This post is about an open problem in learning reductions.

**Background** A reduction might transform a a multiclass prediction problem where there are *k* possible labels into a binary learning problem where there are only 2 possible labels. On this induced binary problem we might learn a binary classifier with some error rate *e*. After subtracting the minimum possible (Bayes) error rate *b*, we get a regret *r = e – b*. The PECOC(Probabilistic Error Correcting Output Code) reduction has the property that binary regret *r* implies multiclass regret at most *4r ^{0.5}*.

**The problem** This is not the “rightest” answer. Consider the *k=2* case, where we reduce binary to binary. There exists a reduction (the identity) with the property that regret *r* implies regret *r*. This is substantially superior to the transform given by the PECOC reduction, which suggests that a better reduction may exist for general *k*. For example, we can not rule out the possibility that a reduction *R* exists with regret transform guaranteeing binary regret *r* implies at most multiclass regret *c(k) r* where *c(k)* is a *k* dependent constant between 1 and 4.

**Difficulty** I believe this is a solvable problem, given some serious thought.

**Impact** The use of some reduction from multiclass to binary is common practice, so a good solution could be widely useful. One thing to be aware of is that there is a common and reasonable concern about the ‘naturalness’ of induced problems. There seems to be no way to address this concern other than via empirical testing. On the theoretical side, a better reduction may help us understand whether classification or *l _{2}* regression is the more natural primitive for reduction. The PECOC reduction essentially first turns a binary classifier into an

*l*regressor and then uses the regressor repeatedly to make multiclass predictions.

_{2}Some background material which may help:

- Dietterich and Bakiri introduce Error Correcting Output Codes.
- Guruswami and Sahai analyze ECOC as an error transform reduction. (see lemma 2)
- Allwein, Schapire, and Singer generalize ECOC to use loss-based decoding.
- Beygelzimer and Langford showed that ECOC is not a regret transform and proved the PECOC regret transform.

[...] This problem has been cracked (but not quite completely solved) by Alina, Pradeep, and I. The problem is essentially finding a better way to reduce multiclass classification to binary classification. The solution is to use a carefully crafted tournament, the simplest version of which is a single elimination tournament where the “players” are the different classes. An example of the structure is here: For the single elimination tournament, we can prove that: For all multiclass problems D, for all learned binary classifiers c, the regret of an induced multiclass classifier is bounded by the regret of the binary classifier times log2 k. Restated: regmulticlass(D,Filter_tree_test(c)) <= regbinary (Filter_tree_train(D),c) Here: [...]

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