interpretation:
Linear SVMs are very easy to interpret--it is the non-linear case that is a bit tricky.  Although any non-linear problem is going to require some work to interpret in that you would need to find a compact basis set to represent the non-linearities.  Heck, If you know the basis set a-priori, you can just project your data onto this basis and run a linear SVM and the problem is trivial.

Having tried linear/poly regression, logistic regression, neural nets, genetic programming, and SVM on a recent large/noisy data project I agree with most of the above.  I‘d add GP (Genetic Programming) to the mix when you can reduce the data via a set of (possibly parameterized) feature detectors, with some simple combination rules.   GP is often slow to converge, and usually hard to understand the results (unless you have some parsimony built-in) - but gives more insight than SVM and Neural Nets.

1. Whats the business problem ? 

what you need to do, transparency and governance, regulatory needs, business interpretation issues and others will outweigh maths in a business.

2. Data determines the result more than the algorithm

if you have lots of good data, all algorithms are good. Multicollinearity is less of a problem with large datasets.