Theoretical Investigations and Practical Enhancements on Tail Task Risk Minimization in Meta Learning

We sincerely thank # Reviewer 8vBx for these helpful comments. The remainders focus on questions to answer.

1. Advantages and disadvantages between group DRO and tail risk minimization in meta-learning

Thanks for this comment. The group DRO method employs risk reweighted algorithm to relax the weights of tasks and assign more weights to the gradient of worst cases. The tail risk minimization principle adopts two-stage optimization strategies to control the worst fast adaptation cases at a certain probabilistic level. (1) Group DRO is advantageous when predefined groups are available to guarantee robust performance across these groups, however, meta training is in a task episodic manner and weakens the applicability of group DRO. Tail risk minimization is more flexible, does not require clearly defined groups, and is suitable for risk-sensitive scenarios. (2) In the experiments, the tail risk minimization method consistently outperforms group DRO, demonstrating the advantages of the two-stage optimization strategy in improving robustness.

2. Application to large models

Thanks for this constructive comment. To answer the scalability question in large NNs, e.g., large models, we directly run experiments with large models.

Experimental setup: CLIP [1] is a recent large vision-language model; hence, we employ "ViT-B/16"-based CLIP as the backbone to enable few-shot learning in the same way as MaPLe (N_CTX: 2, MAX_EPOCH: 30) [2], scaling to large NNs in evaluation. SGD is the default optimizer with LR 0.0035, and A6000 GPUs work for computations. We examine tail task risk minimization effectiveness on three large datasets. The class number split setup in datasets (num train/num val/num test) is TieredImageNet (351/97/160), ImagenetA [3] (128/32/40), and ImagenetSketch (640/160/200).

Result and analysis: We'll include the global rebuttal Figure15-17 and analysis below in the manuscript:

Results are reported in Fig. 15-17. DR-MaPLe and DR-MaPLe+ consistently outperform baselines across both average and ${\rm CVaR}_\alpha$ indicators in $`5-way 1-shot`$ cases, demonstrating the advantage of the two-stage strategy in enhancing the robustness of few-shot learning. DR-MaPLe+ achieves better results as KDE quantiles are more accurate with large batch sizes. These results examine the scalability and compatibility of our method on large models.

3. Opensource plan

Even though this work contributes to more theoretical points, we hope our empirical investigations can provide more insights in developing large model augmented few-shot learners.

We'll opensource codes of large models' experiments to facilitate robust fast adaptation research from the updated manuscript.

Reference

[1] Radford, Alec, et al. "Learning transferable visual models from natural language supervision." ICML 2021.

[2] Khattak, Muhammad Uzair, et al. "Maple: Multi-modal prompt learning." CVPR 2023.

[3] Hendrycks, Dan, et al. "Natural adversarial examples." CVPR 2021.

Finally, we hope your questions are well answered. And thanks for your suggestions in improving the quality of this manuscript.

We sincerely thank # Reviewer 6Ej4 for these insightful comments. The remainder mainly focuses on concerns to address.

1. Application scopes and contribution clarifications

Thanks for the comment. Sorry for confusing you in contributions, and we further summarize:

(1) Regarding the application scopes, this work is meta-learning method agnostic. Apart from MAML, CNP is also used in examinations. Besides, we've taken the advice to conduct experiments on more benchmarks with large models' augmented backbone. See Point 3.

(2) Regarding the contribution, this work leans more on theoretical investigations of tail task risk minimization for meta-learning, which complements the empirical discoveries in Wang et al. (2023a). These include (i) the notion of solutions, (ii) understanding the Stackelberg game, (iii) generalization and asymptotic analysis in tail adaptation risk, and implementation tricks to enhance robustness. The practical enhancement is the theory's side product. See details in Lines 35-39, Lines 602-604, and Table 3.

2. More descriptions on notations

Thank you for your suggestions. We'll add more to Line 96 as follows:

The normalized cumulative distribution $F_{\ell}^{\alpha}(l;\theta)$ is defined as:

We're happy to see most of the concerns are addressed, and we express sincere gratitude for the helpful feedback and suggestions. The following answers other questions:

1. We apologize for not clearly explaining that this work is agnostic to meta-learning methods.

• Eq. (3) is not specifically for DR-MAML but is a general form of the risk function $\ell(\mathfrak{D}\_{\tau}^{Q},\mathfrak{D}\_{\tau}^{S};\theta)$ used in typical meta-learning methods. Eq. (4) is an example of Eq. (3) and is specific to DR-MAML. In this case, the risk function becomes $\ell(\mathfrak{D}\_{\tau}^{Q};\theta-\lambda\nabla\_{\theta}\ell(\mathfrak{D}\_{\tau}^{S};\theta))$, implying the implementation of MAML. Algorithm 2 on Lines 661-662 shows the application of another method CNP, where the loss function is specifically $\ell(\mathfrak{D}\_{{\tau}}^{Q};z,\theta)$, with $z=h\_{\theta\_1}(\mathfrak{D}\_{\tau}^{S})$.

• The theoretical insights in Section 4 are considered in a general form of meta learning with risk function $\ell(\mathfrak{D}\_{\tau}^{Q},\mathfrak{D}\_{\tau}^{S};\theta)$, not limited to MAML cases. Besides, we take MAML as a specific example for conducting theoretical analysis in Appendix Theorem C.1 on Lines 779-782.

2. The practical implementation time/space comparisons. We keep the setup the same as that in [1]: use the same maximum number of meta gradient updates for all baselines in training processes (this means given $\alpha=0.5$, the tail risk minimization principle requires double task batches to evaluate and screen sub-batches). For practical implementation, we take vanilla MaPLe, DR-MaPLe (MC augmented tail risk minimization), and DR-MaPLe+ (KDE augmented tail risk minimization) in tieredImageNet few-shot classification as the example to report. The Table below reports the overall training time and memory (the vanilla MaPLe serves as the anchor point, and + means additional costs from the two-stage operation).

Despite the more complex quantile estimations, DR-MaPLe+ does not show significantly higher time and space consumption compared to DR-MaPLe. It can be seen that both DR-MaPLe and DR-MaPLe+ consume more memories, and the extra training time over MaPLe arises from the evaluation and sub-batch screening in the forward pass. Such additional computations and memory costs bring significant robustness improvement, which can be crucial in risk-sensitive fast adaptation. We'll also include this point as a potential limitation in the Appendix.

[1] Wang, Qi, et al. "A simple yet effective strategy to robustify the meta learning paradigm." NeurIPS 2023.

Feel free to let us know if these questions have been answered well, and we’re happy to engage in further discussions. We’ll incorporate these precious suggestions into the manuscript. It would be appreciated if you could update the scores after the concerns are addressed. Your help means a lot to us.

We sincerely thank # Reviewer ci3Y for these insightful comments. The remainders focus on concerns and questions to address.

1. Additional explanations on the expected tail risk minimization

Thanks for this advice. We'll include more explanations in Line 115 as follows:

Wang et al. [1] minimizes the expected tail risk, or equivalently $\text{CVaR}\_{\alpha}$ risk measure:

Due to no closed form of $p\_{\alpha}(\tau;\theta)$, [1] introduces a slack variable $\xi\in\mathbb{R}$ and reformulates the objective as follows:

where $v\_\beta:=F\_{\ell}^{-1}(\beta)$ denotes the quantile statistics and $\left[\ell(\mathfrak{D}\_{\tau}^{Q},\mathfrak{D}\_{\tau}^{S};\theta)-\xi\right]^{+}:=\max$ {$\ell(\mathfrak{D}\_{\tau}^{Q},\mathfrak{D}\_{\tau}^{S};\theta)-\xi,0$} is the hinge risk.

We hope such descriptions make it easier to understand.

2. Larger font size in figure 1

Thanks for the valuable feedback. We've increased the font size in Figure 1 and will update it in the final manuscript.

3. Questions about hyperparameter $\alpha$

Thanks for your insightful question.

(1) In our experiments, we did not perform extensive hyperparameter $\alpha$ searches. Instead, we adopted hyperparameter settings that are consistent with existing literature [1]. This ensures that our results are fair in comparison to previous studies without introducing additional overhead from hyperparameter tuning.

(2) To further reveal the impact of confidence levels $\alpha$ on model performance, we conducted ablation experiments, as shown in Fig. 12/13. We can observe that in both sinusoid 5-shot and 10-shot tasks, as the confidence level varies, DR-MAML+ exhibits more stable performance than DR-MAML, indicating that DR-MAML+ has a lower sensitivity to confidence levels.

Reference

[1] Wang, Qi, et al. "A simple yet effective strategy to robustify the meta learning paradigm." NeurIPS 2023.

Finally, we hope these questions and concerns are well answered and addressed. Thanks again for your efforts. We are happy to discuss any other questions and provide further clarifications.

I thank the authors for responding to the reviewers comments and feedback, and I appreciate the additional evaluation on larger models.

However, I am inclined to agree with the other reviewers regarding the limited scope of the work. Additionally, the practical benefits of the proposed method is largely tangential to the main theoretical contribution about the Stackerlberg game equivalence, and arises from improving the risk estimates using more sophisticated density estimation, something that is already noted in the original work it builds from [1]. For these reasons, I keep my score as is.

[1] Wang, Qi, et al. "A simple yet effective strategy to robustify the meta learning paradigm." NeurIPS 2023.

Thanks for recognizing our primary theoretical contributions.

You're right; the scope of this work is restricted to a theoretical understanding of tail risk minimization in meta learning.

The practical benefits of quantile estimate are indeed inspired by the hypothesis in [1], and we (i) establish rigorous theoretical analysis in Theorem 4.4 and Remark 1; (ii) empirically conduct large-model few-shot learning experiments and verify the advantage of KDE over MC with larger batch size, complementing the theory.

Ref: [1] Wang, Qi, et al. "A simple yet effective strategy to robustify the meta learning paradigm." NeurIPS 2023.

Finally, we're encouraged by your positive comments, and your valuable suggestions help improve the manuscript greatly. Many thanks.