Machine Learning (Theory)

10/8/2006

Incompatibilities between classical confidence intervals and learning.

Classical confidence intervals satisfy a theorem of the form: For some data sources D,

PrS ~ D(f(D) > g(S)) > 1-d

where f is some function of the distribution (such as the mean) and g is some function of the observed sample S. The constraints on D can vary between “Independent and identically distributed (IID) samples from a gaussian with an unknown mean” to “IID samples from an arbitrary distribution D“. There are even some confidence intervals which do not require IID samples.

Classical confidence intervals often confuse people. They do not say “with high probability, for my observed sample, the bounds holds”. Instead, they tell you that if you reason according to the confidence interval in the future (and the constraints on D are satisfied), then you are not often wrong. Restated, they tell you something about what a safe procedure is in a stochastic world where d is the safety parameter.

There are a number of results in theoretical machine learning which use confidence intervals. For example,

  1. The E3 algorithm uses confidence intervals to learn a near optimal policy for any MDP with high probability.
  2. Set Covering Machines minimize a confidence interval upper bound on the true error rate of a learned classifier.
  3. The A2 uses confidence intervals to safely deal with arbitrary noise while taking advantage of active learning.

Suppose that we want to generalize thse algorithms in a reductive style. The goal would be to train a regressor to predict the output of g(S) for new situations. For example, a good regression prediction of g(S) might allow E3 to be applied to much larger state spaces. Unfortunately, this approach seems to fail badly due to a mismatch between the semantics of learning and the semantics of a classical confidence interval.

  1. It’s difficult to imagine a constructive sampling mechanism. In a large state space, we may never encounter the same state twice, so we can not form meaningful examples of the form “for this state-action, the correct confidence interval is y“.
  2. When we think of succesful learning, we typically think of it in an l1 sense—the expected error rate over the data generating distribution is small. Confidence intervals have a much stronger meaning as we would like to apply them: with high probability, in all applications, the confidence interval holds. This mismatch appears unaddressable.

It is tempting to start plugging in other notions such as Bayesian confidence intervals or quantile regression systems. Making these approaches work at a theoretical level on even simple systems is an open problem, but there is plenty of motivation to do so.

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