Machine Learning (Theory)

3/15/2007

Alternative Machine Learning Reductions Definitions

A type of prediction problem is specified by the type of samples produced by a data source (Example: X x {0,1}, X x [0,1], X x {1,2,3,4,5}, etc…) and a loss function (0/1 loss, squared error loss, cost sensitive losses, etc…). For simplicity, we’ll assume that all losses have a minimum of zero.

For this post, we can think of a learning reduction as

  1. A mapping R from samples of one type T (like multiclass classification) to another type T’ (like binary classification).
  2. A mapping Q from predictors for type T’ to predictors for type T.

The simplest sort of learning reduction is a “loss reduction”. The idea in a loss reduction is to prove a statement of the form:
Theorem For all base predictors b, for all distributions D over examples of type T:

E(x,y) ~ D LT(y,Q(b,x)) <= f(E(x’,y’)~R(D) LT’(y’,b(x’)))

Here LT is the loss for the type T problem and LT’ is the loss for the type T’ problem. Also, R(D) is the distribution over samples induced by first drawing from D and then mapping the sample via R. The function f() is the loss transform function—we try to find reductions R,Q which minimize it’s value.

If R,Q are deterministic, then there always exists a choice of D,b such that the loss rate on the right hand side is 0. However, it’s common to encounter real-world learning problems D which are inherently noisy, implying that the induced problem D’ is often inherently noisy. Distinguishing between errors due to environmental noise and errors due to base predictor mistakes seems important (and experimentally, it has been). Regret transform reductions can get at this. They have theorems of the form:
Theorem For all base predictors b, for all distributions D over examples of type T:

E(x,y) ~ D LT(y,Q(b,x)) – minc E(x,y) ~ D LT(y,c(x)) <= f(E(x’,y’)~R(D) LT’(y’,b(x’)) – minb’ E(x’,y’)~R(D) LT’(y’,b'(x’)))

The essential idea in regret transform reductions is that we subtract off the inherent noise in both the induced and original problem, and bound the excess loss due to suboptimal prediction directly.

The skeletons of the theory for these families of reductions have been layed out at this point. There remain some open problems, but another interesting direction to consider is other families of reductions. The hope is that by placing more stringent requirements on reductions, we limit ourselves to algorithms which tend to perform better in practice. This hope is pretty reasonable—empirically, we have observed a consistent step up in performance going from loss transform to regret transform reductions.

  1. Limited Regret Transform Reductions. The fact that the minimum is taken over all predictors in regret transforms is counterintuitive to some people, who are used to “Empirical Risk Minimization” statements where a minimum is taken over a limited set of predictors. We could imagine theorem statements of the form:
    Theorem For all sets of base predictors B, For all base predictors b, for all distributions D over examples of type T:
    E(x,y) ~ D LT(y,Q(b,x)) – minb’ in B E(x,y) ~ D LT(y,Q(b’,x)) <= f(E(x’,y’)~R(D) LT’(y’,b(x’)) – minb’ in B E(x’,y’)~R(D) LT’(y’,b'(x’)))

    This is a more general statement than a regret transform reduction—when B is the set of all base predictors, we recover standard regret transforms
    One case where it’s easy to see that this kind of statement holds is for the reduction from importance weighted binary classification to binary classification. However, little more is currently known.
  2. Reversible Reductions. This is an idea which Russell Impagliazzo first mentioned to me. Essentially, we limit ourselves to reductions with the property that they are reversible. Reversibility can be tested by mapping from one problem to another, and then back. There are a several variant theorem statements we could imagine. The most tractable variant for analysis might be the following:
    Theorem There exists R-1,Q-1 such that for all base predictors b, for base learning problems D’:
    E(x’,y’)~D’ LT’(y’,b(x’)) = E(x’,y’) ~ R(R-1(D’)) LT’(y’,b(x’))

    and Q-1(Q(b))=b

    Closely related (but different) is the following:
    Theorem There exists R-1,Q-1 such that for all type T predictors h, for all type T distributions D:
    E(x,y) ~ D LT(y,h(x)) = E(x,y)~R-1(R(D)) LT(y,h(x))

    and Q(Q-1(h)) = h
  3. Bayesian Reductions This is an idea which Simon Osindero mentioned. The basic observation is that Bayes Law is pretty important to the process of learning. We would like it to be the case that Bayes Law and reductions compose. A theorem statement of the following form might be about right.
    Theorem For some large family of priors P over distributions D of type T:
    Bayes(P,(x,y)~D~P) = Q(Bayes(R(P),(x’,y’)~D’~R(P)))

    Here “Bayes” is a learning algorithm which takes as input a prior P (or R(P)), and a sample (x,y) drawn by first drawing a D from P and then drawing from D (and similarly for the induced problem). Also, R(P) is the prior induced by mapping D to R(D) after drawing from P.

The two missing components for these kinds of reductions are:

  1. Theoretical evidence that we can satisfy these definitions of reduction between interesting types of learning problems.
  2. Empirical evidence that algorithmic modifications driven by the theory are useful.

My experience is that analyzing reductions has yielded significant insight into how to solve learning problems, so I would encourage anyone with a bit of theoretical inclination in Machine Learning to consider the above (or other) families of reductions.

3/3/2007

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.)

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