Generalization Analysis for Controllable Learning

Thank you for your constructive comments and active interest in helping us improve the quality of the paper.

The following are our responses to the Questions:

1. Response to Suggestions.

The difficulty of theoretical analysis for controllable learning lies in two aspects.

First, the development of controllable learning is driven by real-world applications, and the methods developed are very different. It is difficult to establish a unified theoretical framework to cover all these typical methods. The lack of a unified framework is the most intuitive and primary obstacle to using existing analytical tools to establish an effective theory bound for controllable learning. In addition, controllable learning needs to dynamically and adaptively respond to task requirements and may involve many completely different task targets, which increases the challenge of explicitly reflecting these factors in generalization analysis. In addition, an effective unified theoretical framework should also be extensible and provide interfaces that can cover a wider variety of methods that may be potentially developed in the future. The definition of the controllable learning function class and the modular theoretical results provided for commonly used control functions and prediction functions in controllable learning ensure the establishment of a unified theoretical framework.

Second, about reducing the dependency of the bounds on the number of task targets $c$. The analysis of the number of task targets in the bounds can be traced back to a basic bound with a linear dependency on $c$, which comes from the following inequality:

$\mathbb{E}\left[\sup_{\boldsymbol{f} \in \mathcal{F}} \frac{1}{n} \sum_{i=1}^n \sum_{j=1}^c \epsilon_{ij} f_j\left(\boldsymbol{x}_{i}\right)\right]$

$\leq c \max_j \mathbb{E}\left[\sup_{f_j} \frac{1}{n} \sum_{i=1}^n \epsilon_{ij} f_j\left(\boldsymbol{x}_{i}\right) \right].$

The dependency of the bounds on $c$ can be improved to square-root or logarithmic by preserving the coupling among different components reflected by the constraint, i.e., $\\|\boldsymbol{w}\\| \leq \Lambda$. However, each task target in controllable learning can be completely different, hence the coupling relationship between the weights of the functions corresponding to each task target needs to be decoupled. Hence, we introduce the projection operator. We found that the square-root dependency of the bound on $c$ is inevitable for $\ell_2$ Lipschitz loss, which essentially comes from the $\sqrt{c}$ factor in the radius of the empirical $\ell_2$ cover of the projection function class, but the Lipschitz continuity of the loss function with respect to $\ell_\infty$ norm can eliminate it. Hence, the tight bounds with no dependency on $c$, up to logarithmic terms, can be derived.

Thank you for your constructive comments and active interest in helping us improve the quality of the paper.

The following are our responses to the Questions:

1. Response to the Weakness 1.

The theoretical results in Section 5 can be used on the multi-target bipartite ranking loss. When the specific controllable learning method involves the multi-target bipartite ranking loss, we only need to consider replacing the corresponding constants in Theorem 4.7 (instead of Theorem 4.5) as in the proof process in Subsection 5.3.

2. Response to the Weakness 2 and Question 1.

Controllable learning often involves the selection of control functions and prediction functions corresponding to task targets. Although increasing the capacity of the model can enhance the representation ability of controllable learning, the increase in capacity is not blind. Our theoretical results show that for embedding-based controllable learning methods, the control function should choose to use a large-capacity model, and for hypernetwork-based controllable learning methods, the main model used for classification (rather than the hypernetwork) should choose to use a large-capacity model. These theoretical results are consistent with practical methods and can serve as a general guide for the structural design of controllable learning models.

3. Response to the Question 2.

The proposed vector-contraction inequality only assumes that the loss is Lipschitz continuous with respect to the $\ell_\infty$ norm. In addition, many loss functions in other problem settings also satisfy this assumption, such as multi-class classification [1], [2], multi-label learning [3], clustering [4], etc., which means that our theoretical results can also be extended when it comes to dealing with these problems, which also shows that the assumption on the loss function do not lose generality.

[1] Maksim Lapin, Matthias Hein, Bernt Schiele. "Top-k Multiclass SVM", NIPS 2015.

[2] Yunwen Lei, Ürün Dogan, Ding-Xuan Zhou, Marius Kloft. "Data-dependent generalization bounds for multi-class classification", IEEE TIT 2019.

[3] Yi-Fan Zhang, Min-Ling Zhang. "Generalization Analysis for Multi-Label Learning", ICML 2024.

[4] Shaojie Li, Yong Liu. "Sharper Generalization Bounds for Clustering", ICML 2021.

Thank you for your constructive comments and active interest in helping us improve the quality of the paper.

The following are our responses to the Questions:

1. Response to the Weakness and Suggestion.

All proof sketches and detailed proofs of the theoretical results in this paper have been moved to the appendix due to the limitation of the paper length. We will add all proof sketches to the main body of the paper in the revised version to better demonstrate key theoretical techniques and improve the readability of the paper.

2. Response to the Question.

Although this paper only studies two loss functions, namely weighted Hamming loss and multi-target bipartite ranking loss, in fact our theoretical results are not limited to these two forms of loss. Note that our theoretical results only assume that the loss is Lipschitz continuous with respect to the $\ell_\infty$ norm. This assumption is relatively mild and easy to satisfy. It only requires that the derivative of the loss function is bounded, which can also be guaranteed to some extent by the boundedness of the loss function. Hence, for potential evaluation metrics in the future, it is only necessary to verify that they are Lipschitz continuous with respect to the $\ell_\infty$ norm, and the theoretical results here are also applicable to them, with only the difference of the Lipschitz constants $\mu$, as we showed in Proposition 4.3.

Thank you for your constructive comments and active interest in helping us improve the quality of the paper.

The following are our responses to the Questions:

1. Response to Weakness.

We will add relevant explanations after Eqn. (2). For embedding-based controllable learning methods, the task requirement $\boldsymbol{z}$ can be editable user profiles, the control function $\psi$ often uses Transformers, and the nonlinear mapping $\zeta$ induced by classifier can use FNNs. For hypernetwork-based controllable learning methods, the task requirement can be a task indicator, the model corresponding to the control function is a hypernetwork, and the nonlinear mapping $\zeta$ induced by classifier can use GCN-based structures or Transformer-based models.

2. Response to Suggestion.

We provide two real-world examples of controllable learning scenarios as follows.

1. A news aggregator adjusts article rankings in real time using user-specified rules (e.g., exclude gaming content today) to promote diverse topics. The control function modifies inputs/parameters to enforce filters while keeping recommendations relevant.

2. A trading algorithm adapts portfolio strategies based on real-time goals (e.g., protect capital during market drops) to minimize losses and maximize risk-adjusted returns. The control function tunes parameters dynamically to balance objectives without retraining.

These examples show how our framework enables systems to adapt inputs/parameters at test time to meet evolving task requirements (e.g., content preferences, market conditions) while ensuring generalization guarantees for targets like diversity and risk management.

3. Response to Question.

From the perspective of the model's output, the outputs of both controllable learning and multi-label learning can be expressed as vector-valued outputs. From the perspective of the model itself and the input, they are different. Controllable learning places more emphasis on task requirements, i.e., the learner can adaptively adjust to dynamically respond to the requirements of different task targets, while multi-label learning does not involve task requirements corresponding to different task targets, so controllable learning is not a special case of multi-label learning. However, when the control function $\psi$ in controllable learning is $\emptyset$, controllable learning will degenerate into a specific multi-label method, i.e., multi-label learning based on the label-specific representation learning strategy. At this time, the relevant theoretical results can be generalized to multi-label learning scenarios.