Towards Understanding Why Label Smoothing Degrades Selective Classification and How to Fix It

We thank reviewer 7wzD for their feedback. We are especially thankful for the feedback on the clarity of the presentation in Sec. 4. Upon reflection, we agree that this should be improved. Addressing their specific queries:

1. We agree that the term “regularisation gradient” Eq. (14) is confusing. We originally conceived it in order to associate it with an intuition of “holding the model back” and also because LS can be understood as adding regularisation to CE. However, we understand that it may be confused with the “regularisation” in Eqs. (8,13). “Regularisation” in Eqs. (8,13) is illustrated in Fig. 2 (left), and refers to how label smoothing encourages the softmax output to be uniform. This is informal and mathematically distinct from the “regularisation gradient” in Eq. (14). As such we have renamed “regularisation gradient” to “suppression gradient”, to improve clarity as well as changed “regularisation” to “suppression” across relevant cases in the whole paper. This more directly reflects the idea of the gradient "pushing down" the logits.

2. We are unable to identify the typo, we would be grateful if the reviewer could point it out to us explicitly.

3. • Although we define all notation in Sec. 2, upon reflection we agree that Sec 4 is difficult to parse given notation is not consistently clarified in the text. We have included a glossary at the start of the appendix such that a reader can more easily clarify any notation.

   • "This directly impacts softmax-based U such as MSP". If we consider the max softmax probability $\exp v_\text{max}/\sum_i \exp v_i$ in relation to logits $\boldsymbol v$, we can see that due to the exponentiation, MSP calculation will be dominated by the largest logits, in particular the max logit $v_\text{max}$. Thus suppressing the max logit will tend to reduce the MSP. Other softmax-based uncertainties such as Entropy and DOCTOR (Appendix F.1 of the updated paper, prev E.1) are similarly dominated by the max logit and thus behave similarly. We have added this clarification to Sec. 4.2. We note that it may be helpful to intuit using "confidence" (negative uncertainty), i.e. pushing down on the max logit reduces model output confidence.

4. “v_max is more strongly suppressed for lower P_error”. In more plain language, the max logit is pushed down more on training samples where the model is (likely) right, and less pushed down when it is (likely) wrong. This leads the softmax to be less confident on correct predictions and more confident when it is wrong. Clarifying notation, Eq. (15) shows that for a given sample, compared to CE, LS suppresses/pushes down the maximum logit $v_\text{max}$ more(less) for lower(higher) probability of misclassification $P_\text{error}$.

5. Left refers to the left two cells and right the right two cells. We have separated them to make this clearer and updated the caption.

6. • The purpose of Fig. 5 is to empirically illustrate/verify that training with LS leads to lower max logit values for correct predictions (following point 4.) We report max logit $v_\text{max}$ values given the MSP value $\pi_\text{max}$, because generally max logit (and MSP) for errors will be lower than correct predictions for both CE and LS. By conditioning on MSP we remove this bias. This also reflects that for the same $U$ (-MSP), samples with lower probability of error will have the max logit suppressed more by LS.
   • Fig. 3 shows main selective classification results, which evidences that the ability of the MSP score to rank/separate/distinguish correct vs incorrect predictions is degraded when using LS compared to CE.

We thank you again for your detailed feedback. Please do not hesitate to ask if you have any further queries, or require further clarification on the above.

We thank reviewer s8Gk for their feedback and heartwarming comments.

I think it is also important to quantitatively compare the LS+logit normalisation with other existing solutions under the experimental setting used

If we consider post-training uncertainty scores for selective classification, we consider a number of other existing approaches (with and without LS) in Appendix F.1 (E.1 in previous version) . For softmax-based DOCTOR and Entropy, we find them to be similar but slightly worse than MSP. For OOD score Energy, we find it to be much worse than MSP. Thus LS+logit norm is better than LS+MSP and all the other uncertainty scores we investigate. Please let us know if we have interpreted your comment correctly.

In figure 7(a), it is hard to find the "CE logit norm".

We have revised the figure for improved readability in the updated submission, by updating the colour of "CE logit norm".

Please do not hesitate to ask if you have any further queries, or require further clarification on the above.

We thank you for your service. We are grateful that you enjoyed our paper and for the helpful feedback you provided.

We thank reviewer KVT2 for their feedback and positive comments on our work.

using two specific datasets

We note that Appendix B.3 contains results on CIFAR-100 and the jupyter notebook in the supplementary uses CIFAR-10. These results mirror the main paper's findings.

Could these findings generalize to other domains, such as text or tabular data?

We have added binary classification results on two small-scale tabular UCI binary classification datasets (used by [1], for instance) in section B.4 of the appendix of the updated submission. We find LS degrades SC in this case as well.

Is it possible to identify a threshold at which the effect of LS on SC begins to diminish?

Our interpretation of the reviewer’s query is that it relates to the SC rejection threshold $\tau$/corresponding level of coverage. The rightmost column of Fig. 3 shows the difference in risk between the baseline CE and LS. As you say, it shows that the absolute degradation in risk tends to diminish as coverage increases. After a certain coverage ($\gtrsim 75\%$) the LS models may have lower risk than CE. This is due to the regularisation of LS improving the base accuracy (@100 coverage) of the model.

The right of Fig. 3 shows that as coverage decreases LS consistently degrades risk relative to CE. This demonstrates that LS is worse at separating errors from correct preds, even if it has fewer errors @100 coverage

We emphasise that LS prevents SC from being effective at low risks (Fig. 1 bottom), which need lower coverages to be achieved, which is especially important for safety-critical applications.

We thank the reviewer again and would be grateful if the reviewer could clarify in the case we have misunderstood the above.

Please do not hesitate to ask if you have any further queries, or require further clarification on the above.

References:

[1] Huang, X., Khetan, A., Cvitkovic, M., & Karnin, Z. (2020). Tabtransformer: Tabular data modeling using contextual embeddings. arXiv preprint arXiv:2012.06678.

I thank the authors for their detailed responses. Most of my questions have been answered. I will keep a positive score and raise my confidence to 4.

We thank reviewer wn1r for their feedback.

Novelty/Significance

As noted by the reviewer, we are honest about the position of our contribution in the literature. We firmly believe that the additional new knowledge provided by our work will be of value to the research community of ICLR (and practitioners in general).

Label smoothing degrades selective classification

Although Zhu et al. (2022) empirically discover this, their empirical results are limited (it is not the focus of their work) and there is no way of knowing whether these empirical results would generalise or whether they are only true in specific instances. By providing extended empirical evidence, as well as a clear analytical explanation rooted in the mathematical loss, we provide strong evidence that this behaviour is generalisable. We believe that this insight is useful to researchers and practitioners of selective classification.

Logit normalisation improves the SC of LS-trained models

We provide an effective and well-motivated solution to the above problem, which is useful to practitioners.

In the original logit normalisation paper (Cattelan & Silva, 2024) they do not provide a clear explanation for why the approach is effective sometimes and isn’t effective in other instances. They suggest simply trying it out on a validation dataset and falling back to the MSP if it is not effective. The approach is presented like a black box. This may reduce the confidence of practitioners interested in using it, especially in safety-critical applications for which selective classification is relevant.

By analytically investigating the mechanism of logit-normalisation, and linking it directly to our previous analysis/experiments on LS we are able to elucidate this black box. This provides clear guidance on when and why to use logit normalisation. This will give practitioners confidence in the effectiveness of the approach, when they previously may have chosen to avoid it in a safety-critical application.

Additional benefits of our analysis

The novel analysis in Sec. 4 raises questions for future work beyond selective classification, thus we believe it is of interest to the broader ICLR community: Does this aspect of label smoothing (suppressing the max logit less for incorrect predictions) affect generalisation? What about behaviour on OOD data? Could it help explain the behaviour in (Kornblith et al., 2021) where LS results in worse transfer/representation learning? Can this insight lead to a modification of LS to improve it?

Our analysis of logit-normalisation also generalises beyond LS – we simply demonstrate that logit normalisation reduces confidence when the max logit is higher. Thus, this knowledge can be potentially applied to other uncertainty scenarios (e.g. OOD detection).

As authors, we believe such research, that aims to explain and understand behaviour in deep learning is valuable at ICLR.

Questions

1. We investigated NLS and found training with it to be unstable. We remark that it appears worth investigating since negative $\alpha$ does reverse the logit suppression of Eq. (15). However, an analysis of the gradients (similar to the main paper) reveals that since NLS training targets can exist outside of the interval [0,1], in these cases the NLS logit gradients can never become zero (training minimum), leading to training instability. The discussion can be found in Appendix F.6 of the updated submission. We remark that NLS may still be useful for learning with label noise as presented in the original paper.
2. We have included the definition of the Kronecker delta in the paper (as well as the newly added notation glossary in the appendix)

Please do not hesitate to ask if you have any further queries, or require further clarification on the above.

Dear reviewers, we have further updated the submission pdf with LSTM experiments on text data in Appendix B.5. They again show that LS degrades SC performance across different modalities and architectures.

For reviewers who have yet to reply to our comments, we acknowledge the challenges of responding during this short time period. However, we are still eager to hear from you so that we can improve our manuscript and address any further issues/queries you may have.