On the Limitations of Temperature Scaling for Distributions with Overlaps

Response to Reviewer JXm6

We would like to thank Reviewer JXm6 for reviewing our paper, and we are grateful that they found our work to be clear and technically well-founded. We hope to address the raised weaknesses and questions below. 1. **Scope of discussed methods.** We definitely agree that it would be possible to extend the ideas from our theory to other data augmentation/training-time regularization approaches, and we have now emphasized this more in Section 1.1 (outline) in addition to stressing it in the conclusion. 2. **Real data experiments.** We also agree that having a more canonical "overlap" setting would be useful, however we felt that label noise was a natural way to introduce this overlap that also is readily encountered in practice (since samples are often mislabeled). Our results in Sections 5.2 and B.3 also include the 0\% label noise setting, which involves no artificial modification. 3. **Explaining regularization effect of $d$-Mixup.** When mixing several points with a uniform mixing distribution, mixtures in which one point gets a label very close to its original label (i.e. a mixing weight very close to 1) become less likely, and as a result we get even less spiky predictions (much closer to uniform probabilities). 4. **NLL gets worse but ECE/ACE get better.** It is possible to have very poor NLL while still having good (or even perfect) ECE, so worse NLL does not necessarily imply worse ECE performance. For example, a model that predicts uniformly at random can achieve zero ECE in the balanced class regime despite having poor NLL. Additionally, the ECE results in Table 4 for Mixup on CIFAR-100 are roughly within the 1 standard deviation error bounds of each other (so this does not represent a marked improvement), which we have also now pointed out in the surrounding discussion. Thank you again for your feedback and comments and we are happy to answer any further questions you may have.

Response to Reviewer dtwE

We would like to thank Reviewer dtwE for reviewing our paper, and we are happy to see that they found our work to be well-written and our claims to be well-evidenced. We hope to address the mentioned weaknesses and questions below. 1. **Amount of mixing in $d$-Mixup.** We definitely agree that the optimal mixing choice depends on the underlying data distribution being considered, and it is certainly true that mixing more points is not necessarily better (as we mention, this can lead to over-regularization of the predicted probabilities). However, it is tricky to analyze the optimal mixing choice in practice outside of treating it as an additional hyperparemeter. This is due to the fact that we do not know the underlying ground truth distribution - if we did have some way to quantify the amount of overlap in the ground truth, then it is certainly conceivable that we could develop theoretical guidelines for the optimal mixing number (although this would be separate from the theory in our work, which mixes $d + 1$ points since we do not make assumptions on the model class). Additionally, even when only two classes are overlapping there can still be significant benefit to mixing more than two points, which is demonstrated by the high overlap case in our experiments in Section 5.1 (4-Mixup performs the best in terms of NLL and ECE). The intuition here is again that the number of mixed points corresponds to the amount of regularization, and in the cases of high overlap we want to make sure the predicted probabilities are far away from being spiky (i.e. close to point masses). 2. **Generating additional class labels.** Generating synthetic labels as a means of regularization is definitely a useful idea, and in fact this is already done by methods such as label smoothing and Mixup (which drives the theory behind Mixup in our paper). The main difficulties with what you are suggesting (constructing a new class entirely) is the lack of knowledge of the ground truth distribution required to do so (that is, the input data does not specify which samples are in the overlapping part between class 1 \& 2). We need to specify some strategy for determining which samples get split off into the newly constructed class, and any such strategy will necessarily come with some baked-in biases about how we expect the ground truth distribution to behave. Naive strategies such as randomly assigning samples to the new class probably fall within the realm of existing label augmentation techniques, such as variants of label smoothing and Mixup. 3. **ERM interpolator drawbacks.** It is true that the properties of the ERM interpolators that we study are not ideal for calibration, which also drives the failure of temperature scaling in this case. However, we focus on this class of interpolating models as they are incredibly common in practice, and for these models we observe empirically the kind of assumptions we make in our theory (spiky probabilities that are well-separated). It is definitely possible that training ERM models with early stopping can provide sufficient regularization for better calibration (especially when combined with temperature scaling), which would be an interesting avenue to explore. If you have any further questions we are happy to answer them; thank you again for your useful feedback and remarks.

Response to Reviewer UALj

We would like to thank Reviewer UALj for taking the time to review our paper, and we are glad they found the work to be well-motivated and interesting. We hope to address their main questions below. 1. **Reconciling theory with common temperature scaling wisdom.** As you correctly point out, temperature scaling has been demonstrated repeatedly since [1] to be useful for calibration performance. Our theory does not contradict the *relative benefit* of temperature scaling (i.e. ERM + TS will almost certainly be better than ERM alone), but rather says that in *absolute* terms that there are classes of data distributions for which the improvement due to temperature scaling is simply not enough to achieve good calibration. This is made clearer by a new set of comparisons (Tables 6-8) that we have added as part of Appendix Section B.4, which show the relative gain from temperature scaling for both ERM and Mixup. As can be seen in this table, temperature scaling improves ERM calibration performance, but nowhere near enough to be even close to competitive with Mixup alone. Also worth pointing out is that in our setting, we do not train our ERM models with additional regularization (data augmentation, weight decay, etc.) as was done in prior work (so as to stay within the setting of our theory), and these almost certainly play a non-trivial role in how much temperature scaling can improve the base model (since the aforementioned modifications can lead to less spiky predictions and also better test accuracy). 2. **Non-temperature-scaling comparisons and effect of hyperparameters.** Thank you for the useful point; as mentioned above, we have added ERM vs ERM + TS and Mixup vs Mixup + TS comparisons to the Appendix as part of Section B.4. The main obversation from these comparisons is that while TS can significantly improve ERM performance (at least with respect to negative log-likelihood), it simply does not make a substantial difference with respect to the gap in performance between ERM and Mixup (exactly as suggested by our theory). With regards to the question about hyperparameters - calibration performance will certainly be affected heavily by hyperparameter tuning, with early stopping (among the modifications you mention) in particular having a significant effect (since this can prevent predicted probabilities from becoming too spiky for ERM). Our results are constrained to the common interpolation regime in which models are trained to have very low training loss, and in this regime we found that minor tweaks to hyperparameters such as learning rate did not have a significant effect (since we are training for a long enough time horizon). Additionally, as mentioned above, in this regime it is also harder to improve calibration performance since model predictions are even closer to point masses (which drives the failure of temperature scaling in our theoretical results). We are happy to answer any further questions you may have - thank you again for your review and helpful comments. [1] https://arxiv.org/abs/1706.04599