Machine Learning (Theory)

6/18/2005

Lower Bounds for Learning Reductions

Tags: Problems,Reductions jl@ 10:40 pm

Learning reductions transform a solver of one type of learning problem into a solver of another type of learning problem. When we analyze these for robustness we can make statement of the form “Reduction R has the property that regret r (or loss) on subproblems of type A implies regret at most f ( r ) on the original problem of type B“.

A lower bound for a learning reduction would have the form “for all reductions R, there exists a learning problem of type B and learning algorithm for problems of type A where regret r on induced problems implies at least regret f ( r ) for B“.

The pursuit of lower bounds is often questionable because, unlike upper bounds, they do not yield practical algorithms. Nevertheless, they may be helpful as a tool for thinking about what is learnable and how learnable it is. This has already come up here and here.

At the moment, there is no coherent theory of lower bounds for learning reductions, and we have little understanding of how feasible they are or which techniques may be useful in proving them. Here is a rough summary of what I know:

  1. For structured prediction, we have a partially worked out lower bound for all reductions using the structure to only carry single bits. A proof for reductions using the structure in others ways seems tricky at the moment.
  2. For Reinforcement learning it may (this is unclear) be possible to prove a lower bound showing that prediction ability alone can not solve RL well.
  3. There are various results which can be thought of as lower bounds for more limited families of reductions. One example is analyzing exactly how badly margin optimization can underperform for 0-1 loss when there is noise.

Overall, this is a moderately interesting direction of research which has not been much investigated.

6/17/2005

Reopening RL->Classification

In research, it’s often the case that solving a problem helps you realize that it wasn’t the right problem to solve. This is the case for the “reduce RL to classification” problem with the solution hinted at here and turned into a paper here.

The essential difficulty is that the method of stating and analyzing reductions ends up being nonalgorithmic (unlike previous reductions) unless you work with learning from teleoperated robots as Greg Grudic does. The difficulty here is due to the reduction being dependent on the optimal policy (which a human teleoperator might simulate, but which is otherwise unavailable).

So, this problem is “open” again with the caveat that this time we want a more algorithmic solution.

Whether or not this is feasible at all is still unclear and evidence in either direction would greatly interest me. A positive answer might have many practical implications in the long run.

5/21/2005

What is the right form of modularity in structured prediction?

Tags: Problems,Reductions,structured jl@ 10:57 pm

Suppose you are given a sequence of observations x1,…,xT from some space and wish to predict a sequence of labels y1,…,yT so as to minimize the Hamming loss: sumi=1 to T I(yi != c(x1,…,xT)i) where c(x1,…,xT)i is the ith predicted component. For simplicity, suppose each label yi is in {0,1}.

We can optimize the Hamming loss by simply optimizing the error rate in predicting each individual component yi independently since the loss is a linear combination of losses on each individual component i. From a learning reductions viewpoint, we can learn a different classifier for each individual component. An average error rate of e over these classifiers implies an expected Hamming loss of Te. This breakup into T different prediction problems is not the standard form of modularity in structured prediction.

A more typical form of modularity is to predict yi given xi, yi-1, yi+1 where the circularity (predicting given other predictions) is handled in various ways. This is often represented with a graphical model like so:

This form of modularity seems to be preferred for several reasons:

  1. Graphical models of this sort are a natural language for expressing what we know (or believe we know) about a problem in advance.
  2. There may be computational advantages to learning to predict from fewer features. (But note that handling the circularity is sometimes computationally difficult.)
  3. There may be sample complexity advantages to learning to predict from fewer features. This is particularly true for many common learning algorithms.

The difficulty with this approach is that “errors accumulate”. In particular, an average error rate of e for each of the predictors can easily imply a hamming loss of O(eT2). Matti Kaariainen convinced me this is not improvable for predictors of this form.

So, we have two forms of modularity. One is driven by the loss function while the other driven by simplicity of prediction descriptions. Each has advantages and disadvantages from a practical viewpoint. Can these different approaches be reconciled? Is there a compelling algorithm for solving structured prediction which incorporated both intuitions?

5/16/2005

Regret minimizing vs error limiting reductions

Tags: Problems,Reductions jl@ 2:21 pm

This post is about a reductions-related problem that I find mysterious. There are two kinds of reductions analysis currently under consideration.

  1. Error limiting reductions. Here, the goal is to bound the error rate of the created classifier in terms of the error rate of the binary classifiers that you reduce to. A very simple example of this is that error correcting output codes where it is possible to prove that for certain codes, the multiclass error rate is at most 4 * the binary classifier error rate.
  2. Regret minimizing reductions. Here, the goal is to bound the regret of the created classifier in terms of the regret of the binary classifiers reduced to. The regret is the error rate minus the minimum error rate. When the learning problem is noisy the minimum error rate may not be 0. An analagous result for reget is that for a probabilistic error correcting output code, multiclass regret is at most 4 * (binary regret)0.5.

The use of “regret” is more desirable than the use of error rates, because (for example) the ECOC error rate bound implies nothing when there is enough noise so that the binary classifiers always have error rate 0.25. However the square root dependence introduced when analyzing regret is not desirable. A basic question is: Can we have the best of both worlds? Can we find some algorithm doing multiclass classification with binary classifiers such that average regret r for the binary classifiers implies average regret bounded by 4r for the multiclass classifier?

If the answer is “yes”, that reduction algorithm may be empirically superior to the one we use now.
If the answer is “no”, that is a sharp and unexpected distinction between error rate analysis and regret analysis.

4/27/2005

DARPA project: LAGR

Tags: Funding,Problems,Robots jl@ 9:41 am

Larry Jackal has set up the LAGR (“Learning Applied to Ground Robotics”) project (and competition) which seems to be quite well designed. Features include:

  1. Many participants (8 going on 12?)
  2. Standardized hardware. In the DARPA grand challenge contestants entering with motorcycles are at a severe disadvantage to those entering with a Hummer. Similarly, contestants using more powerful sensors can gain huge advantages.
  3. Monthly contests, with full feedback (but since the hardware is standardized, only code is shipped). One of the premises of the program is that robust systems are desired. Monthly evaluations at different locations can help measure this and provide data.
  4. Attacks a known hard problem. (cross country driving)

3/18/2005

Binomial Weighting

Tags: Online,Papers,Problems jl@ 8:16 pm

Suppose we have a set of classifiers c making binary predictions from an input x and we see examples in an online fashion. In particular, we repeatedly see an unlabeled example x, make a prediction y’(possibly based on the classifiers c), and then see the correct label y.

When one of these classifiers is perfect, there is a great algorithm available: predict according to the majority vote over every classifier consistent with every previous example. This is called the Halving algorithm. It makes at most log2 |c| mistakes since on any mistake, at least half of the classifiers are eliminated.

Obviously, we can’t generally hope that the there exists a classifier which never errs. The Binomial Weighting algorithm is an elegant technique allowing a variant Halving algorithm to cope with errors by creating a set of virtual classifiers for every classifier which occasionally disagree with the original classifier. The Halving algorithm on this set of virtual classifiers satisfies a theorem of the form:

errors of binomial weighting algorithm less than minc f(number of errors of c, number of experts)

The Binomial weighting algorithm takes as a parameter the maximal minimal number of mistakes of a classifier. By introducing a “prior” over the number of mistakes, it can be made parameter free. Similarly, introducing a “prior” over the set of classifiers is easy and makes the algorithm sufficiently flexible for common use.

However, there is a problem. The minimal value of f() is 2 times the number of errors of any classifier, regardless of the number of classifiers. This is frustrating because a parameter-free learning algorithm taking an arbitrary “prior” and achieving good performance on an arbitrary (not even IID) set of examples is compelling for implementation and use, if we had a good technique for removing the factor of 2. How can we do that?

See the weighted majority algorithm for an example of a similar algorithm which can remove a factor of 2 using randomization and at the expense of introducing a parameter. There are known techniques for eliminating this parameter, but they appear not as tight (and therefore practically useful) as introducing a “prior” over the number of errors.

2/26/2005

Problem: Reductions and Relative Ranking Metrics

Tags: Problems,Reductions jl@ 11:34 am

This, again, is something of a research direction rather than a single problem.

There are several metrics people care about which depend upon the relative ranking of examples and there are sometimes good reasons to care about such metrics. Examples include AROC, “F1″, the proportion of the time that the top ranked element is in some class, the proportion of the top 10 examples in some class (google‘s problem), the lowest ranked example of some class, and the “sort distance” from a predicted ranking to a correct ranking. See here for an example of some of these.

Problem What does the ability to classify well imply about performance under these metrics?

Past Work

  1. Probabilistic classification under squared error can be solved with a classifier. A counterexample shows this does not imply a good AROC.
  2. Sample complexity bounds for AROC (and here).
  3. A paper on “Learning to Order Things“.

Difficulty Several of these may be easy. Some of them may be hard.

Impact Positive or negative results will broaden our understanding of the relationship between different learning goals. It might also yield new algorithms (via the reduction) for solving these problems.

2/25/2005

Problem: Online Learning

Tags: Online,Problems jl@ 8:27 am

Despite my best intentions, this is not a fully specified problem, but rather a research direction.

Competitive online learning is one of the more compelling pieces of learning theory because typical statements of the form “this algorithm will perform almost as well as a large set of other algorithms” rely only on fully-observable quantities, and are therefore applicable in many situations. Examples include Winnow, Weighted Majority, and Binomial Weighting. Algorithms with this property haven’t taken over the world yet. Here might be some reasons:

  1. Lack of caring. Many people working on learning theory don’t care about particular applications much. This means constants in the algorithm are not optimized, usable code is often not produced, and empirical studies aren’t done.
  2. Inefficiency. Viewed from the perspective of other learning algorithms, online learning is terribly inefficient. It requires that every hypothesis (called an expert in the online learning setting) be enumerated and tested on every example. (This is similar to the inefficiency of using Bayes law as an algorithm directly, and there are strong similarities in the algorithms.)

For an example of (1), there is a mysterious factor of 2 in the Binomial Weighting algorithm which has not been fully resolved. Some succesful applications also exist such as those based on SNoW.

The way to combat problem (2) is to introduce more structure into the hypothesis/experts. Some efforts have already been made in this direction. For example, it’s generally feasible to introduce an arbitrary bias or “prior” over the experts in the form of some probability distribution, and perform well with respect to that bias. Another nice piece of work by Adam Kalai and Santosh Vempala discusses how to efficiently handle several forms of structured experts. At an intuitive level, further development discovering how to efficiently work with new forms of structure might payoff well.

The basic research direction here is to address the gap between theory and practice, possibly by solving the above problems and possibly by discovering and addressing other problems.

2/23/2005

Problem: Reinforcement Learning with Classification

Tags: Problems,Reductions,Reinforcement jl@ 12:49 pm

At an intuitive level, the question here is “Can reinforcement learning be solved with classification?”

Problem Construct a reinforcement learning algorithm with near-optimal expected sum of rewards in the direct experience model given access to a classifier learning algorithm which has a small error rate or regret on all posed classification problems. The definition of “posed” here is slightly murky. I consider a problem “posed” if there is an algorithm for constructing labeled classification examples.

Past Work

  1. There exists a reduction of reinforcement learning to classification given a generative model. A generative model is an inherently stronger assumption than the direct experience model.
  2. Other work on learning reductions may be important.
  3. Several algorithms for solving reinforcement learning in the direct experience model exist. Most, such as E3, Factored-E3, and metric-E3 and Rmax require that the observation be the state. Recent work extends this approach to POMDPs.
  4. This problem is related to predictive state representations, because we are trying to solve reinforcement learning with prediction ability.

Difficulty It is not clear whether this problem is solvable or not. A proof that it is not solvable would be extremely interesting, and even partial success one way or another could be important.

Impact At the theoretical level, it would be very nice to know if the ability to generalize implies the ability to solve reinforcement learning because (in a general sense) all problems can be cast as reinforcement learning. Even if the solution is exponential in the horizon time it can only motivate relaxations of the core algorithm like RLgen.

2/21/2005

Problem: Cross Validation

Tags: Prediction Theory,Problems jl@ 10:47 am

The essential problem here is the large gap between experimental observation and theoretical understanding.

Method K-fold cross validation is a commonly used technique which takes a set of m examples and partitions them into K sets (“folds”) of size m/K. For each fold, a classifier is trained on the other folds and then test on the fold.

Problem Assume only independent samples. Derive a classifier from the K classifiers with a small bound on the true error rate.

Past Work (I’ll add more as I remember/learn.)

  1. Devroye, Rogers, and Wagner analyzed cross validation and found algorithm specific bounds. Not all of this is online, but here is one paper.
  2. Michael Kearns and Dana Ron analyzed cross validation and found that under additional stability assumptions the bound for the classifier which learns on all the data is not much worse than for a test set of size m/K.
  3. Avrim Blum, Adam Kalai, and myself analyzed cross validation and found that you can do at least as well as a test set of size m/K with no additional assumptions using the randomized classifier which draws uniformly from the set of size K.
  4. Yoshua Bengio and Yves Grandvalet analyzed cross validation and concluded that there was no unbiased estimator of variance.
  5. Matti Kääriäinen noted that you can safely derandomize a stochastic classifier (such as one that randomizes over the K folds) using unlabeled data without additional assumptions.

Some Extreme Cases to Sharpen Intuition

  1. Suppose on every fold the learned classifier is the same. Then, the cross-validation error should behave something like a test set of size m. This is radically superior to a test set of size m/K.
  2. Consider leave-one-out cross validation. Suppose we have a “learning” algorithm that uses the classification rule “always predict the parity of the labels on the training set”. Suppose the learning problem is defined by a distribution which picks y=1 with probability 0.5. Then, with probability 0.5, all leave-one-out errors will be 0 and otherwise 1 (like a single coin flip).

(some discussion is here)

Difficulty I consider this a hard problem. I’ve worked on it without success and it’s an obvious problem (due to the pervasive use of cross validation) that I suspect other people have considered. Analyzing the dependency structure of cross validation is quite difficult.

Impact On any individual problem, solving this might have only have a small impact due to slightly improved judgement of success. But, because cross validation is used extensively, the overall impact of a good solution might be very significant.

2/15/2005

ESPgame and image labeling

Tags: Problems,Vision jl@ 8:21 am

Luis von Ahn has been running the espgame for awhile now. The espgame provides a picture to two randomly paired people across the web, and asks them to agree on a label. It hasn’t managed to label the web yet, but it has produced a large dataset of (image, label) pairs. I organized the dataset so you could explore the implied bipartite graph (requires much bandwidth).

Relative to other image datasets, this one is quite large—67000 images, 358,000 labels (average of 5/image with variation from 1 to 19), and 22,000 unique labels (one every 3 images). The dataset is also very ‘natural’, consisting of images spidered from the internet. The multiple label characteristic is intriguing because ‘learning to learn’ and metalearning techniques may be applicable. The ‘natural’ quality means that this dataset varies greatly in difficulty from easy (predicting “red”) to hard (predicting “funny”) and potentially more rewarding to tackle.

The open problem here is, of course, to make an internet image labeling program. At a minimum this might be useful for blind people and image search. Solving this problem well seems likely to require new learning methods.

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