Machine Learning (Theory)


All Models of Learning have Flaws

Attempts to abstract and study machine learning are within some given framework or mathematical model. It turns out that all of these models are significantly flawed for the purpose of studying machine learning. I’ve created a table (below) outlining the major flaws in some common models of machine learning.

The point here is not simply “woe unto us”. There are several implications which seem important.

  1. The multitude of models is a point of continuing confusion. It is common for people to learn about machine learning within one framework which often becomes there “home framework” through which they attempt to filter all machine learning. (Have you met people who can only think in terms of kernels? Only via Bayes Law? Only via PAC Learning?) Explicitly understanding the existence of these other frameworks can help resolve the confusion. This is particularly important when reviewing and particularly important for students.
  2. Algorithms which conform to multiple approaches can have substantial value. “I don’t really understand it yet, because I only understand it one way”. Reinterpretation alone is not the goal—we want algorithmic guidance.
  3. We need to remain constantly open to new mathematical models of machine learning. It’s common to forget the flaws of the model that you are most familiar with in evaluating other models while the flaws of new models get exaggerated. The best way to avoid this is simply education.
  4. The value of theory alone is more limited than many theoreticians may be aware. Theories need to be tested to see if they correctly predict the underlying phenomena.

Here is a summary what is wrong with various frameworks for learning. To avoid being entirely negative, I added a column about what’s right as well.

Name Methodology What’s right What’s wrong
Bayesian Learning You specify a prior probability distribution over data-makers, P(datamaker) then use Bayes law to find a posterior P(datamaker|x). True Bayesians integrate over the posterior to make predictions while many simply use the world with largest posterior directly. Handles the small data limit. Very flexible. Interpolates to engineering.
  1. Information theoretically problematic. Explicitly specifying a reasonable prior is often hard.
  2. Computationally difficult problems are commonly encountered.
  3. Human intensive. Partly due to the difficulties above and partly because “first specify a prior” is built into framework this approach is not very automatable.
Graphical/generative Models Sometimes Bayesian and sometimes not. Data-makers are typically assumed to be IID samples of fixed or varying length data. Data-makers are represented graphically with conditional independencies encoded in the graph. For some graphs, fast algorithms for making (or approximately making) predictions exist. Relative to pure Bayesian systems, this approach is sometimes computationally tractable. More importantly, the graph language is natural, which aids prior elicitation.
  1. Often (still) fails to fix problems with the Bayesian approach.
  2. In real world applications, true conditional independence is rare, and results degrade rapidly with systematic misspecification of conditional independence.
Convex Loss Optimization Specify a loss function related to the world-imposed loss fucntion which is convex on some parametric predictive system. Optimize the parametric predictive system to find the global optima. Mathematically clean solutions where computational tractability is partly taken into account. Relatively automatable.
  1. The temptation to forget that the world imposes nonconvex loss functions is sometimes overwhelming, and the mismatch is always dangerous.
  2. Limited models. Although switching to a convex loss means that some optimizations become convex, optimization on representations which aren’t single layer linear combinations is often difficult.
Gradient Descent Specify an architecture with free parameters and use gradient descent with respect to data to tune the parameters. Relatively computationally tractable due to (a) modularity of gradient descent (b) directly optimizing the quantity you want to predict.
  1. Finicky. There are issues with paremeter initialization, step size, and representation. It helps a great deal to have accumulated experience using this sort of system and there is little theoretical guidance.
  2. Overfitting is a significant issue.
Kernel-based learning You chose a kernel K(x,x’) between datapoints that satisfies certain conditions, and then use it as a measure of similarity when learning. People often find the specification of a similarity function between objects a natural way to incorporate prior information for machine learning problems. Algorithms (like SVMs) for training are reasonably practical—O(n2) for instance. Specification of the kernel is not easy for some applications (this is another example of prior elicitation). O(n2) is not efficient enough when there is much data.
Boosting You create a learning algorithm that may be imperfect but which has some predictive edge, then apply it repeatedly in various ways to make a final predictor. A focus on getting something that works quickly is natural. This approach is relatively automated and (hence) easy to apply for beginners. The boosting framework tells you nothing about how to build that initial algorithm. The weak learning assumption becomes violated at some point in the iterative process.
Online Learning with Experts You make many base predictors and then a master algorithm automatically switches between the use of these predictors so as to minimize regret. This is an effective automated method to extract performance from a pool of predictors. Computational intractability can be a problem. This approach lives and dies on the effectiveness of the experts, but it provides little or no guidance in their construction.
Learning Reductions You solve complex machine learning problems by reducing them to well-studied base problems in a robust manner. The reductions approach can yield highly automated learning algorithms. The existence of an algorithm satisfying reduction guarantees is not sufficient to guarantee success. Reductions tell you little or nothing about the design of the base learning algorithm.
PAC Learning You assume that samples are drawn IID from an unknown distribution D. You think of learning as finding a near-best hypothesis amongst a given set of hypotheses in a computationally tractable manner. The focus on computation is pretty right-headed, because we are ultimately limited by what we can compute. There are not many substantial positive results, particularly when D is noisy. Data isn’t IID in practice anyways.
Statistical Learning Theory You assume that samples are drawn IID from an unknown distribution D. You think of learning as figuring out the number of samples required to distinguish a near-best hypothesis from a set of hypotheses. There are substantially more positive results than for PAC Learning, and there are a few examples of practical algorithms directly motivated by this analysis. The data is not IID. Ignorance of computational difficulties often results in difficulty of application. More importantly, the bounds are often loose (sometimes to the point of vacuousness).
Decision tree learning Learning is a process of cutting up the input space and assigning predictions to pieces of the space. Decision tree algorithms are well automated and can be quite fast. There are learning problems which can not be solved by decision trees, but which are solvable. It’s common to find that other approaches give you a bit more performance. A theoretical grounding for many choices in these algorithms is lacking.
Algorithmic complexity Learning is about finding a program which correctly predicts the outputs given the inputs. Any reasonable problem is learnable with a number of samples related to the description length of the program. The theory literally suggests solving halting problems to solve machine learning.
RL, MDP learning Learning is about finding and acting according to a near optimal policy in an unknown Markov Decision Process. We can learn and act with an amount of summed regret related to O(SA) where S is the number of states and A is the number of actions per state. Has anyone counted the number of states in real world problems? We can’t afford to wait that long. Discretizing the states creates a POMDP (see below). In the real world, we often have to deal with a POMDP anyways.
RL, POMDP learning Learning is about finding and acting according to a near optimaly policy in a Partially Observed Markov Decision Process In a sense, we’ve made no assumptions, so algorithms have wide applicability. All known algorithms scale badly with the number of hidden states.

This set is incomplete of course, but it forms a starting point for understanding what’s out there. (Please fill in the what/pro/con of anything I missed.)


Continuizing Solutions

This post is about a general technique for problem solving which I’ve never seen taught (in full generality), but which I’ve found very useful.

Many problems in computer science turn out to be discretely difficult. The best known version of such problems are NP-hard problems, but I mean ‘discretely difficult’ in a much more general way, which I only know how to capture by examples.

  1. ERM In empirical risk minimization, you choose a minimum error rate classifier from a set of classifiers. This is NP hard for common sets, but it can be much harder, depending on the set.
  2. Experts In the online learning with experts setting, you try to predict well so as to compete with a set of (adversarial) experts. Here the alternating quantifiers of you and an adversary playing out a game can yield a dynamic programming problem that grows exponentially.
  3. Policy Iteration The problem with policy iteration is that you learn a new policy with respect to an old policy, which implies that simply adopting the new policy can go very wrong.

For each of these problems, there are “continuized” solutions which can yield smaller computation, more elegant mathematics, or both.

  1. ERM By shifting from choosing a single classifier to choosing a stochastic classifier we can prove a new style of bound which is significantly tighter, easier to state, and easier to understand than traditional bounds in the traditional setting. This is the PAC-Bayes bound idea.
  2. Experts By giving the adversary slightly more power—the ability to split experts and have them fractionally predict one way vs. another, the optimal policy becomes much easier to compute (quadratic in the horizon, or maybe less). This is the continuous experts idea.
  3. Policy Iteration For policy iteration, by stochastically mixing the old and the new policy, we can find a new policy better than the old policy. This is the conservative policy iteration idea.

There is some danger to continuizing. The first and second examples both involve a setting shift, which may not be valid—in general your setting should reflect your real problem rather than the thing which is easy to solve. However, even with the setting shift, the solutions appear so compellingly more elegant that it is hard to not hope to use them in a solution to the original setting.

I have not seen a good formulation of the general approach of continuizing. Nevertheless, I expect to see continuizing in more places and to use it in the future. By making it explicit, perhaps this can be made eaesier.


Regularization = Robustness

The Gibbs-Jaynes theorem is a classical result that tells us that the highest entropy distribution (most uncertain, least committed, etc.) subject to expectation constraints on a set of features is an exponential family distribution with the features as sufficient statistics. In math,

argmax_p H(p)
s.t. E_p[f_i] = c_i

is given by e^{\sum \lambda_i f_i}/Z. (Z here is the necessary normalization constraint, and the lambdas are free parameters we set to meet the expectation constraints).

A great deal of statistical mechanics flows from this result, and it has proven very fruitful in learning as well. (Motivating work in models in text learning and Conditional Random Fields, for instance. ) The result has been demonstrated a number of ways. One of the most elegant is the “geometric” version here.

In the case when the expectation constraints come from data, this tells us that the maximum entropy distribution is exactly the maximum likelihood distribution in the exponential family. It’s a surprising connection and the duality it flows from appears in a wide variety of work. (For instance, Martin Wainwright’s approximate inference techniques rely (in essence) on this result.)

In practice, we know that Maximum Likelihood with a lot of features is bound to overfit. The traditional trick is to pull a sleight of hand in the derivation. We start with the primal entropy problem, move to the dual, and in the dual add a “prior” that penalizes the lambdas. (Typically an l_1 or l_2 penalty or constraint.) This game is played in a variety of papers, and it’s a sleight of hand because the penalties don’t come from the motivating problem (the primal) but rather get tacked on at the end. In short: it’s a hack.

So I realized a few months back, that the primal (entropy) problem that regularization relates to is remarkably natural. Basically, it tells us that regularization in the dual corresponds directly to uncertainty (mini-max) about the constraints in the primal. What we end up with is a distribution p that is robust in the sense that it maximizes the entropy subject to a large set of potential constraints. More recently, I realized that I’m not even close to having been the first to figure that out. Miroslav Dudík, Steven J. Phillips and Robert E. Schapire, have a paper that derives this relation and then goes a step further to show what performance guarantees the method provides. It’s a great paper and I hope you get a chance to check it out:

Performance guarantees for regularized maximum entropy density estimation.

(Even better: if you’re attending ICML this year, I believe you will see Rob Schapire talk about some of this and related material as an invited speaker.)

It turns out the idea generalizes quite a bit. In Robust design of biological experiments. P. Flaherty, M. I. Jordan and A. P. Arkin show a related result where regularization directly follows from a robustness or uncertainty guarantee. And if you want the whole, beautiful framework you’re in luck. Yasemin Altun and Alex Smola have a paper (that I haven’t yet finished, but at least begins very well) that generalizes the regularized maximum entropy duality to a whole class of statistical inference procedures. If you’re at COLT, you can check this out as well.

Unifying Divergence Minimization and Statistical Inference via Convex Duality

The deep, unifying result seems to be what the title of the post says: robustness = regularization. This viewpoint makes regularization seem like much less of a hack, and goes further in suggesting just what range of constants might be reasonable. The work is very relevant to learning, but the general idea goes beyond to various problems where we only approximately know constraints.


Bounds greater than 1

Nati Srebro and Shai Ben-David have a paper at COLT which, in the appendix, proves something very striking: several previous error bounds are always greater than 1.

Background One branch of learning theory focuses on theorems which

  1. Assume samples are drawn IID from an unknown distribution D.
  2. Fix a set of classifiers
  3. Find a high probability bound on the maximum true error rate (with respect to D) as a function of the empirical error rate on the training set.

Many of these bounds become extremely complex and hairy.

Current Everyone working on this subject wants “tighter bounds”, however there are different definitions of “tighter”. Some groups focus on “functional tightness” (getting the right functional dependency between the size of the training set and a parameterization of the hypothesis space) while others focus on “practical tightness” (finding bounds which work well on practical problems). (I am definitely in the second camp.)

One of the dangers of striving for “functional tightness” is that the bound can depend on strangely interrelated parameters. In fact, apparently these strange interrelations can become so complex that they end up always larger than 1 (some bounds here and here).

It seems we should ask the question: “Why are we doing the math?” If it is just done to get a paper accepted under review, perhaps this is unsatisfying. The real value of math comes when it guides us in designing learning algorithms. Math from bounds greater than 1 is a dangerously weak motivation for learning algorithm design.

There is a significant danger in taking this “oops” too strongly.

  1. There exist some reasonable arguments (not made here) for aiming at functional tightness.
  2. The value of the research a person does is more related to the best they have done than the worst.


Multiplication of Learned Probabilities is Dangerous

This is about a design flaw in several learning algorithms such as the Naive Bayes classifier and Hidden Markov Models. A number of people are aware of it, but it seems that not everyone is.

Several learning systems have the property that they estimate some conditional probabilities P(event | other events) either explicitly or implicitly. Then, at prediction time, these learned probabilities are multiplied together according to some formula to produce a final prediction. The Naive Bayes classifier for binary data is the simplest of these, so it seems like a good example.

When Naive Bayes is used, a set of probabilities of the form Pr'(feature i | label) are estimated via counting statistics and some prior. Predictions are made according to the label maximizing:

Pr'(label) * Productfeatures i Pr'(feature i | label)

(The Pr’ notation indicates these are estimated values.)

There is nothing wrong with this method as long as (a) the prior for the sample counts is very strong and (b) the prior (on the conditional independences and the sample counts) is “correct”—the actual problem is drawn from it. However, (a) seems to never be true and (b) is often not true.

At this point, we can think a bit from a estimation perspective. When trying to estimate a coin with bias Pr(feature i | label), after observing n IID samples, the estimate is accurate to (at most) c/m for some constant c. (Actually, it’s c/m0.5 in the general case c/m for coins with bias near 0 or 1.) Given this observation, we should expect the estimates Pr’ to differ by c/m or more when the prior on the sample counts is weak.

The problem to notice is that errors of c/m can quickly accumulate. The final product in the naive bayes classifier is n-way linear in the error terms where n is the number of features. If every features true value happens to be v and we happen to have a 1/2 + 1/n0.5 feature fraction estimate too large and 1/2 – 1/n0.5 fraction estimate too small (as might happen with a reasonable chance), the value of the product might be overestimated by:

(v – c/m)n/2 + n^0.5(v + c/m)n/2 + n^0.5 – vn

When c/m is very small, this approximates as c n0.5 /m, which suggests problems must arise when the number of features n is greater than the number of samples squared n > m2. This can actually happen in the text classification settings where Naive Bayes is often applied.

All of the above is under the assumption that the conditional independences encoded in the Naive Bayes classifier are correct for the problem. When these aren’t correct, as is often true in practice, the estimation errors can be systematic rather than stochastic implying much more brittle behavior.

In all of the above, note that we used Naive bayes as a simple example—this brittleness can be found in a number of other common prediction systems.

An important question is “What can you do about this brittleness?” There are several answers:

  1. Use a different system for prediction (there are many).
  2. Get much more serious about following Bayes law here. (a) The process of integrating over a posterior rather than taking the maximum likelihood element of a posterior tends to reduce the sampling effects. (b) Realize that the conditional independence assumptions producing the multiplication are probably excessively strong and design softer priors which better fit reasonable beliefs.
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