C-Adapter: Adapting Deep Classifiers for Efficient Conformal Prediction Sets

Thank you for the valuable comments and detailed feedback on our manuscript. Please find our response below.

1. Broader application of C-Adapter

Thank you for the suggestion. We extended our experiments to the task of text classification using large language models. We present the experimental setting and results in Appendix I. In particular, we adopt conformal prediction in the 14-class classification of dbpedia 14 using LLama3-8B. The results in Table 7 of Appendix I show that C-Adapter can work well in this application. We also put our results here for your reference. The results are organized by (w/o C-Adapter) / (w/ C-Adapter).

In addition, we'd like to emphasize the model-agnostic advantage of C-Adapter in real applications. For example, C-Adapter can be applied to CLIP models (as shown in Table 1), which are multi-modal vision and language models. C-Adapter requires access only to the model outputs and integrates effortlessly with any classifier (even black-box models), regardless of the network architecture or pre-training strategy.

2. Theoretical insight of C-Adapter? [W1]

First, we clarify that our method is orthogonal to ConfTr and can be used to improve ConfTr. In particular, we can append the C-Adapter to models trained by ConfTr and further enhance the performance of conformal prediction (See Figure 4). Therefore, the contribution of this work does not necessarily depend on the simple comparison between C-Adapter and ConfTr.

Cost of accuracy. As presented in the motivation (Figure 1 and Figure 11), the performance of ConfTr is limited because its regularization inevitably deteriorates the classifier accuracy. While ConfTr may improve the efficiency with a specific $\alpha$, the cost of accuracy is generally unacceptable and potentially limits the improvements. In the revised paper, we provide a theoretical analysis in Appendix K to show the effect of topK accuracy on the bounds of the expected set size: the lower bound of the expected set size is negatively related to the top-k accuracy. Therefore, the cost of accuracy introduced by ConfTr will increase the lower bound of the expected size, leading to suboptimal performance in efficiency. This highlights the importance of preserving top-k accuracy, which establishes the advantage of our method. Notably, our method can adapt pre-trained classifiers for conformal prediction while preserving the top-k accuracy of the classifiers unchanged, rather than mitigating accuracy drops (as demonstrated in Theorem 1).

Overall efficiency. Our loss function is novel and is superior to the Size loss of ConfTr (See Table 2). While ConfTr only optimizes the efficiency at a pre-defined error rate $\alpha$, they cannot ensure the optimization of the overall efficiency in Eq.(4), which limits the application of ConfTr after training. Differently, our method does not require to define a specific $\alpha$ during training and directly optimize the discriminability of non-conformity scores between correctly and randomly matched data-label pairs. Through Proposition 1, we show that optimizing our loss function is theoretically equivalent to minimizing the overall efficiency.

Flexibility. C-Adapter requires access only to the model outputs and seamlessly integrates with any classifier, including black-box models. In contrast, ConfTr necessitates modifying model weights, which limits its applicability.

In summary, we provide theoretical analysis (Propositions 1, 2, 3 and Theorem 1) to demonstrate the advantages of our method in accuracy preserving and overall efficiency, respectively. Moreover, we'd like to clarify that our work is not a simple combination of two previous works. In this work, we implement the adapter layer for accuracy preserving and proposes a novel loss function for its optimization, which establish the new SOTA training method for conformal prediction.

3. Clarification of distribution shifts [W2, Q3]

There might be some misunderstandings. We clarify that the distribution shifts we mentioned happen between the training set and calibration/test sets. Thus, coverage is not affected under this setting, as the calibration and test sets is sampled from the same dataset (e.g., ImageNet-V2). We present this analysis to show that C-Adapter can work well even though it is trained on a different distribution from the calibration/test set. Specifically, C-Adapter is tuned to enhance the efficiency of conformal predictors on ImageNet, while still improving efficiency on ImageNetV2. In the revised paper, we rewrite this paragraph to avoid any misunderstandings.

4. Clarification of Figure 1 [Q1]

Thank you for pointing out the ambiguous description. We clarify that the decreasing of classifier accuracy would limits the improvement of conformal prediction from ConfTr. We have corrected the description in the revised paper. In Figure 1a, we show that the improvement of ConfTr is reduced after the accuracy is decreased with a large $\lambda$ (for THR, ConfTr consistently has a negative effect due to the accuracy drop). In Figure 1b, we show that ConfTr cannot improve the efficiency of prediction sets on ImageNet with different $\lambda$. The results consistently demonstrate the suboptimal performance of ConfTr on different scales of application. This motivates our approach to adapting pre-trained classifiers for conformal prediction without compromising accuracy.

As for the difference of performance between the datasets, we conjecture that it might be related to the class numbers of datasets: ImageNet contains 1000 classes, leading to a large Size Loss and making it more challenging to balance the classification loss with Size Loss. A larger $\lambda$ causes a significant drop in model accuracy, whereas a smaller $\lambda$ fails to enhance efficiency. Consequently, applying ConfTr to large-scale datasets such as ImageNet becomes more challenging.

5. Clarification of the proposed loss function [Q2]

Yes, this is exactly the advantage of our proposed loss function over the Size loss of ConfTr. In particular, ConfTr optimizes the average set size at a pre-defined error rate $\alpha$ but hope it can generalize to all error rates. Instead, our loss function is not designed for a specific error rate. To optimize the overall efficiency defined in Eq.(4), we translate it into an equivalent form as shown in Eq.(5). In proposition 1, we formally prove that minimizing the probability in Eq.(5) is equivalent to optimizing the overall efficiency defined in Eq.(4). Intuitively, our goal is to enhance the discriminability of non-conformity scores between correctly and randomly matched data-label pairs, which translates to more efficient conformal prediction sets at various coverage rates. We provide an illustration of score distribution in Figure 3 and validate that C-Adapter promotes highly distinguishable scores between correct and incorrect labels. In addition, we also provides an ablation study of loss functions in Table 2 and we implement the size loss of ConfTr in C-Adapter for comparison. The results show that the size loss (only optimizing for one \alpha value) is inferior to our proposed loss in the performance of conformal prediction.

6. Will the code and data be publicly available? [Q4]

Yes, we will definitely make our code and data publicly available once the paper is published.

Thank you for your thorough response and updated manuscript. Based on your clarifications and comments, I have decided to raise your score.

Thank you for your positive feedback and valuable comments on our manuscript. Please find our response below.

1. Clarification of methodological novelty [W1]

There might be some misunderstandings. We clarify that our loss function in Eq.(7) is novel as one of main contributions in this work (See the 2nd contribution in Introduction). It is totally different from the Size loss used in ConfTr. While the Size loss in ConfTr minimizes the prediction sets at a pre-defined error rate (e.g., $\alpha=0.01$), our loss function is design to separate the scores of correctly and incorrectly matched data-label pairs, which is theoretically equivalent to optimizing the overall efficiency in Eq.(4) (See Proposition 1). Thus, the methodological novelty of this work is sufficient with the accuracy-preserved design for conformal prediction and the new loss function, as appreciated by reviewer itRT.

In the revised paper, we also provide a new theoretical analysis in Appendix K to show the effect of accuracy cost on the efficiency of prediction sets. The analysis formally shows that the cost of accuracy introduced by ConfTr will increase the lower bound of the expected size, leading to suboptimal performance in efficiency. The theoretical results provide deep insights for designing effective training methods in conformal prediction.

2. Why C-Adapter is insensitive to $T$ [W2]:

Sorry for the confusion. We clarify that the insensitivity is due to the fact that the value of $T$ (in the range of $[10^{-6}, 10^{-2}]$) is sufficiently small to ensure a high-quality approximation. The previous analysis of $T$ presented in Figure 7 is limited to the the range $[10^{-6}, 10^{-2}]$, which are too small to show the effect of $T$. To avoid any misunderstanding, we extend the range of $T$ to $[10^{-6}, 10^1]$ in the revised version. As presented in Figure 7, a large $T$ (e.g., 1, 10) leads to poor performance in the efficiency of prediction sets. With the decrease of $T$, the performance of C-Adapter is improved because the sigmoid function gradually approximates the indicator function. Note that the performance is converged when $T$ is sufficiently small (e.g., 0.01), our method does not require a heavy tuning for $T$ as we can simply set a small $T$.

3. Clarification of using Sigmoid as the surrogate function [W2]

Yes, using the sigmoid function cannot remain strictly convex over the entire domain. Yet, we argue that the sigmoid function is still commonly used in deep learning due to its non-linearity and differentiability. For example, it is generally adopted as the activation function in deep neural networks [1], the "switch" of neurons, and the gating mechanism controlling information flow in recurrent neural networks (RNNs) and long short-term memory networks (LSTMs) [2]. Notably, Sigmoid is also common used as the surrogate functions for the indicator function in AUC maximization [3,4].

As for the convergence, we provide a empirical analysis in Appendix F to show the fast convergence of C-Adapter, approaching nearly optimal performance in just 50 iterations. This advantage of C-Adapter may arise from the intra order-preserving functions, which significantly reduce the hypothesis space in learning.

Furthermore, we provide an empirical comparison to show the advantage of Sigmoid in Table 8 (Appendix I of the revised paper). We compare the performance of the most commonly used surrogates for the indicator function [4]: Square, Hinge, and Sigmoid. The results show that Sigmoid consistently outperforms the other two functions across all non-conformity scores.

We also present the results here for your reference:

The experiment is conducted on ImageNet with DenseNet121. The error rate $\alpha$ is set to 0.05. Since all methods achieve the desired coverage, only Size is reported here.

[1] Sharma, Sagar, Simone Sharma, and Anidhya Athaiya. "Activation functions in neural networks." Towards Data Sci 6.12 (2017): 310-316.

[2] Sherstinsky, Alex. "Fundamentals of recurrent neural network (RNN) and long short-term memory (LSTM) network." Physica D: Nonlinear Phenomena 404 (2020): 132306.

[3] Yan, Lian, et al. "Optimizing classifier performance via an approximation to the Wilcoxon-Mann-Whitney statistic." Proceedings of the 20th international conference on machine learning (icml-03). 2003.

[4] Yang, Tianbao, and Yiming Ying. "AUC maximization in the era of big data and AI: A survey." ACM Computing Surveys 55.8 (2022): 1-37.

Thank you for your positive feedback and valuable comments. Please find our response below.

1. Simplified overview of Intra Order-Preserving Function [W1]

Thank you for the suggestion. The core idea of intra order-preserving function we implemented is to decouple the label ranking and the logit values in the tuning. In particular, we begin by preserving a duplicate of the label ranking, and then transmit the logit values to the linear layer for processing. Finally, we recover the label ranking in the final output, keeping the order unchanged. In the revised manuscript, we present this core idea in the section 3 and offer a detailed explanation (includes an illustration) of intra order-preserving functions in Appendix D.

2. More results of C-Adapter under data shifts [W2]

Thank you for the suggestion. In the revised version, we provide additional results on ImageNet-A (Hendrycks et al., 2021a) and ImageNet-R (Hendrycks et al., 2021b). In particular, ImageNet-A focusing on adversarial examples that are modified to mislead models, and ImageNet-R consisting of images transformed by various artistic styles and visual changes to test models' adaptability to different visual distributions. We present the detailed results of these two benchmarks in Table 6 of Appendix I. Specifically, C-Adapter can significantly improve the performance of popular non-conformity scores in both the two datasets. The reults further confirms the effectiveness of our methods in the scenarios of distribution shifts between the training set and test/calibration set.

3. Clarification of C-Adapter's sensitivity to $T$ [W3]

To extensively analyze the effect of $T$, we extend the range of $T$ to $[10^{-6}, 10]$ in the hyparameter sensitivity analysis. In Figure 7 of the revised manuscript, We show that C-Adapter achieves better performance with a smaller value of $T$. And, C-Adapter with a sufficiently small $T$ (e.g., 0.01) can effectively improve the effciency of conformal prediction. It is because Sigmoid function in Eq.(7) can approximate the indicator function with a small $T$. Thus, our method does not require heavy tuning of hyperparameter, as we can simply set a small $T$.

4. Clarification of relations between C-Adapter and ConfTr [Q1]

We clarify that our method is complementary to ConfTr, as C-Adapter can be implemented after training with ConfTr. The results in Figure 4 show that our method can outperform and improve ConfTr. As for the potential benefits of ConfTr to our method, current results show that C-Adapter+ConfTr does not outperform applying C-Adapter alone in all cases. It might be because the regularization term of ConfTr normally damages the classification accuracy, leading to suboptimal performance in conformal prediction. In future work, training methods may benefits C-Adapter if a new training objective is proposed to improve both accuracy and conformal prediction.

5. How C-Adapter benefits the conditional coverage [Q2]

Thank you for the suggestion. In Appendix J, we add a gradient analysis of the proposed loss function to explain the benefits of C-Adapter. The challenge of poor conditional coverage metrics typically arises from the performance variation across sub-groups of data, leading to disparities in score distributions among these groups. C-Adapter mitigates the discrepancies through optimizing the proposed loss function in Eq.(8). In particular, the gradient analysis shows that our loss function enables to put more focus on samples with high scores, decreasing the variation in non-conformity scores among data samples. In this way, our method can improve the conditional coverage with consistent performance across different data sub-groups.

As for previous methods to improve the conditional coverage, they normally alleviate this issue by computing thresholds for different sub-groups. For example, Clustered Conformal Prediction (CCP) [1] clusters the classes based on their similarities and calculates group-specific thresholds to perform conformal prediction. While CCP can benefits the conditional coverage, they usually leads to larger prediction sets than vanilla conformla prediction (See Table 2 in their paper [1]). In addition, the improvement of CCP relies heavily on the quality of clustering, which requires a large caibration set. This highlights the advantage of our method, which can improve both the conditional coverage and the average set size.

[1] Ding, Tiffany, et al. "Class-conditional conformal prediction with many classes." Advances in Neural Information Processing Systems 36 (2024).

6. Evalutions on non-standard score functions [Q3]

Thank you for the suggestion. We are more than willing to provide extra results with non-standard score functions. We will highly appreciate it if you could provide an example of such score functions or related works.

Thank you for the valuable comments and detailed feedback on our manuscript. Please find our response below.

1. Comparison with related methods [W1 and Q1]

Thank you for the suggestion. However, we'd like to clarify that our method is orthogonal to current methods of conformal prediction, as the first adapter-based method in this area. Therefore, we provide extensive experiments to show that our method can enhance the performance of existing methods, including non-conformity scores -- THR, APS, and RAPS (see Table 1) and training algorithm -- ConfTr (see Figure 4). We are willing to supplement more results, if you can explicitly provide some approaches we missed in the paper.

2. Why employing the adapter module [W2 and Q2]

Thank you for the suggestion. In the revised version, we update the related work (Appendix A) to emphasize the distinctions of our method and adapters in other tasks. In the literature, adapters are generally designed as an efficient method to adapt pretrained models for downstreaming tasks [1-5], which plays a similar role as LORA in LLM. While our method shares the same concept of adapter, its underlying insight is totally different from previous adapters.

As the training objective of conformal prediction may deteriorate the accuracy, we hope to preserve the label ranking in the model output. Therefore, our C-Adapter only appends an adapter layer to the output layer of original models, enabling the implementation of intra order-preserving functions. Differently, previous adapters generally insert adapter layers between the existing layers of a neural network. Therefore, they cannot preserve the label ranking (as well as other PEFT methods, like LORA), making it suboptimal for conformal prediction. In the ablation study (Page 8), we empirically show that C-Adapter outperforms other fine-tuning methods, including retraining and linear probing.

In summary, we employ C-Adapter to enable the efficient adaptation of trained classifiers for conformal prediction without sacrificing classification accuracy (line 45). Compared to ConfTr and other PEFT methods, our method does not required to update the parameters of original models (high efficiency) and maintain the classification accuracy (more effective).

[1] Sylvestre-Alvise Rebuffi, Hakan Bilen, and Andrea Vedaldi. Learning multiple visual domains with residual adapters. arXiv:1705.08045 [cs, stat], November 2017.

[2] Houlsby, Neil, et al. "Parameter-efficient transfer learning for NLP." International conference on machine learning. PMLR, 2019.

[3] Zhaojiang Lin, Andrea Madotto, and Pascale Fung. Exploring versatile generative language model via parameter-efficient transfer learning. In Findings of the Association for Computational Linguistics: EMNLP 2020, pp. 441–459, Online, November 2020. Association for Computational Linguistics. doi: 10.18653/v1/2020.findings-emnlp.41.

[4] Hu, Edward J., et al. "Lora: Low-rank adaptation of large language models." arXiv preprint arXiv:2106.09685 (2021).

[5] Sung, Yi-Lin, Jaemin Cho, and Mohit Bansal. "Vl-adapter: Parameter-efficient transfer learning for vision-and-language tasks." Proceedings of the IEEE/CVF conference on computer vision and pattern recognition. 2022.