Plugin estimators for selective classification with out-of-distribution detection

We thank the reviewer for the detailed comments and questions.

Could the authors show the steps to go from (3) to (4)

We have updated Appendix G with a more detailed derivation going from Equation 3 to Equation 4. See also Appendix B, which provides an argument for the appropriateness of solving the Lagrangian under mild distributional assumptions.

Could the excessive loss bound in Lemma 4.1 be rewritten by considering the estimation error?

Thanks for the suggestion. In this setting, we assume access to existing SC and OOD detection scorers, and we bound the SCOD error in terms of how well these scores do on their respective tasks. Importantly, we don’t make specific assumptions about how these scores are obtained.

If however we made more assumptions (e.g., we assume that they are the result of performing ERM with a certain surrogate loss), one could have an analogue to the generalisation analysis in Appendix C, which further bounds the estimation error of the individual scores. We will add a comment on this.

The constraint considered in (3) takes into account the rejection rate on the test distribution. Why not consider the rejection rate only on the in-distribution like (1) and (2)?

This is a reasonable question. Appendix F discusses a potential alternative formulation for SCOD which indeed considers a constraint on the in-distribution. Note that this does not change the general form of the Bayes-optimal scorer, and thus does not change the general plug-in SCOD strategy. Indeed, this is equivalent to (3) with a particular (distribution-dependent) choice of constraint $b_{\rm rej}$.

Regarding why constraining the rejection rate on the test distribution can make more sense: we are motivated by scenarios where the practitioner allocates a budget of the total fraction of abstentions they are comfortable making on test data. These abstentions could either be due to samples being close to the decision boundary, or due to samples being OOD.

This formulation is natural in applications where an abstention results in the sample being deferred to a larger model or a human expert at an additional monetary cost. In this case, the abstention typically incurs the same cost whether it was an ID or OOD sample that was rejected, and it is critical that the total proportion of abstentions be within a fixed budget

Ensuring Assumption (A2) in page 6 is satisfied

In the case of image classification with a fixed set of classes (as considered in Footnote 1), the proposal is to collect “canonical” samples for each class (e.g., unambiguous cat images, unambiguous dog images). Since out-of-distribution samples are, by definition, those drawn from a wholly distinct distribution than the training sample, it should be the case that $\mathbb{P}_{\rm out}(x) = 0$ for such “canonical” samples.

For CIFAR, the proposed SCOD learning in algorithm 1 does not seem to yield better results than training only on the CE loss and using heuristic scores to perform SCOD. Could the authors elaborate on potential limitations on why this is the case?

We believe the reviewer is referring to Table 2, where the OOD samples seen during training are different from those used during test time. In this case, our loss-based estimators (Algorithm 1) may sometimes perform worse than our black-box estimators, despite the latter being trained only on ID samples. This is primarily due to the mismatch between the OOD datasets used during training and testing.

The reviewer’s observation does not however hold with the CIFAR100 experiments in Table 3, where we use samples from the same OOD dataset during both training and testing (albeit a part of a noisy “wild” dataset). Here, Algorithm 1 (last row) is seen to provide substantial improvements in 4 out 6 cases.

Suggestions

How does the inlier rejection option perform on this task compared to the proposed method and existing OOD detectors?... The AUC-RC is compared against OOD methods but not against selective classification methods

Please note that in Appendix I.4, we provided a comparison against a representative baseline from the SC literature, namely, the cost-sensitive softmax cross-entropy loss of (Mozannar and Sontag, 2020). We find similar conclusions to the results reported in the body. Additionally, the MSP method we compare against in all tables is the same as Chow’s rule in the SC literature (Chow, 1970).

In the interest of further expanding the comparison against SC methods, we have added results for the selective classification method of DOCTOR method of Granese et al. (2021) [3] in Tables 2, 3 and 13. This method yields similar performance as the MSP, MaxLogit and Energy scorers that use a base model trained with ID samples alone.

We thank the reviewer for the encouraging comments and questions.

How to choose hyperparameters is crucial for this combined method, and it would be beneficial if the authors could provide more detail.

Our method requires the specification of two parameters $c_{\rm in}$ and $c_{\rm out}$. As noted in Section 4.3, these are in turn chosen based on the abstention budget $b_{\rm rej}$ and the cost of non-rejection $c_{\rm fn}$ provided by the practitioner.

Specifically, for Lagrange multiplier $\lambda$, we may expand $c_{\rm out} = c_{\rm fn} - \lambda \cdot (1 - \pi^*_{\rm in})$ and $c_{\rm in} = \lambda \cdot \pi^*_{\rm in}$, where $\pi^*_{\rm in}$ is the proportion of ID samples we expect in the test population.

The Lagrange multiplier $\lambda$ is the only hyper-parameter that needs to be tuned. We tune $\lambda$ so that the resulting rejector has a rejection rate of $b_{\rm rej}$ on a held-out set. Table 5 in the appendix summarizes the tuned and derived parameters.

How to determine the $\pi^*_{\rm in}$ through inspection of logged data?

Sorry for the confusion. We simply meant that if one has a sample of historical data, it could be possible to perform manual labelling on a subset to identify the proportion of OOD samples, or equally, $1 - \pi^{*}_{\rm in}$.

Evaluating performance on datasets that are exclusively OOD or SC is meaningful, as the proportion of each may vary across different scenarios.

In Table 7, we include results for two extreme settings, where the test set has 1% OOD samples and 99% OOD samples (to avoid numerical overflow issues with the metrics, we retain a small fraction of ID/OOD samples in each case).

When all (or most) of the samples are ID, we find that the MSP method becomes very competitive, as this is known to be a strong baseline for selective classification tasks.

When all (or most) of the samples are OOD, all the methods yield similar performance. This is because the optimal rejection strategy in this case is to declare almost all samples as OOD, and rejecting fewer samples is bound to incur a high cost. So irrespective of the method used, the AUC-RC metric, which averages the risk over different rejection rates, yields the same value.

We warmly thank the reviewer for their encouraging comments and detailed feedback.

In the formulation of selective classification, would it not be better to use the conditional probability of misclassification given the input is accepted

Indeed, there are two closely related formulations for classification problems with an abstention option: one uses the conditional error $\mathbb{P}(y \neq h(x) \mid r(x) = 0)$ (as exemplified in, e.g., SelectiveNet), while the other uses the joint error $\mathbb{P}(y \neq h(x), r(x) = 0)$ (as exemplified in, e.g., Ramaswamy et al. 2018).

There is a one-to-correspondence between the two formulations: owing to the constraint on $\mathbb{P}(r(x) = 1)$ in Equation 1, it is not hard to see that min_{h, r}~ \mathbb{P}(y \neq h(x) \mid r(x) = 0) \colon \~ \mathbb{P}(r(x) = 1) \leq b min_{h, r}~ \frac{\mathbb{P}(y \neq h(x), r(x) = 0)}{P(r(x) = 0)} \colon \~ \mathbb{P}(r(x) = 1) \leq b min_{h, r, a}~ \frac{\mathbb{P}(y \neq h(x), r(x) = 0)}{a} \colon \~ \mathbb{P}(r(x) = 1) \leq b,\~ \mathbb{P}(r(x) = 0) \geq a min_{h, r, a}~ \frac{\mathbb{P}(y \neq h(x), r(x) = 0)}{a} \colon \~ \mathbb{P}(r(x) = 1) \leq \min(b, 1 - a)

Thus, for a fixed a, the problem is equivalent to Equation 1, with a modified choice of the constraint on $\mathbb{P}(r(x) = 1)$. The Bayes-optimal classifier is thus unaffected.

We agree that discussion of this point is worthwhile, and have added this to Appendix F. We note also that Appendix F had a discussion of a slightly different SCOD formulation.

It is not clear to me how to arrive at the Lagrangian in Eqn (4) from the objective (3)... it seems like $c_{\rm in}$ and $c_{\rm out}$ specified here may have been swapped?

Thanks for the catch! The reviewer is correct: there was a typo with the constants $c_{\rm in}, c_{\rm out}$ being swapped. This has been fixed. We have also updated Appendix G with a more detailed derivation of going from Equation 3 to Equation 4, and updated the discussion in Section 4.3.

Need for strictly-inlier dataset

The reviewer is correct that the strictly-inlier dataset $S^*_{\rm in}$ is only needed to estimate the mixing weight $\pi_{\rm mix}$.

Per Lemma E.1, we use the assumption that $\\mathbb{P}\_{\rm out}(x) = 0$ for $x \in S^*_{\rm in}$ to estimate $\pi_{\rm mix}$. Note that if we just used $x \in S_{\rm in}$, we may not necessarily have $\\mathbb{P}\_{\rm out}(x) = 0$: the latter would only hold under the assumption that the support of $\\mathbb{P}\_{\rm in}$ and $\\mathbb{P}\_{\rm out}$ are disjoint. We allow for the two distributions to have overlapping support in general (e.g., outliers might correspond to draws from a low (but non-zero) density region of $\\mathbb{P}\_{\rm in}$).

Please clarify if $\pi^*_{\rm in}$ used in the formulation for SCOD Eqn (3) can be estimated using $\hat{\pi}_{mix}$?

$\pi^*_{\rm in}$ is the proportion of inliers in the test set, while $\pi_{\rm mix}$ is the proportion in the wild set. As the reviewer points out, indeed $\pi^*_{\rm in}$ may be equal to $\pi_{\rm mix}$ in practice, as the wild set is drawn from a production system.

In our experiments, we additionally consider cases where $\pi^*_{\rm in}$ can be different from $\pi_{\rm mix}$; in such cases, we will have to obtain estimates $\pi^*_{\rm in}$ through other means, such as through the practitioner or through manual inspection of historical data.

In the proof of Lemma 3.1 (Appendix A), it seems like the reject class is allowed to be part of the classifier output. Please clarify this point.

Sorry for the confusion. One can either work with a classifier-rejector pair $(h, r)$, or an augmented classifier $\bar{h} \colon \mathcal{X} \to [ L ] \cup \{ \perp \}$. We have updated the proof of Lemma 3.1 to rewrite statements involving $h(x) = \perp$ to $r(x) = 1$.

In Table 2, why do methods such as MSP, MaxLogit, and Energy (which do not use OOD training data) have different performance under the two settings: Random300K and OpenImages?

We thank the reviewer for spotting this discrepancy between the two sets of results. This difference is due to a subtle implementation detail that we had overlooked.

All our models are trained on the same dataset, with each batch comprising both ID and OOD samples. When training the base model for MSP, MaxLogit, and Energy, we compute the loss only on the ID samples from a batch, and ignore the OOD samples. However, because the base ResNet model may internally perform other operations such as batch normalization that, for example, compute averages across the entire batch of ID + OOD samples, we find that the trained models still have a mild dependence on the OOD samples in the training data.

We have now re-run the MSP, MaxLogit, and Energy results in Tables 2 and 3 on a dataset consisting exclusively of only ID samples. The conclusions remain the same, with one or more of the proposed plug-in methods often performing better than these baselines.

We are glad your concerns were satisfactorily addressed! Thanks for the positive and encouraging feedback.