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.

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