## COLT/ICML Open Questions and ICML Instructions

Sasha is the open problems chair for both COLT and ICML. Open problems will be presented in a joint session in the evening of the COLT/ICML overlap day. COLT has a history of open sessions, but this is new for ICML. If you have a difficult theoretically definable problem in machine learning, consider submitting it for review, due March 16. You’ll benefit three ways:

1. The effort of writing down a precise formulation of what you want often helps you understand the nature of the problem.
2. Your problem will be officially published and citable.
3. You might have it solved by some very intelligent bored people.

The general idea could easily be applied to any problem which can be crisply stated with an easily verifiable solution, and we may consider expanding this in later years, but for this year all problems need to be of a theoretical variety.

Joelle and I (and Mahdi, and Laurent) finished an initial assignment of Program Committee and Area Chairs to papers. We’ll be updating instructions for the PC and ACs as we field questions. Feel free to comment here on things of plausible general interest, but email us directly with specific concerns.

## The Efficient Robust Conditional Probability Estimation Problem

I’m offering a reward of \$1000 for a solution to this problem. This joins the cross validation problem which I’m offering a \$500 reward for. I believe both of these problems are hard but plausibly solvable, and plausibly with a solution of substantial practical value. While it’s unlikely these rewards are worth your time on an hourly wage basis, the recognition for solving them definitely should be 🙂

## The Problem

The problem is finding a general, robust, and efficient mechanism for estimating a conditional probability P(y|x) where robustness and efficiency are measured using techniques from learning reductions.

In particular, suppose we have access to a binary regression oracle B which has two interfaces—one for specifying training information and one for testing. Training information is specified as B(x’,y’) where x’ is a feature vector and y’ is a scalar in [0,1] with no value returned. Testing is done according to B(x’) with a value in [0,1] returned.

A learning reduction consists of two algorithms R and R-1 which transform examples from the original input problem into examples for the oracle and then transform the oracle’s predictions into a prediction for the original problem.

The algorithm R takes as input a single example (x,y) where x is an feature vector and y is a discrete variable taking values in {1,…,k}. R then specifies a training example (x’,y’) for the oracle B. R can then create another training example for B based on all available information. This process repeats some finite number of times before halting without returning information.

A basic observation is that for any oracle algorithm, a distribution D(x,y) over multiclass examples and a reduction R induces a distribution over a sequence (x’,y’)* of oracle examples. We collapse this into a distribution D'(x’,y’) over oracle examples by drawing uniformly from the sequence.

The algorithm R-1 takes as input a single example (x,y) and returns a value in [0,1] after using (only) the testing interface of B zero or more times.

We measure the power of an oracle and a reduction according to squared-loss regret. In particular we have:

reg(D,R-1)=E(x,y)~ D[(R-1(x,y)-D(y|x))2]

and similarly letting mx’=E(x’,y’)~ D’[y’].

reg(D’,B)=E(x’,y’)~ D’(B(x’) – mx’)2

The open problem is to specify R and R-1 satisfying the following theorem:

For all multiclass distributions D(x,y), for all binary oracles B: The computational complexity of R and R-1 are O(log k)
and

reg(D,R-1) < = C reg(D’,B)

where C is a universal constant.

Alternatively, this open problem is satisfied by proving there exists no deterministic algorithms R,R-1 satisfying the above theorem statement.

## Motivation

The problem of conditional probability estimation is endemic to machine learning applications. In fact, in some branches of machine learning, this is simply considered “the problem”. Typically conditional probability estimation is done in situations where the conditional probability of only one bit is required, however there are a growing number of applications where a well-estimated conditional probability over a more complex object is required. For example, all known methods for solving general contextual bandit problems require knowledge of or good estimation of P(a | x) where a is an action.

There is a second intrinsic motivation which is matching the lower bound. No method faster than O(log k) can be imagined because the label y requires log2 k bits to specify and hence read. Similarly it’s easy to prove no learning reduction can provide a regret ratio with C<1.

The motivation for using the learning reduction framework to specify this problem is a combination of generality and the empirical effectiveness in application of learning reductions. Any solution to this will be general because any oracle B can be plugged in, even ones which use many strange kinds of prior information, features, and active multitask hierachical (insert your favorite adjective here) structure.

## Related Results

The state of the art is summarized here which shows it’s possible to have a learning reduction satisfying the above theorem with either:

1. C replaced by (log2 k)2 (using a binary tree structure)
2. or the computational time increased to O(k) (using an error correcting code structure).

Hence, answering this open problem in the negative shows that there is an inherent computation vs. robustness tradeoff.

There are two other closely related problems, where similar analysis can be done.

1. For multiclass classification, where the goal is predicting the most likely class, a result analogous to the open problem is provable using error correcting tournaments.
2. For multiclass classification in a partial label setting, no learning reduction can provide a constant regret guarantee.

## Silly tricks that don’t work

Because Learning reductions are not familiar to everyone, It’s helpful to note certain tricks which do not work here to prevent false leads and provide some intuition.

This doesn’t work, because the quantification is for all D. Any specified learning algorithm will have some D on which it has nonzero regret. On the other hand, because R calls the oracle at least once, there is a defined induced distribution D’. Since the theorem must hold for all D and B, it must hold for a D your specified learning algorithm fails on and for a B for which reg(D’,B)=0 implying the theorem is not satisfied.

Feed random examples into B and vacuously satisfy the theorem by making sure that the right hand side is larger than a constant.

This doesn’t work because the theorem is stated in terms of squared loss regret rather than squared loss. In particular, if the oracle is given examples of the form (x’,y’) where y’ is uniformly at random either 0 or 1, any oracle specifying B(x’)=0.5 has zero regret.

Feed pseudorandom examples into B and vacuously satisfy the theorem by making sure that the right hand side is larger than a constant.

This doesn’t work, because the quantification is “for all binary oracles B”, and there exists one which, knowing the pseudorandom seed, can achieve zero loss (and hence zero regret).

Just use Boosting to drive the LHS to zero.

Boosting theorems require a stronger oracle—one which provides an edge over some constant baseline for each invocation. The oracle here is not limited in this fashion since it could completely err for a small fraction of invocations.

Take an existing structure, parameterize it, randomize over the parameterization, and then average over the random elements.

Employing this approach is not straightforward, because the average in D’ is over an increased number of oracle examples. Hence, at a fixed expected (over oracle examples) regret, the number of examples allowed to have a large regret is increased.

## Static vs. Dynamic multiclass prediction

I have had interesting discussions about distinction between static vs. dynamic classes with Kishore and Hal.

The distinction arises in multiclass prediction settings. A static set of classes is given by a set of labels {1,…,k} and the goal is generally to choose the most likely label given features. The static approach is the one that we typically analyze and think about in machine learning.

The dynamic setting is one that is often used in practice. The basic idea is that the number of classes is not fixed, varying on a per example basis. These different classes are generally defined by a choice of features.

The distinction between these two settings as far as theory goes, appears to be very substantial. For example, in the static setting, in learning reductions land, we have techniques now for robust O(log(k)) time prediction in many multiclass setting variants. In the dynamic setting, the best techniques known are O(k), and furthermore this exponential gap may be essential, at least without further assumptions.

Are there techniques for converting from dynamic multiclass to static multiclass? For example, we could embed a dynamic set of classes within a much larger static set ranging over all possible dynamic classes while eliminating all class-dependent features. In some cases, this approach may work well, but I’ve also seen it fail, with the basic problem being that a learning algorithm might easily choose an invalid class. We could of course force a learning algorithm to choose amongst the dynamically valid set, but I don’t know a general way to do that without making the running time at least scale with the number of valid classes.

So, a basic question that’s bothering me is: When and how can we effectively predict amongst a set of dynamically defined classes in sublinear time? A quick answer is “it’s not possible because simply reading off the set of dynamically defined classes require O(class count) time”. This answer isn’t satisfying, because there are many ways to implicitly specify a set in sublinear time. So the modified question is “Are there natural ways to dynamically define classes in sublinear time? And can these be used for sublinear prediction?”

Centmail is a scheme which makes charity donations have a secondary value, as a stamp for email. When discussed on newscientist, slashdot, and others, some of the comments make the academic review process appear thoughtful :). Some prominent fallacies are:

1. Costing money fallacy. Some commenters appear to believe the system charges money per email. Instead, the basic idea is that users get an extra benefit from donations to a charity and participation is strictly voluntary. The solution to this fallacy is simply reading the details.
2. Single solution fallacy. Some commenters seem to think this is proposed as a complete solution to spam, and since not everyone will opt to participate, it won’t work. But a complete solution is not at all necessary or even possible given the flag-day problem. Deployed machine learning systems for fighting spam are great at taking advantage of a partial solution. The solution to this fallacy is learning about machine learning. In the current state of affairs, informed comment about spam fighting without knowing machine learning is difficult to imagine.
3. Broken crypto fallacy. Some commenters seem to think that stamps can be reused arbitrarily on emails. This ignores the existence of strong hashes. The solution to this fallacy is simply checking the details and possibly learning about cryptographics hashes.

Dan Reeves made a very detailed FAQ trying to address all the failure modes seen in comments, and there is a bit more discussion at messy matters.

My personal opinion is that Centmail is an interesting idea that might work, avoids the failure modes of many other ideas, hasn’t failed yet, and hence is worth trying. It’s a better approach than my earlier thoughts on economic solutions to spam.

## Asymmophobia

One striking feature of many machine learning algorithms is the gymnastics that designers go through to avoid symmetry breaking. In the most basic form of machine learning, there are labeled examples composed of features. Each of these can be treated symmetrically or asymmetrically by algorithms.

1. feature symmetry Every feature is treated the same. In gradient update rules, the same update is applied whether the feature is first or last. In metric-based predictions, every feature is just as important in computing the distance.
2. example symmetry Every example is treated the same. Batch learning algorithms are great exemplars of this approach.
3. label symmetry Every label is treated the same. This is particularly noticeable in multiclass classification systems which predict according to arg maxl wl x but it occurs in many other places as well.

Empirically, breaking symmetry well seems to yield great algorithms.

1. feature asymmetry For those who like the “boosting is stepwise additive regression on exponential loss” viewpoint (I don’t entirely), boosting is an example of symmetry breaking on features.
2. example asymmetry Online learning introduces an example asymmetry. Aside from providing a mechanism for large scale learning, it also enables learning in entirely new (online) settings.
3. label asymmetry Tree structured algorithms are good instances of example asymmetry. This includes both the older decision tree approaches like C4.5 and some newer ones we’ve worked on. These approaches are exponentially faster in the number of labels than more symmetric approaches.

The examples above are notably important, with good symmetry breaking approaches yielding substantially improved prediction or computational performance. Given such strong evidence that symmetry breaking is a desirable property, a basic question is: Why isn’t it more prevalent, and more thoroughly studied? One reasonable answer is that doing symmetry breaking well requires more serious thought about learning algorithm design, so researchers simply haven’t gotten to it. This answer appears incomplete.

A more complete answer is that many researchers seem to reflexively avoid symmetry breaking. A simple reason for this is the now pervasive use of Matlab in machine learning. Matlab is a handy tool for fast prototyping of learning algorithms, but it has an intrinsic language-level bias towards symmetric approaches since there are builtin primitives for matrix operations. A more complex reason is a pervasive reflex belief in fairness. While this is admirable when reviewing papers, it seems less so when designing learning algorithms. A third related reason seems to be a fear of doing unmotivated things. Anytime symmetry breaking is undertaken, the method for symmetry breaking is in question, and many people feel uncomfortable without a theorem suggesting the method is the right one. Since there are few theorems motivating symmetry breaking methods, it is often avoided.

What methods for symmetry breaking exist?

1. Randomization. Neural Network learning algorithms which initialize the weights randomly exemplify this. I consider the randomization approach particularly weak. It makes experiments non-repeatable, and it seems like the sort of solution that someone with asymmophobia would come up with if they were forced to do something asymmetric.
2. Arbitrary. Arbitrary symmetry breaking is something like random, except there is no randomness—you simply declare this feature/label/example comes first and that one second. This seems mildly better than the randomized approach, but still not inspiring.
3. Data-driven. Boosting is a good example where a data-driven approach drives symmetry breaking (over features). Data-driven approaches for symmetry breaking seem the most sound, as they can result in improved performance.

While there are examples of learning algorithms doing symmetry breaking for features, labels, and examples individually, there aren’t any I know which do all three, well. What would such an algorithm look like?