Generalization Analysis for Controllable Learning
Authors: Yi-Fan Zhang
Xiao Zhang
Min-Ling Zhang
Venue: ICML 2025 poster (2025)
Keywords: Controllability
Generalization Analysis
Primary area: theory->learning_theory

Abstract: Controllability has become a critical issue in trustworthy machine learning, as a controllable learner allows for dynamic model adaptation to task requirements during testing. However, existing research lacks a comprehensive understanding of how to effectively measure and analyze the generalization performance of controllable learning methods. In an attempt to move towards this goal from a generalization perspective, we first establish a unified framework for controllable learning. Then, we develop a novel vector-contraction inequality and derive a tight generalization bound for general controllable learning classes, which is independent of the number of task targets except for logarithmic factors and represents the current best-in-class theoretical result. Furthermore, we derive generalization bounds for two typical controllable learning methods: embedding-based and hypernetwork-based methods. We also upper bound the Rademacher complexities of commonly used control and prediction functions, which serve as modular theoretical components for deriving generalization bounds for specific controllable learning methods in practical applications such as recommender systems. Our theoretical results without strong assumptions provide general theoretical guarantees for controllable learning methods and offer new insights into understanding controllability in machine learning.

Generalization Analysis for Controllable Learning
Yi-Fan Zhang 12
Xiao Zhang
+34
Min-Ling Zhang 255
Abstract
1. Introduction
Controllability has emerged as a crucial aspect of Al (Yam- polskiy, 2020; Artificial Intelligence Safety Summit, 2023; A167, 2025), enabling Al models to be more reliable and adaptable to user needs in dynamic environments. In the context of machine learning, controllability ensures that learners can dynamically adapt to evolving task require- ments at test time, a concept known as controllable learn- ing. In many real-world applications, such as information retrieval and recommender systems, the ability to control model behavior at test time is essential for accommodating diverse user preferences, balancing competing objectives, and adapting to dynamic information-sseking needs (Chen et al., 2023; Mysore et al., 2023; Gao et al., 2024; Chang et al., 2023; He et al., 2022). Existing methods for con- trollable learning including embedding- based methods that adjust model inputs, and hypernetwork- generate task-specific model parameters, have demonstrated empirical success (Chen et al., 2023; Chang et al., 2023; Shen et al., 2025; Xie et al., 2025).
Controllability has become critical issue in trust- worthy machine learning, as controllable learner allows for dynamic model adaptation to task re- quirements during testing. However, existing re- search lacks a comprehensive understanding of how to effectively measure and analyze the gen- eralization performance of controllable learning methods. In an attempt to move towards this goal from generalization perspective, we first estab- lish a unified framework for controllable learning. Then, we develop a novel vector-contractin in- equality and derive a tight generalization bound for general controllable learning classes, which is independent of the number of task targets ex- cept for logarithmic factors and represents the cur- rent best-in-class theoretical result. Furthermore, we derive generalization bounds for two typical controllable learning methods: embedding-basse methods. We also upper bound the Rademacher complexities of commonly used control and prediction functions, which serve as modular theoretical components for deriving generalization bounds for specific controllable learning methods in practical applications such as recommender systems. Our theoretical results without strong assumptions provide general theo- retical guarantees for controllable learning meth- ods and offer new insights into understanding con- trollability machine learning.
methods that
and hypernetworkk-aseee
However, theoretically understanding controllability in ma- chine learning remains an important open question in ma- chine learning theory. Theoretical analysis of controllable learning methods from the perspective of generalization is important research avenue. A comprehensive gener- alization analysis of controllable learning faces two main challenges: 1) How to establish the relationship between the generalization bounds and controllability? 2) How to reduce the dependency of the generalization bounds on the number of task targets? First, controllable learning often needs to dy- namically respond to the task requirement and often involves multiple task targets. How to establish a unified theoretical framework that can formally characterize these factors is the primary issue of theoretical analysis. In addition, the ideal theoretical framework should be general enough to cover existing controllable learning methods and facilitate the de- velopment of general analysis methods and theoretical tools for generalization analysis controllable learning. With an effective unified theoretical framework, we can explicitly introduce controllability in generalization analysis. Second, intuitively, the multiple task targets involved in controllable learning suggest that learning will become more difficult as the number of task targets increases, but the empirical success of controllable learning methods suggests that the
Equal contribution 'School of Cyber Science and Engineering, Southeast University, Nanjing 210096, China Key Laboratory of Computer Network and Information Integration (Southeast Uni- versity), Ministry of Education, China 'Gaoling School of Artifi cial Intelligence, Reamin University of China, Beijing, China 'Engineering Research Center of Next-Generation Intelligent Search and Recommendation, MOE "Shoool of Computer Science and Engincering, Southeast University, Nanjing 2 210096, China. Correspondence to: Min-Ling Zhang <zhangml @ ecu...
Proceedings of the nnnd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 20255 by the author(s).

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Generalization Analysis for Controllable Learning
impact of increased difficulty should be limited (Chen al., 2023; He et al., 2022; Chang et al., 2023), meaning that the effective generalization bound is weakly dependent the number of task targets. It is clear that generalization analysis can promote a better understanding of controllable learning methods.
introduced LACE, an embedding- -basii controllable recom- mendation model that constructs editable user profiles from human-readable concepts, allowing users to modify them for adaptive recommendations without retraining. Chang et al. (2023) proposed PEPNet to address the imperfectly double seesaw problem by incorporating controllable parameters in dynamic, multi-domain recommendation scenarios. This ap- proach facilitates controllable and personalized predictions of user interactions across diverse tasks and domains.
In this paper. we establish an effective unified theoretical framework and derive tight bounds for controllable learn- ing. Specifically, we develop a novel vector-contraction inequality, which induces the state-of-the-art bound with no dependency on the number of task targets except for logarithmic factors for general controllable learning classes. In addition, we derive general bounds for two typical con- trollable learning methods, and we also develop modular theoretical results for commonly used control and prediction functions and show that the bounds for the specific control- lable learning methods are flexible combinations of these modular theoretical results.
Plypernetwork-basee controllable learning generates (partially) new model parameters through a control function based on the given task requirements, replacing the origi- nal model parameters to enable adaptive adjustment (Chen et al., 2023; Shen et al., 2023; He et al., 2022; Li et al., 2023; Yan al., 2022). Since this approach modifies the mapping between task requirements and model parameters during testing, it can also be considered an in-processing method. Chen et al. (2023) proposed novel framework named CMR, which leverages a feedforward neural network as a hypernetwork to dynamically generate parameters for a re- re-ranking model based on varying preference weights. This approach enables online adaptability without retraining and has demonstrated positive gains in online A/B tests within e- e-commerce scenarios. He et al. (2022) presented Hyper- prompt, a technique offering controllable task-conditioning of transformers by dynamically adjusting prompts using a hypernetwork for diverse tasks.
Our goal is to deepen our understanding of controllable learning methods through the systematic generalization anal- ysis. Major contributions of the paper include:
We establish a unified theoretical framework and de- velop novel vector-contraction inequality for control- lable learning, which exploits the Lipschitz continuity of losses w.r.t. lo norm and can induce tight bounds.
We derive tight bounds for general controllable learn- ing classes with no dependency on the number of task targets, which provides general theoretical guarantees for controllable learning methods.
Controllable learning has broad applications in information access. Existing theoretical work has focused solely on the approximation properties of controllable learning (Galanti & Wolf, 2020), while the study of its learning properties, par- ticularly generalization analysis, remains an open problem. This paper aims to bridge the gap in the generalization anal- ysis of controllable learning and provide theoretical tools for a broader exploration of its learning properties.
We derive bounds for two typical controllable learning methods and reveal that different manipulation meth- ods based the input and control function will lead to significant differences in the bounds. The theoretical techniques and modular results on FNN and Trans- former here serve as promising theoretical tools for the generalization analysis of controllable learning.
3. Preliminaries
Let [r] == {I,...,m) n) for any natural number n. In the context of controllable learning, given a dataset D = {(x1, ((r),,)).....E......... yn)) with n examples which are iden- tically and independently distributed (i.i.d)) from a proba- bility distribution P on x V, where X' C Rd denotes the d-dimensional input space and y denotes the label space. x E X. y y.
2. Related Work
The theoretical analysis in this paper focuses on controllable machine learning methods. Controllable learning empha- sizes the model's ability to adapt dynamically at test time based on task requirements to meet specific task targets. can be broadly categorized into the following two cate- gories.
3.1. Controllable Learning
lanbedding-bssed controllable learning focuses map- ping task requirements into embeddings (Gao et al., 2024; Mysore et al., 2023; Penaloza et al., 2024; Chang et al., 2023; Kong et al., 2024), which are then integrated with other inputs into the original model, serving as a pre- processing method during testing. Mysore et al. (2023)
Unlike traditional learning methods, where the learning ob- jective is fixed during the training phase and cannot change in the testing phase, controllable learning aims to dynami- cally adapt to newly arrived task requirements during testing, thereby achieving learner controllability. More specifically,
2

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Generalization Analysis for Controllable Learning
in controllable learning, the learner receives not only the input a: X' but also the task requirement E 2 during the testing phase. It adjusts adaptively based on any given task requirement x E 2, ensuring that the outputs for c task targets are responsive to the dynamic changes in 2.
the dataset D by optimizing the loss l. The generalization
performance is measured by the expected risk: R(f*) We denote the empirical risk w.r.t. training dataset D as RD(f*) = + (f In addition, we define the loss function space as L = {((f"(x), (z), : f E F). However, the above mentioned loss is typically the 0-1 loss, which is hard to handle in practice. Hence, one usually consider its surrogate losses.
E E(u.))~r((((" (x))
yr).
Formally, for y/ C IR" the task of controllable learning to solve the problem of multi-output regression with a given task requirement z, i.c., the j-th task targer of the learner f² (ft ............. f() X + Re corresponds to the regression of f7. E [c]. For } C [[1,++1) +13. each (y1...., Ve) is binary vector and y/ = 1(-1) denotes that the j-th label is (ir)relevant, j E [c]. and the task of controllable learning is to learn a classifier corresponding to the task requirement& z, which assigns each instance with set of relevant labels. A common strategy is to learn vector-valued function f = ,ff f2) ))= X' + I and derive the classifier by thresholding function which divides the label space into relevant and irrelevant label sets.
3.2. Related Evaluation Metrics
Although controllable learning has been implicitly used in modern information retrieval. However, there is still lack of specific evaluation metrics to measure the generaliza- tion performance of different controllable learning methods. Here we focus on two commonly used evaluation metrics in controllable learning, i.e., weighted Hamming loss and bipartite ranking loss, and define their surrogate losses. The surrogate loss for weighted Hamming loss is denoted by:
We consider the prediction function (also called controllable learner) for task target j E [c] of the general form: ff fx) = (z)))))) where the nonlinear mapping 1/y serves as a control function, while the manipulation function Φ represents the nonlinear transformation that integrates the input and control function into the learner, and the nonlinear mapping Sj corresponds to the classifier learned on the controllable representation generated by Φy.. We define a vector-valued function class of controllable learning as:
Ew (f = Σ Egly (1)()) j=1
{w., (, (ax,
where Ej is the loss of the j-th task target, 1y is the weight for the j-th task target and is bounded by |يy| S V for any E [c].
For bipartite ranking problems in controllable learning, since they involve pairwise losses, we need to addition- ally define the corresponding risks. Let be the probability that the samples are relevant to the j-th label. D denotes the conditional distribution of the samples over X' given that the samples are relevant to the j-th label, and Dj denotes the conditional distribution of the samples over X' given that the samples are irrelevant to the j-th label. We define the expected risk w.r.t. multi-target bipartite ranking for controllable learning as follows:
F = (-+ (z) f(x)
(x) w. (())(,,,)))))
w = (wi,...,, we) E dR0 aa()) S A, (((()) S (,)((())))) & B, (()())))) S C, # E E 2,j A, 0),
A, B, G
(1)
R(F*) ED)
where a represents a functional that constrains weights, 33 represents a functional that constrains nonlinear map- pings Çy. 7 represents a functional that constrains nonlin- ear mappings 6y. K represents functional that constrains control functions V/y. For embedding-barte based controllable learning methods, the task requirement& z can be editable user profiles, the control function & often uses Transform- ers, and the nonlinear mapping Ç induced by classifier can controllable learning
Σ p,E D
Σ
00 (1)(2))
BIND a/NDD ==1
In addition, the empirical risk w.r.t. multi-target bipartite ranking is defined as
use FNNs. For hypernetwork- methods, the task requirement& can be a task indicator, the model corresponding to the control function is hypernet- work, and the nonlinear mapping induced by classifier can use Transformer-based models.
RD(f*) = Σ |
RD(
11)
(2)
|
Σ
6 (17))))
( (2))),
n X |
x xxEX,
For any function fz X' + V, the prediction quality on the example (x. y) is measured by a loss function e: X x y + R.. The goal of controllable learning is to find a hypoth- esis fz E F with good generalization performance from
where X = {za y = +1,1 E [n]] (Xj = {2| ر/ =
- 1,, E [m])) corresponds to the set of the samples that are
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Generalization Analysis for Controllable Learning
relevant (irrelevant) the j-th label. The surrogate loss for multi-target hipartite ranking loss is denoted by:
Assumption 4.2. Assume that the loss function & is j
Lipschitz continuous w.r.t. the &o norm, that is:
(a, = Σ e (ff(m))) - j=1
[<(f(x);)) e((' -) s - (o))llo
where 14 > O, ||E||oc maxze(d] |tyl for t (f1...., tc).
where xi (xx) corresponds to the instances that are relevant (irrelevant) to the j-th label.
Assumption 4.1 and 4.2 are mild assumptions. For Assump- tion 4.1, The boundedness of the input features is easy to satisfy since in practice normalization is often applied to the input features. In addition, for the function class of control- lable learning (I), we often use the assumptions ||w, l2 S A, (-)l22 S A for any j E [c] to replace the boundedness of each task target of the learner, i.e., F == AA, and A can be further refined by the specific model used in control- lable learning. For Assumption 4.2, the Lipschitz continuity w.r.t. the & norm has been considered in several literature (Foster & Rakhlin, 2019; Lei et al., 2019; Wu et al., 2021; Zhang & Zhang, 2023; 2024a). The following Proposition 4.3 further illustrates that the commonly used loss functions in controllable learning actually satisfy Assumption 4.2.
3.3. Related Complexity Measures
Here we introduce the related complexity measures involved in our theoretical results. The Rademacher complexity is used to perform generalization analysis for controllable learning.
Definition 3.1 (Rademacher complexity). Let F be class of real-valued functions mapping from X' to R. Let D = {x)...., } be a set with n i.i.d. samples. The empirical Rademacher complexity over F is defined by
RD( = E sup fEr
Σ of (a:) (a:)
Proposition 4.3. Assume that the loss of each output task target é defined Subsection 3.2 p-Lipschitz continuous, then the surrogate weighted Hamming Loss is -V--ippchitz w.r.t. the loe norm, the surrogate multi-target bipartite ranking loss is p-Lipschitz w.r.f. the foo norm.
where 61, Cn are i.i.d. Rademacher random variables. In addition, we define the worst-case Rademacher complexity as RR, (F) suppex" (F).
The vector-valued function class F of controllable learn- ing makes traditional analysis methods developed for the Rademacher complexity of scalar-valued function class in- valid. Hence, we need to new analysis tools to convert the Rademacher complexity of loss function space associated with F into the Rademacher complexity of a tractable scalar- valued function class. The Rademacher complexity can be bounded by other scale-sensitive complexity measures, e.g. covering number and fat-shattering dimension (Srebro et al., 2010; Zhang& & Zhang, 2023). The relevant definitions are provided in the appendix.
In fact, each task target in controllable learning can be com- pletely different, hence the coupling relationship between the weights of the functions corresponding to each task target needs to be decoupled. To this end, we define a pro- jection operator Pj IR + E to project the c-dimensional output vector onto the j-th coordinate, Pj P, J [c]. Then, we have the projection function class P(F) (i)z) + (ff((x)) Ppff = (z). E F, (j.x) E E x X'), which decouples the relationship among dif- ferent task targets. With Assumption 4.2 and the above definitions, we develop the following novel vector- contraction inequality for controllable learning to show that the Rademacher complexity of the loss function space L can be bounded by the worst-case Rademacher complexity of the projection function class P(F):
4. General Bounds for Controllable Learning
In this section, we first introduce the assumptions used. Then, we develop novel vector-contraction inequality for the Rademacher complexity of the vector- function class F. Finally, with the novel vector-contraction inequal- ity, we derive bounds for general function classes of control- lable learning with no dependency on the number of task targets, up to logarithmic terms, which achieve the state of the art. The proof sketches and detailed proofs of the theoretical results in this paper are provided in the appendix.
Lemma 4.4. Let F be function class of controllable learn- ing defined by (1). Let Assumptions 4.1 and 4.2 hold. Given a dataset D of size n. Then, we have
12M
RD) <
+96 / (P(F))
Vii
Mvn
(1 + (nen In μE
Assumption 4.1. Assume that the input features, the loss function and each task target of the learner are bounded: | 122 < R. ((..-) < M, (+) < E for E [m]. j e [c], where R, M, F. > are constants.
where RD((C) = E. "suptec.f=E3 1 Ei ell (f (x:))
is the empirical Rademacher complexity of the loss function space associated with F, and Rne(P(F) is the worst-case
Rademacher complexity of the projection function class.
4

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Generalization Analysis for Controllable Learning
Remark 4.5. The difficulty of theoretical analysis for con- trollable learning lies in two aspects. First, the development of controllable learning is driven by real-world applications, and the methods developed are quite different. It is diffi- cult 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 exist- ing analytical tools to establish an effective theory bound for controllable learning. The definition of the controllable learning function class and the modular theoretical results provided ensure the establishment of unified theoretical framework. Second, about reducing the dependency on 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 typical pector-oontrcction lemma in (Maurer, 2016). The dependency on e can be improved to square-root logarithmic by preserving the coupling among different components, i.e., ||w||| S A. However, each task target in controllable learning can be completely different, and the coupling relationship needs to be decou- pled. Hence, we introduce the projection operator. We found that the square-root dependency on e is inevitable for l2 Lipschitz loss, which essentially comes from the vc fac- tor in the radius of the empirical &2 cover of the projection function class, but fo Lipschitz continuity of the loss can eliminate it. Hence, the tight bounds with no dependency on c, up to logarithmic terms, can be derived.
complexity of the loss function space L actually indepen- dent on c, and results in tight ((1/vn) bound with no dependency on the number of task targets. The projection function class combined with the foo norm Lipschitz conti- nuity of loss functions is the key to inducing a generalization bound with no dependent on c except for logarithmic factors. Theorem 4.7 shows that when the number of task targets in- creases, the generalization bound will be slightly larger, i.c. only a logarithmic increase. This means that although the in- crease in the number of task targets will affect the difficulty of learning, since Theorem 4.7 implies that the increase in difficulty is logarithmic, the real impact may only come from a few important task targets. Therefore, a controllable learning method can eventually obtain a learner with good generalization performance as long as it can learn these task targets well. Our theoretical results can provide general theoretical guarantees for controllable learning methods that can handle many task targets well in practice (Chen et al., 2023; He et al., 2022; Chang et al., 2023).
Since the surrogate multi-target bipartite ranking loss in- volves pairwise functions, a sequence of pairs of i.i.d. individual observation in (2) is no longer independent, which makes standard techniques in the i.i.d case for tra- ditional Rademacher complexity inapplicable. Inspired by (Clémençon et al., 2008), we convert the non-sum-of-i.it pairwise function sum-of-i.i.d form by using permuta- tions in U-process. Hence, for each task target, we define the construction method of the set of i.i.d disjoint positive and negative sample pairs for the j-th class label as follows:
Remark 4.6. The construction of the auxiliary dataset in- duced by projection functions can reduce the dependency the output dimension. For the construction of other types of auxiliary datasets under different problem settings and simi- lar ideas, please refer to (Lei et al., 2019; Wu et al., 2021; Mustafa et al., 2021; Hieu et al., 2024; Lei et al., 2023; Graf et al., 2022). Regarding the more desirable constant, the use of discretized variant of Dudley's 's integral inequality in (Lei al., 2023) suggests that this can be achieved.
1. We denote |x and |X,] as Lj and uj and (j + n for any j E [c]. We denote the number of disjoint pos- itive and negative sample pairs as sj = min(tj, 4z}.
2.
We uniformly select a positive sample from the set of the samples that are relevant to the j-th label and select a negative sample from the set of the samples that are irrelevant to the j-th label, then construct the selected positive and negative samples into a pair (xx x )
With the contra-ontttttton inequality above, we can de-
rive the following tight bound for the surrogate weighted Hamming loss:
Theorem 4.7. Let F be a function class of controllable learning defined by (1). Let Assumptions 4. I and 4.2 hold. Given a dataset D of size n. Then, for the surrogate weighted Hamming loss, for any 0 < 8 1, with prob- ability at least 1 5, 8, the following holds for any f E F:
3.
We construct the set of positive and negative sample pairs by matching the samples from the set of the sam- ples that are relevant to the j-th label with the samples from the set of the samples that are irrelevant to the j-th label until one of the sets of the positive samples and the negative samples exhausts its available samples for selection. We denote the set of i.i.d disjoint positive and negative sample pairs for the j-th class label as Dj. and D $z-
2 log 2n
24M Vn
R(f*) <Ro(F*) + 3M
192 + logo ( 4en pp?) In Mvn
With these definitions, we then derive the tight bound for the surrogate multi-target bipartite ranking loss as follows: Theorem 4.9. Let F be a function class of controllable learning defined by (l). Let Assumptions 4. I and 4.2 hold.
Remark 4.8. Although Lemma 4.4 contains a /c factor, the term Rie (P(F)) S Vxc. which makes the Rademacher
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Generalization Analysis for Controllable Learning
Given a datasel D size n. Then, for the surrogate multi- target bipartite ranking loss, for any O < 5 < 1, with probability at least 1 - 8, the following holds for any f* e F:
corresponding to the specific controllable learning meth- ods are flexible combinations of these modular theoretical components.
Specifically, in Subsection 5.1, we establish refined formal definitions for embedding- based and hypernetworkk-aade methods (i.e., oj in class (1)) and give their general theo- retical analysis methods. In Subsection 5.2, we give the formal definitions of commonly used control and predic- tion functions (i.e., 1y and Cj in class (1)), and derive their Rademacher complexity. In Subsection 5.3, we derive the corresponding generalization bounds for specific control- lable learning methods. This process shows how to flexibly apply the modular theoretical results developed in this pa- per. We show that different manipulation methods based on the input and control function will lead significant differences in the constant A of the generalization bound in Theorem 4.7 and 4.9.
24V2Mg R(F*) (fD
In 8
+ 22M4y
+
V8D
80
384 x (1+ + log2 1082(4es0cc (4Eage²
M V In
Vso
and = E Pj. Remark 4.10. The above bound shows a logarithmic depen- dency on the number of task targets with a faster conver- gence rate Ö(L/v (D)). When the examples of positive and negative classes are balanced, which is the ideal situation, we have f; = U, for any j E [c] 50 = 5. then the order of the bound is ((1/vn) When class-imbalance occurs, it is obvious that so will be smaller than T. If class-imbalance is more serious, so will be smaller, which will lead to a looser bound for the surrogate multi-target bipartite ranking loss. It means that when imsalaneee becomes more and more serious, if the learned classifier cannot handle the problem of dlas-iimbllance well, then its performance on the multi-targe nulti-target bipartite ranking loss will be worse.
where so = minmmin, Pj. min,(1-13))
5.1. Theoretical Analysis Methods for Embedding and Hypernetwork-basse Methods
Embedding- Methods
Embedding-baace methods aim to map the task requirement z into a latent space of inputs æ. This can be seen as a pre-processing technique at test time, where controllable learners adapt to the task requirement solely by modifying the input features, making them easy to implement in tasks like recommender systems classification.
Remark 4.11. The term Σ, Py is involved in the above bound, which indicates that each sample associated with multiple labels, and each task target can correspond to label in controllable learning. Real-world scenarios and practical tasks imply that the presence of the term E, Pj introduces two challenges. First, in label-sparse tasks, en- suring the applicability of the above theoretical result re- quires establishing a connection between the term E, Pj and appropriate sparsity conditions, thereby revealing how sparsity influences generalization bounds. Second, in recom- mendation systems, while users may simultaneously prefer multiple product categories, excessive probability summa- tion could lead to excessively long recommendation lists, thereby diminishing personalization efficacy. Therefore, de- signing suitable regularization terms may be necessary to address this issue. future work, we will further explore solutions to the limitations induced by the term Σ, Pj from these two perspectives.
Formally, for each task target, we denote the output of the control function as a² := (2) E Rdd, Φj is a concatenate function denoted as Concatel:,)) which concatenate x and a' ie.. Concate(æ;, a²) = [II,..., ,d, .i..., a²-1" E Rd+d', Then, a family of classifiers fy with p-Lipschitz nonlinear mapping (i.e., Çj is p-L-ipschiz)) are learned on the generated concatenate representations.
With the above definitions, we can derive the upper bound of the worst-case Rademacher complexity for embedding- based controllable learning methods:
Theorem 5.1. Let F be a function class of embedding- based controllable learning methods defined by (I). Let Assumptions 4.1 and 4.2 hold. Given a dataset D of size n.. Then, the worst-case Rademacher complexity of the corresponding projection function class can be bounded as: (P(FF) S 2pA(R+C)
5. General Bounds for Typical Controllable Learning Methods
vnc
In this section, we analyze the generalization bounds for two typical controllable learning methods, i.e., embedding- based and snpernetwork-basele controllable learning methods. In order to improve the readability and enhance the applica- bility of the developed theoretical tools, we decompose the process of generalization analysis of these controllable learn- ing methods into multiple modules. In this way, the bounds
The constant A of the generalization bound in Theorem 4.7 and 4.9 corresponds to 2p(R + C*) here.
Hypernetwork-based Methods
Hypernetorr--oe based methods use a network as the control function & to generate the model parameters of the predic- tion function, effectively allowing the control function to
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Generalization Analysis for Controllable Learning
control and adjust the parameters of the prediction function. These methods can be seen as in-processing technique at test time, offering flexibility in adapting model parameters to various task requirements. This makes hypernetworks an appealing approach for applications that require dynamic and task-specific parameter adjustments.
bounds of their Rademacher complexities. These theoretical results can be used components to derive the generaliza- tion bounds for specific controllable learning methods.
Feedforward Neural Network (FNN)
We define a feedforward neural network as follows:
Formally, for each task target, the weights of the nonlinear function by are generated by a hypernttork 15 (i.e.. the model corresponding to the control function is hypernet- work), we denote the generated vector as wwy. and Φb can be denoted as ( (æ; wv, ). Furthermore, ww, is determined by minimizing the Metn-Lipschiiz loss hourl defined on the control function wy., and we denoted the loss function space corresponding to fetrl as Letrl. We also define the class of control function f as Gj. j E [c], and use g e refer the c-fold Cartesian product of the function classes Gj.
f(x) = w a
where Wt are the parameter metrices, / E [1]. and a is the ReLU activation. With the above definitions, we can derive the upper bound of the Rademacher complexity for FNN:
Theorem 5.4. Let F be a function class of FNN. Assume that the parameter matrices in FNN are bounded, i.e., |Will s for any I E [L]. where BF > 0 is constant. Given a dataset D size n. Then, the Rademacher complex- ity of layer FNN can be bounded as: RD(F) S B..BBRR v'
With the above definitions, we can derive the upper bound of the worst-case Rademacher complexity for hypernetwork- based controllable learning methods:
Remark 5.5. According to the proof process. it is obvi- ous that RFF S 2 2/ Vn for I, layer FNN. The capacity- generalization analysis of deep models in- volved in Subsection 5.2 mainly uses the "peeling" argu- ment, which a commonly used method in capacity-bast based theoretical analysis. The main idea of "peeling" is to reduce the complexity bound for / layer networks to complexity bound for 1 - 1 layer networks, and for each reduction, a product factor of a Lipschitz constant for the activation func- tion and an upper bound for the weight matrix norm will be introduced. After applying / reductions, the multiplica- tion of product factors with exponential dependency on the depth may make the bound vacuous (Neyshabur et al., 2015; Bartlett et al., 2017: 2019). How to develop non-vacuous capacity- -aaed generalization analysis methods for deep models is still an open question (Neyshabur et al., 2018; Golowich et al., 2018; Zhang & Zhang, 2023) and we will further explore
Theorem 5.2. Let F be function class of hypernetwork- based controllable learning methods defined by (1). Let Assumptions 4.1 and 4.2 hold. Given a dataset D of size Th. Then, the worst-case Rademacher complexity of the corresponding projection function class can be bounded as: (Rnc(((H)) S 2PAB
</n + 2teemn, where H F
v/nc"
fear g e is the whole function class corresponding to the hypernetwork-basse methods.
Remark 5.3. Since hypernetwork-baseeee controllable learn- ing method is two-stage, the hypernetwor is used gener- ate vector of weights (i.e., model parameters) for each task target in the first stage, and then these generated parameter vectors are used in the learner of the second stage. Therefore, in order to fully consider the capacity of the hypernetwork corresponding to the first stage, it is necessary and appro- priate to define the whole function class as the sum of the function classes H. = F Letel g corresponding to the models of these two stages. The bound of the hypernetwork- based method can be obtained by combining Theorem 5.2 with Lemma 4.4, but (P( in Lemma 4.4 should be replaced by (P.L)). The constant A of the bound in Theorem 4.7 and 4.9 corresponds to 2pB here. However, unlike the embedding- based methods, the introduction of the hypernetwork class Gj leads to an additional increase in complexity, i.e., the last term in Theorem 5.2.
the future. However, our goal here is to provide modular theoretical results for generalization anal- ysis of controllable learning, and the depth of the models used in controllable learning is often limited or even shallow, so the theoretical results in Subsection 5.2 are valid.
Transformer
We follow the definition and notation of self-attention and Transformer in (Edelman et al., 2022). Let We e Rexd Wv E Bdxk and Wo, Wx E Rdxk be trainable weight matrices. Let X == [x1, ".....,,]]] E xd be the input, i.e., a sequence of d-dimensional tokens. Let a be an element-wise L.- L-pLisschitz activation func- tion with (0)) = O. Then, Transformer layer is defined as (RowSoftmax LXWQWXX' XWv) Wc, where RowSoftmat applies softmax on each row of its input. For the convenience of analysis, we represent WoW T a single matrix Wek E Rdddd Here we focus on the scalar output from the Transformer. We construct a special extra input in the length-7 sequence with a index [CT.S), which
5.2. The Rademacher Complexities for Commonly-used Control and Prediction Functions
In this Subsection, we first formally introduce the commonly used control or prediction functions in controllable learn- ing, mainly including two models, i.c., Feedforward Neural Network (FNN) and Transformer, and then derive the upper
K with
7

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Generalization Analysis for Controllable Learning
fixed or trainable vector F[CLs] E Rd, Finally, the output at index [CLS] is a linear function w' T "jCCLs). for a trainable parameter w E F.. This setup for scalar output is used in BERT (Devlin et al., 2019) and all of its derivatives. The scalar one layer Transformer is w "[CT.S] where ViCT,s] = W a (WiJ2 X ssftmmax (XWQ (ICLS))) . Then, we have the bound of the Rademacher complexity for Transformer:
controllable learning methods in recommender systems as examples concretize the nonlinear mapping and the control function & of controllable learner in (I), and further provide generalization bounds for embedding -baiin and hypernetwork- based controllable learning, which are flexi- ble combinations of the above modular theoretical results.
Bounds for Embedding- Controllable Learning
Theorem 5.6. Let F be a function class of Transformer. Assume that the parameter matrices in Transformer are bounded, i... ||wl|| S Bw ||W'll S. Bw [| S Bw [|Wakll I Bwok and ||,||| I for any t E T. Given a dataset D of size n. Then, the Rademacher complexity of the scalar one layer Transformer can be bounded as: ABn Bwy Bwok R.
In embedding -bsed controllable recommendation models, the task description z is often expressed in natural language (e.g., editable user profiles). After being processed by a Transformer language model encoder, the obtained embed- dings are used for the subsequent recommendation task (i.e., the control function a/ is typically implemented using a Transformer) (Gao al., 2024; Mysore et al., 2023; Kong et al., 2024). The nonlinear mapping C is often an FNN (Shen et al., 2021; Chang al., 2023; Penaloza et al., 2024), which receives the embeddings generated by & and aligns them with the user's preference patterns. Then, we have:
(F(F)= S
v"
Remark 5.7. Theorem 5.6 implies that the order of the bound for Transformer is O (z)) with no dependency on the se- quence length T, which is tight w.r.t. the dependency on T and matches the recent theoretical bound for Transformer (Trauger & Tewari, 2024). The key to reducing the depen- dency on T' is to avoid summation in the norm, where the Lipschitz property of softmax is used, i.c. Corollary A.7 in (Edelman et al., 2022). The symbol ||- || denotes Frobenius norm. (Edelman et al., 2022) mainly derives bounds for Transformer through covering numbers. It uses the Lips- chitz property of functions to convert the covering number of Transformer function class into those of vecuur- lin- ear classes, and uses the upper bounds of covering numbers for vector-valued linear classes obtain the result for Trans- formers. The ||,, norm involved therein originates from the upper bounds of covering numbers for vector-valued linear classes. Specifically, when upper bounding the cover- ing number of the vector-valued linear class, the summation of covering numbers for scalar-valued linear classes cor- responding to each component in the vector-valued linear function induces the || Il norm in the || ll2,1 norm. Our result is derived using the standard Rademacher analysis, following the conventional neural network analysis method, which naturally induce Frobenius norm. We do not empha- size the tightness regarding norm constraints here. Our main purpose here to reduce the dependency on T, so we focus on analyzing a single layer Transformer as an example. For the bound of the deep case, similar to other deep models, the peeling argument is also used for analysis. The dependency of induced hound on depth is similar to existing theoretical results (Edelman et al., 2022; Trauger& & Tewari, 2024).
Theorem 5.8. Let F be a function class of embedding- based controllable learning methods defined by (1), where the control function is a Transformer and the classifier is induced by a three-layer FNN. Let Assumptions 4.1 and 4.2 hold. Given a dataset D of size nT, Then, for the surrogate weighted Hamming loss, for any < 8 < 1, with probabilit at least 1 8, the following holds for any f E F:
2 log 0
24M +
R(F*) {FD + +3M
2n
Vn
M
VTi
(1 + (1+ (oog((err))) 2 22)) -ln
)x
192nV B (R + 4Bw Bwy Bwax Lo RS)
Vn
where By, Bw Bwy, BW Lo > 0 are constants.
Proof. According to Theorem 5.4 and 5.6, 2p and C in The- orem 5.1 correspond 2 B7. and 4Bw, Bw, Bwox L, R3. This is obvious since the processes of upper bounding these Rademacher complexities are similar. Therefore, we only need to find the upper bounds for Ç and & corresponding to the specific models in Theorem 5.1 and replace them. Then, according to Theorem 5.1, A = 2p(R + +C). Finally, the desired bound can be derived by replacing A in Theorem 4.7 with 2 B + Bwr
5.3. Generalization Bounds for Specific Controllable Learning Methods
Bounds for Hypernetwork-based Controllable Learning
In hypernetwork-baseee controllable recommendation mod- els, the task description z is often represented as a task indicator or a preference weight vector for various metrics. In such cases, a hypernetwokk is typically chosen to be an FNN (Chen et al., 2023; Shen et al., 2023; He al., 2022; Li
Recently, the controllability of AI has become key issue in many applications, particularly in information access ap- plications such as recommender systems (Joachims, 2024; Shen et al., 2024). This subsection will use otate-of-
8

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Generalization Analysis for Controllable Learning
al., 2023; Yan et al., 2022). The hypernetork takes as input and outputs model parameters, which replace certain parameters in the main recommendation model (i.e., nonlin- ear mapping ς). In this hypernetwork- -aaeor paradigm, the method is model-agnostic, meaning the main recommenda- tion model can be any representative architecture, including models for modeling user behavior se- quences (Kang& & McAuley, 2018: Zhou et al., 2020; Wu al., 2020; de Souza Pereira Moreira et al., 2021). Then, we have the following bound for hypernetwork-based models:
ods and can provide an explanation for the models selected in the actual methods cited in Subsection 5.3.
6. Discussion
To understand controllable learning more clearly and intu- itively, we provide two real-world examples of controllable learning scenarios: 1) A news aggregator adjusts article rankings in real time using user-specified rules (e.g., ex- clude gaming content today) to promote diverse topics. The control function modifies inputs/parameters to enforce filters while keeping recommendations relevant. 2) A trading al- gorithm 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 with- out 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.
Transformer-ban
Theorem 5.9. Let F be a function class of hypernetwork- based controllable learning methods defined by (1), where the control function is three-layer FNN and the classifier is induced by Transformer. Let Assumptions 4.1 and 4.2 hold. Given a dataset D of size n. Then, for the surrogate weighted Hamming loss, for any 0- < 5 < 1, with probability at least - ô, the following holds for any f E FF:
log
2
24M
R(f*) <RD(f) + 3M
+
+
2n
vn
M Ti )x
19 + logg( (ee 2x²)) Inn
Controllable learning and multi-label learning are different learning settings, but there some connections between them. Specifically, from the perspective of the model 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 the requirements of different task targets, while multi-label learning does not involve task requirements corresponding to different task targets, so con- trollable learning is not a special case of multi-label learning. However, when the control function to in controllable learn- ing is 0, controllable learning will degenerate into a specific multi- learning method, i.e., multi-label learning based on the label-specific representation learning strategy (Zhang & Zhang, 2024b). At this time, the relevant theoretical results can be generalized to multi-label learning scenarios.
P.F.
4Bwe Bwy Bwax Lo R&A + 2 Meart A A
vn
where Bp, BW Bw BWox. La > O are constants.
Proof. The proof idea is the same as Theorem 5.8, accord- ing to Theorem 5.6, 2pB and C in Theorem 5.2 corre- spond to 4Bw, Bw Bwox R3 R3 and 2 AB..RR Then,
according to Theorem 5.2, A 2pB. However, the in- troduction of the hypernetwork class leads to an additional last term in Theorem 5.2, hence, for hypernetwork-baste methods, AA in Theorem 4.7 should actually be 2pA B + 2/sctei C. Finally, the desired bound can be derived by replac- ing AA in Theorem 4.7 with 4Bwe Bwy BWak Lo R² A + 2 2AM
Remark 5.10. Theorems 5.8 and 5.9 show that the order of the bounds for specific embedding-based and hypernetwork- based methods is ((1/ However, their constants are different. The more complex constant term in Theorem 5.8 corresponds to C in Theorem 5.1, while the more complex constant term in Theorem 5.9 corresponds to the first term in Theorem 5.2. They show that to improve the capacity of models, for embedding- based methods, the control func- tion often chooses models with higher complexity, while for hypernetwork-basee
7. Conclusion
In this paper, we first establish a unified theoretical frame- work for controllable learning. Then, we propose a novel vector-contraction inequality and derive tight bound with no dependency on c for general controllable learning classes. In addition, we derive general bounds for two typical con- trollable learning methods and develop modular theoretical results for commonly used control and prediction functions.
methods, since the main model is used for learning, the complexity of models selected for the main model is higher than that of the hypernetwork. This can improve the representation ability of models, so it is easier to learn hypotheses with better generalization in the class. These theoretical results are consistent with the actual meth-
In future work, we will extend the analysis to more control- lable learning methods, e.g., controllable generation models, and derive theoretical results for broader range of models to enrich our modular theoretical tool set.

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Generalization Analysis for Controllable Learning
Acknowledgements
Chen, S., Wang, Y., Wen, Z.,, Li, Z.,, Zhang, C., Zhang, X., Lin, Q. Zhu, C., and Xu, J. Controllable multi-objective re-ranking with policy hypemetworks. In Proceedings of the 29th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, pp. 3855-3864, 2023.
The authors wish to thank the anonymous reviewers and the area chair for their helpful comments and suggestions, especially the area chair for his dedicated efforts and self- less assistance in clarifying all ambiguities, ensuring the soundness and enhancing the quality of the paper. This work was supported by the National Science Foundation of China (62225602, 62376275).
Clémençon, S., Lugosi, G., and Vayatis, N. Ranking and empirical minimization of u-statistics. The Annals of Statistics, 36(2):844-874, 2008.
de Souza Pereira Moreira, G., Rabhi, S., Lee, J. M., Ak, R., and Oldridge, E. Transformers4Rec: Bridging the gap between NLP and sequential/session bbase/ recommen- dation. In Proceedings of the 15th ACM Conference on Recommender Systems, pp. 143-153, 2021.
Impact Statement
This paper presents work whose goal is advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
Devlin, J., Chang, M., Lee, K., and Toutanova, K. BERT: pre-training of deep bidirectional transformers for lan- guage understanding. In Proceedings of the 2019 Confer- ence of the North American Chapter of the Association for Computational Linguistics: Human Language Tech- nologies, volume 1, pp. 4171-4186, 2019.
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Generalization Analysis for Controllable Learning
A. Appendix
A.I. Appendix Outline
In the appendix, we give the detailed proofs of our theoretical results in the main paper. Our main proofs include:
The 2o Lipschitz continuity of the commonly used losses in controllable learning (Proposition 4.3).
The novel vector-contraction inequality for &e Lipschitz surrogate loss (Lemma 4.4).
The tight bound of the general controllable learning class for the surrogate weighted Hamming loss (Theorem 4.7).
The tight bound of the general controllable learning class for the surrogate multi- taltittarget bipartite ranking loss (Theorem 4.9).
The upper bound of the Rademacher complexity for embedding- based controllable learning methods (Theorem 5.1).
The upper bound of the Rademacher complexity for hypernetorkk-based controllable learning methods (Theorem 5.2).
The upper bound of the Rademacher complexity for FNN (Theorem 5.4).
The upper bound of the Rademacher complexity for Transformer (Theorem 5.6).
A.2. Preliminaries
A.2.1. DEFINITIONS OF THE CORRESPONDING COMPLEXITY MEASURES
Definition A.1 (Eoo norm covering number). Let JF be a class of real-valued functions mapping from X' to IR Let
D (xi,..., ]n be set with n i...d. samples. For any > O, the empirical foo norm covering number Noc F. F; D) w.r.t.
D is defined as the minimal number m collection of vectors v! o'...., Tt R" such that maXic(n] f (xx) oa I e
(v, is the i-th component of the vector v²). In this case, call {u² ..... wn (oover of F with respect to D. We also define (c,
(e,
Definition A.2 (Fat-shattering dimension). Let FF be class of real-valued functions mapping from X' to R. We define the fat-sh shattering dimension fFF at scale > 0 as the largest E N such that there exist points 21,..., Ep p E X and witnesses ;
8p E IR satisfying: for any 6,, ,,...., 6p E {-1,+1 +1} there exists f E F with
5z (J,;)) 8,) 2 Vi 1....p..
A.2.2. THE BOUND FOR THE LOSS FUNCTION SPACE
According to McDiarmid's inequality (McDiarmid et al., 1989) and the symmetrization technique, it is easy to obtain that for any training dataset = {(x, y:) [n]], E(;,-) < M, with probability at least 1 - 8, the following holds:
log 2
R(A) S Rn(()) (c) 3M
(3)
2n
A.3. General Bounds for Controllable Learning
1. PROOF OF PROPOSITION 4.3
We first prove that the surrogate weighted Hamming loss μV -Lisschitz continuous with respect the loo norm.
Lw (f) lw (f*((r)) (x))
Σ (17()))) Σ tzly (z),)))
13

---

Generalization Analysis for Controllable Learning
2/ |4, (1)))) (f7
<V
p |5 (xx) - 1f
j=1
1 <V pcmax ff (x) jEld
=vV 1 (a) fz (ax) ||x
Then, prove that the surrogate surrogate multi- bipartite ranking loss is p-Lipschitz continuous with respect the to norm.
R ( (,,,))), 3) (
(x (x,
1 Σ (f) - 1z(1 - Σ
l
j=1
Σ 1 (z) ff(x²) - -, (z,) - (z(()
p (f.)))) ff (x)(
- 1f (z)))
pemax (17 (x:)) 17 (a!)) (z:) - (=)(
jEld
= (xx, - f (x;, 22)
A.3.2. PROOF OF LEMMA 4.4
Proof Sketch: First, with the refined Dudley's entropy integral inequality (Ledent et al., 2021), the Rademacher complexity of the loss function space L can be bounded by the empirical &oo norm covering number. Second, according to the Lipschitz continuity w.r.t the Lo norm, the empirical & norm covering number of F can be bounded by that of P(F). Third, the empirical &oo norm covering number of P(F) can be bounded by the worst-case Rademacher complexity of P(F) through the fat-shattering dimension. Hence, the problem is transferred to the estimation of the worst-case Rademacher complexity. Finally, we estimate the lower bound of the worst-case Rademacher complexity of P(F), and then combined with the above steps, the Rademacher complexity of the loss function space L associated with F can be bounded.
We first introduce the following lemmas:
Lemma A.3 (Khintchine-Kahian inequality (Lust-Piquard & Pisier, 1991)). Let Un E H. where H. is Hilbert space with| || being the associated p-th norm. Let el,..., en be a sequence of independent Rademacher variables. Then, for any p 2 1 there holds
min( VD- - 1,1)
<
E
< maxx( - 1,1)
|u |
iml
and
n
n
Eg
c
2
11
Lemma A.4 (Lemma A.2 in (Srebro et al., 2010)). For any function class F, any S with a finite sample of size n and any Rs(F), we have that
4
(F) <
2
14

---

Generalization Analysis for Controllable Learning
Lemma A.S (Theorem 12.8 in (Anthony & Bartlett 2009), (Lei al., 2023)). any function in class F takes values in [- B, Bl. then for any S with finite sample of size any with (F) < n, have
4eBn 4nB² de
Go (c,F,)) - + doog
log e²
where d = (F))
Lemma A.6 (Refined Dudley's entropy integral inequality (Ledent al., 2021)). Let F be real-valued function class with f < B, f E F, B > O, and assume that 0 F. Let S be finite sample of size n. For any 2 S p S co, we have the following relationship between the Rademacher complexity Rs(F) and the covering number N,,
12 vn
Rs(F)
inf a>0
4a
F,
Step 1:
We first derive the relationship between the empirical so norm covering number No (c, L, D) and the empirical So norm covering number (e, PFF), x D).
For the dataset D ((11111))) (z mn with i.idd examples:
max (((ff((;)) y:) - [ (z:), yal
Su [ fzd (x:) (Use Assumption 4.2)
Samaxmax | (xx)) - 17' (2:;) Sumaxmax by ( (x,)) P3 (x:(())))
(The definition the projection function class P(F))
Then, according to the definition of the empirical loo covering number, we have that an empirical foo cover of P(F) at radius e/ j is also an empirical for cover of the loss function space C at radius e, and we can conclude that:
Noc L, No0 (H
(4)
Step 2: We show that the empirical lo norm covering number P(F) can be bounded by the fat-shattering dimension, and the fat-shattering dimension can be bounded by the worst-case Rademacher complexity of P(F).
According to Lemma A.4, for any (P()))(((FF)), we have
fat, (P(FF) 4nch? Cc) (P(F))
Then, combining with Lemma A.5, for any E (0,2E), we have
S
lo (e,P((F, [c] x D) - + fate// log?
2² TxIDD ( (P(F)
64nc
Se Se nc
< 1 +
log 2
2
2
R4 (P(F))
Ee
< 1 +
log
(5)
2
6
Step 3: According to Assumption 4.1 in the main paper, obtain the lower bound of the worst-case Rademacher complexity (PFF) by the Chintchine-Kahand inequality with p 1:
Ric (P(F))
R (c(xD(P(F))
sup [c)x [c))xx())x>*'
15

---

Generalization Analysis for Controllable Learning
sup Ee x
sup
Gips (f (x:))
py (E)))())))
TCC
1=1 ==1
1 n1
sup [dxDEl<)xx'*
Ee
sup
CjfJJ (ax)
F,EF,
nc
jml
sup
Ee
sup
Eig (5
(1)0))/,..,,,,))))):::444 ne
A
sup
Eel
E 2,,,
ne
i=1 5=1
A
sup
(Use Lemma A.3)
i=1 j=1 (ecee ne v2
E Ej- |1s((d) (;)))))+++
Since (k (y(())) s A, set siipilclee....eee//aaaaaaaaaaaa
vnc So,
AA E
Rno (P(F))
(6)
/2c
v 2nc
Step According Lemma A.6 and combined with the above steps, we have
(C)
AM
< inf a>0
4a +
No
Vn a
12
inf a>0
4a +
V
Nog
P(F), [c] x D)de
(Use inequality (4))
Vn a
p
12
64ncu R2 (F(F)
2eh de 22 (P(F))
inf
4a +
1+
log
(Use inequality (5)
Vn
22
12 M
64ncu R2 (P(F)
inf
4a +
V
11
log 24e c2
Vn
12M
I M c' de
inf
4a +
+
Vn
12M
( Inn M
+ inf
4a
logg In +
Vn
12M
+ (
Vn
(Choose a= / (P(F))
12M
M vn
+ OR (og, 2o In
μE
16

---

Generalization Analysis for Controllable Learning
A.3.3. PROOF OF THEOREM 4.7
Proof Sketch: We upper bound the worst-case Rademacher complexity (P(F), and then combined with Lemma 4.4 and Proposition 4.3 (Le.,, substitute μV into μ in Lemma 4.4), the desired bound can be derived.
We upper bound the worst-case Rademacher complexity (PF(P(F)) as the following:
R Rnn(P(FF)
sup [d]DE|[]xx*
sup
Eg
sup
Gipyy (S (x:))
(E) ne
sup
Ec
sup
Gij (xi)
[[] x
S9E3,
j=1
sup
E.c
sup
Eig (w (10
(6j00/ nc
||,
1=1 j=1
A
n2
sup
Eell
(
ne
A
<
E.
(6 (,)x,))())))1
(Use Jensen's Inequality)
IKG ( nc
1=
==
AA Vnc'
<
sup (m., (aega ne
(0 (5)
(Use Lemma A.3)
1= 1=
Then, combining with Proposition 4.3, we have
12M
+ 96μV Venne (P(F)) +og, (4n M Myn
Rb(C) v
Vn
uE
96pVAA(1 + loga wlwgog/ennene (2 .nn
MV
12M
nE
Vn
Vn
Combining with (3), then
192μV logg 24 In Mva
R(S*) < RD(
24M
nF
+ +3M V
log 2
Vn
Vn
2n
A.3.4. PROOF OF THEOREM 4.9
First, for the surrogate multi-target bipartite ranking loss, by using the U-process technique, we define the empirical Rademacher complexity of loss function space associated with the controllable learning class F over the set of i.i.d disjoint positive and negative sample pairs for each j-th label, then with two-sided multiplicative Chernoff bound, the generalization error can be bounded by RD (f) = E, sup 1 Ei _ ES G, (ff ff where [Pol = 50 the number of positive and negative sample pairs that can be constructed for any label. Second, combining with Lemma 4.4 and Proposition 4.3, have RD, (C) 12M + /Rn x (1 + log, (4es}e² " /
Proof Sketch:
4²) In M /ea
FE
Finally, we upper bound the worst-case Rademacher complexity (P(F)) S 2AA the desired bound can be derived. Vage*
According to the definitions of bipartite ranking problems in controllable learning in Subection 3.2, for the surrogate multi-target bipartite ranking loss, we have
17

---

Generalization Analysis for Controllable Learning
- RD(ff)
-D) - Σ 1 RD((()
Ep; Σ Ep,Ro((*(1) + E [P;RD(ff()- - Ed
- (t((1))
EpS (R($* - Ro((f15) + Σ(r (o)(
ΣpS Ps R( - RD( + Σ l,, $ IM
(8)
Rademacher complexity has proved be a powerful data-dependent measure of hypothesis space complexity. However, since the surrogate multi-target bipartite ranking loss involves pairwise functions, sequence of pairs of i.idd individual observation in (2) is no longer independent, which makes standard techniques in the i.i.d case for traditional Rademacher complexity inapplicable. We convert the non-sum-offi...d pairwise function to sum-of-i.i.d form by using permutations in U-process (Clémençon et al., 2008).
For each task target, we define the empirical Rademacher complexity of loss function space associated with the controllable learning class FF over the of i.i.d disjoint positive and negative sample pairs for the j-th label as follows:
RD (C) Eelj
sup J=EF 8j
1
<la(f=(x ot ad!1))
Eels
sup J=EF sj
eely -1504)))5((11)
(9)
where each e; is an independent Rademacher random variable. The corresponding expected Rademacher complexity is defined as R.y (4) = RD, (4).
We then proof the following lemma:
Lemma A.7. Let dr X' X R be real-valued functions indexed by T where T is some set. If #1,.... Et and ay...., x'y are i.i.d. min {t, u), then for any corvex non-decreasing function 1b,
Exs
sup TET
Σ Or (#))
<
Exb
sup FET
Σ dr (at att) at)). i=
Proof. The proof of this lemma is inspired by (Clémençon et al., 2008).
1
u ar (I:; 2²/)
Enb
sup TET
tu
1=1
EEb
sup
gr [T(i): ((()))((())) (())
TET
u! TTa
8
<Ei
Σ 9r x+))): (x)())= 2l(0)) TT TET
sup
1
(tb is nondecreasing)
Egb
sup
Σ9 9r xx +x()))
(Jensen's 'ss inequality)
TET
1=1
1 8
=Epb
sup
Σ gr (x) x())
rET
8
1=1
18

---

Generalization Analysis for Controllable Learning
According to Lemma A.7 and the symmetrization technique, we can obtain
EDD
sup FAE
R(f - R - EDIEE
2 sup SREF $j f=1
(aln
(10)
by Ei GlR(F (a,t, at as H..
We denote FFEr
The details of inequality (10) are follows:
R(} RD( (1)
EDDU
E
sup FEEF
<'DD
sup f*ET
Σ
2 exx
ok EX₁
<EDDy
sup f*EF
Σ
(R(ff(
(1))))))
(00)) (000) R(F R(S 16))
ty!-ug!
uu!
1
8j (en(f( (0)) xx- R(F*18)) 111)
<EnD
sup JaET ty! uy!
8j
<EDD
ug!.
sup FAEF
(0) 12( 12($
(vb is nondecreasing)
1
ED
( ) (RR(F x)-)) ( -
sup f=EF
8j
(n((f
x(1)) 2 -
(Jensen's 's inequality)
tyl uz!
8j
T--
i=1
1
sup <EDD
S*EF
8j
<EDD
sup P*EF
1
(enlf (aft, xe - ED ( ao!
8j
1
<EDU
sup f*ET
Enls (aft ax[ ED ((((((((())))) (actt axl
8j
(D'lj is the set of samples with the same distribution as D1j)
(alff* (eettxx()) (n (xi xx] 1))
<
EDI.D'I
sup FEEF
(Jensen's inequality)
8j
<EDDDDAAG
sup f=EF
G (en(f*(a)) ad! ln(s (arft oo
((rf"
8)
1
Σ G (x{" .of
+ E
1
1
2 sup JEEF
2 sup JCEF
GRR
84
1
Σ Gln(f (xx x m
sup F=EF
(D'lj with the same distribution as D(j).
First, since the maximum difference caused by replacing one element in Dj or eili 2M ",
according to McDiarmid's
19

---

Generalization Analysis for Controllable Learning
inequality, have
P(I Re, S 2C 2M
Then, according to the tail bound for sub-Gaussian random variables and Theorem 2.1 (Boucheron al., 2013), 2 is sub-Gaussian random variable with variance proxy 16 M² With the definition of the sub-Gaussian random variable, we have 8,
ED s. D I 1R, 12)+ 913
Vt 0.
(11)
Then, for any 6 > 0, we have
P sup R(f*()) RD(F*1)) |1) Elj
f*EF
d (1) 100 c
E RR(MM) E
(Use Markov's Inequality)
ele
E EDiielje22HH
(Use incquality (10) with vxx) = e²x))
ete
2tR2 (C)+ 2ad
(Use inequality (11)).
ele
We set (2)) (2)4 2013
8, then we have
els
32LM2
2R,, (C) +
85
t
Hence, we upper bound the term with probability at least 1-6 8
R(F - RD(
sup JEEF
$2R,, (C) + 32tM²
In
8j
t
2 1
<2R,, + 8M
(12)
3j
Next, we upper bound the term E, 1y 1 First, since ~ Binomial(n, p.), we have E = mpy. With the two-sided multiplicative Chernoff bound, we have
2 Vr E (0, 1).
P(t, - npyl 2 rmp,) S 2e
2ee
20

---

Generalization Analysis for Controllable Learning
Then, have
P
Σ
4
pyl
2r
1
Ug{
pyl rps}
1 psl 2 rpj
(Use Union Bound Inequality)
20
8
2 Pummin Pr
220
An ming ps
3 + min,
We set 20e
6, then we have r
V
Hence, the following holds with probability least 6:
In 2e min, Pj
Σ
psl S
(13)
Since P (122 - p,1 2 rpj)
S (U - p,l 2 rp,))
according to the proof process of inequality (13), we also have
the following holds with probability at least - ō:
tj n
I
pol S Ps n ming
3 30 min ). Similarly, we have by (1 p3)] S (1 p
3 mini,
Solving the above inequality yields tú 2 (1
and solving this inequality yields uj 2 n21 - p1)(1
3lm 4 min, ps
Hence, we have the following holds with probability at least 1-5 5:
8 = min u,} 2 min(np,(1 3 2c min, Pj
3 3 zf
- n.,((1,-))) P.)
mimnn Pj }}.
(14)
In order to ensure that for every j-th label, disjoint positive and negative sample pairs can be constructed, we need to derive the lower bound for min, { to obtain the number of disjoint positive and negative sample pairs that can be constructed for every j-th label. Since
Sj 2 min(s) [sy} min (min (ty, 1y}}
3 In 2e
In 26 min, Pj }}}
2 min min(min(npp(111
}} min(n11 - -P;)(1
min, Pj
3
In
= mmiffmmmppp
n min, Pj n minmi - P;)
V min, Pj
12 2c
3
2 min[
n min Pj. 2 n min(i - p3)} (Assume that n
miny Pj
min minfmin py. min(1 p,)}
2 8o (Define su = n min min fmin p,. min(1 p;)1),
21

---

Generalization Analysis for Controllable Learning
then have 8, 2 200 with probability at least 8.
According to inequality (12) and Union bound Inequality, have the following holds with probability at least 1 8:
2 c
E ps - - Σp Pj 2R,, (C) + 8.M
V
(15)
8j
Combining inequalities (8), (15), (13) and Union bound Inequality, we have the following holds with probability least 6-
R(f*) - RD(f*)
ΣDs Ps
R(f*))) RD -()())))RD((($1)) + Σ
2 M n
In 2c
I 4c 5
$ Σ Pj
2R, (C) + 8M V
8
+ M
$j
nmmn Pj
2
Σ PrRRy + 8M Σ Pj
2 ln
2e
8
M
3 In 4/ n min, Pj
(16)
Next, we transform R. following holds with probability at least 6:
(C) (C) in inequality (16) into RD (4), according to McDiarmid's inequality, is easy to obtain that the
In
R2 (c) (D, M
2sj
With Union bound Inequality, we have the following holds with probability at least 8:
Σ R,RR,,(2 (2)
In +h 28g
Σni Pj (in,(()) + My
In
Σ RD (C) + M Σ Pj
(17)
2sj
Combining inequalities (16), (17), and Union bound Inequality, we have the following holds with probability at least 1 5:
R(F²) - RD(
2 In
2 5
3 In & 8
< Σ Pj PDRD (C) +2M Σ pi
5
+ 8.M Σ Pjj.
+ M1
25j
85
minn Pa
<2v2 Σ pj P. (L) 2M
Σ
In
20
8M Σ
///
41n
4e
M V
3m Se
(Use sj 2 su and n mimpp pj 2 so)
1
80
so
so
2
In Nc
22v2 Σ PiR (2) + 22M Σ
P5
(18)
3
where (L) =B
Be S 1 Eee
(f = Ee supperr 80
1 E ES til, (17 -
22

---

Generalization Analysis for Controllable Learning
Then, according Lemma 4.4 and Proposition 4.3, have
12M
M V
DD. (C)
+ (6c(E (1 logg (Les 12) In
50
We then upper bound the worst-case Rademacher cotplexty (x(F( the following:
Moe (P(F)
sup
R Mc)xD. (P(F)
Do xxxx
=
sup
E
sup Pj (a()))))))P(
EbP, (f (20,, x(i))
Do x
80C i= j=1
80
sup
Eg
sup
Ei (1)( - (3
8pC
j=l
40
sup
EE
sup || Amsll22A
Ezy (, (3,,) (,, [ttp://00000a///1e-0--//////////
80C
A Eell
G (3,,, (1) (x))))
sup
suc
2A
80
sup
(6 (a.,, (2)))
Spc
sup
Ee
( (2., (3 1,
(Use Jensen's Inequality)
[esll/dc....de)lllla///ee/oo//o Suc
1=1
2A
8g
2AA
<
sup
<
(Use Lemma A.3)
(19)
(ks/6//..ryydelllllsssseaaaalled Suc
/SuC'
Then, have
192,AA x (1 + log 2 In Mv30) n
12M
RD. (f) S
/80
/8D
Combining with (18), then
M 384 /84/2quAA (1 log, (4esge (12) .In
24/2Mg
In
R(F*) < RD(
DE + 22M
/50
Vsp
80
where q - -E Pj.
B. General Bounds for Typical Controllable Learning Methods
B.I. Proof of Theorem 5.1
Proof Sketch: First, according to the definition of & norm, we have that the upper bound of (æ,, (z)()) is R + C' for any E [n). j E [c]. Then, using Jensen's Inequality and Khintchine- Kahane inequality, the desired bound can be derived.
First, according to the definitions in Subsection 5.1 and & norm, we have
23

---

Generalization Analysis for Controllable Learning
(a, (z))]] = ||Concale(ar, v=())] == [[Concate(x, a!)ll
1x,, Tel ai......0 1
d'
la 12
i=1
p=11
1a 2
i=1
| + ||
æ| + (z)
Since ||a s R, (=)]0
C for any E [m]. j [c]. have that for any i [n]. [c]:
1,0y (z)) s R + C.
(20)
According to the proof of Theorem 4.7 (i.e., the first five equations in (7)), we have
Rnd (P(F))
sup ne Ee ΣΣ
(21)
1=1
Then, we upper bound the worst-case Rademacher complexity Rnn(P((F) as the following:
Rno(P(FF)
A
sup
Ee
E (o) (y,,,
nC
2Ap
sup
E2
Gy (o; (3)())
(Use Co is p-Lipschitz)
ni
/=1
2pA
sup
Bel
(a 2y)))
(Use Jensen's 's Inequality)
i=1
2pA
sup
Σ
(x 15 (=))11
(Use Lemma A.3)
(x)
ne
2pA(R+ (Use inequality (20)) Vnc
(22)
B.2. Proof of Theorem 5.2
Proof Sketch: Since Sypernetwork-based controllable learning method is two-stage, the hypernetwork is used generate vector of weights for each task target in the first stage, and then these generated parameter vectors are used in the second stage learning. Therefore, the corresponding whole function class is actually denoted as H. = F Lpen G c Since the generated parameter vectors in the first stage are actually used as fixed parameters rather than inputs in the second stage, in order to fully consider the capacity of the hypernetwork corresponding to the first stage, it is necessary and appropriate to define the whole function class as the sum of the funetion classes F+ Ereat o g corresponding to the models of these two stages. Then, the upper bound of the worst-case Rademacher complexity of the whole function class transformed into upper bounding the worst-case Rademacher complexity Rne(( and Rie P(g respectively. Finally, using Jensen's 's Inequality and Khintchine- Kahane inequality, the desired upper bounds can be derived respectively.
24

---

Generalization Analysis for Controllable Learning
Since sopernetwork-basee controllable learning method is two-stage, the hypernetwork used to generate vector of weights for each task target in the first stage, and then the generated parameter vectors used in the second stage learning. Therefore, the corresponding whole function class is actually denoted as H = F + Cetn G Then, we have
Rnn(P(H)
Rne(P(FF + Cetel o gg
Rno RRno(P(F) P(Corel go))
(P(FF + Leul o P(G
Rncc(((F) Rnd (etrrl P(g@))) (Use Theorem 12 in (Bartlett & Mendelson, 2002))
Hence, the upper bound of the worst-case Rademacher complexity for hypernetwork-basee controllable learning methods is transformed into upper bounding the worst-case Rademacher complexity (P((( and o pP((@ respectively.
first upper bound the worst-case Rademacher complexity Rin(P as the following:
A
(P(F))
sup A
Eel
Σ>>
w DIl
(Use inequality (21))
IIs, (w:
nc
i=1 j=1
2Ap
sup (anlaaiw,,llss..i..e
Eell
By (2,; w )l
(Use Çj is p-Lipschitz)
nc
1=1 j=1
2pA
n
sup
Eell Σ
1 (a:; way 1|
(Use Jensen's 's Inequality)
(,,y Raie [ev
nc
1=
1/2
2pA
x
2pAB
sup
Σ
10y w,,; )112
(Use Lemma A.3) (23)
1 (ly
nc
/ n
We then upper bound the worst-case Rademacher complexity o P(g@ as the following:
LEert o 9)
sup Er [c]xZE(c)x2*
sup
Es Eorn (1,))
(x)
nc
im
< sup (=;)EE
BE
E (z)))))
(Use letrl is pete--Lppschizz)
nc
i=1
7
sup (e
2/tetll
Σ
3y (2,))
(Use Lemma A.3)
ne
i=1
1=1
C
Vnc
Hence, we can derive the upper bound of the worst-case Rademacher complexity for hypernetwork based controllable learning methods:
Rne(P( < 2pAB
2pennC
Vnc
Vnc
25

---

Generalization Analysis for Controllable Learning
B.3. Proof of Theorem 5.4
We upper bound the Rademacher complexity (F) of 5 layer FNN the following:
RD(F)= = Ee
sup [EF n
ef (z:)
=Ee
sup
cw nauTole ooWaooWW.. .......-0000000 (W(z)))
SA,0
n2
i=1
1
<AlE, sup n2
<2AE, sup B|P'l
Σ oWa (ReLU activation is I-Lipschitz)
n
i=1
<2ABrE. sup
Go(W, (o[[))--(W[a;))))
i=1
sup <2>ABpB
e WWg ...
B
1 <2AABB.. sup n
B
sup Bn,
<2 <BBABBBe sup
n2 e,
<BBABEB.
sup
||
A2AABBBB sup
(Wi
<BAABB..
sup
c Wi
|| By
<2" <2"ABBE.
25 ABB
Bell Σ
E |
(Use Jensen's 's Inequality)
i=1
25 ABB
|
(Use Lemma A.3)
25 AAB R
B.4. Proof of Theorem 5.6
We first introduce the following lemma:
Lemma B.1 (Corollary A.7 in (Edelman et al., 2022)). For vectors 0,,02 E HR?
|softmax (f1) softmax (02)ll 2 ||0 Ozl1a-
26

---

Generalization Analysis for Controllable Learning
Then, upper bound the Rademacher complexity (F) of the scalar one layer Transformer as the following:
12 6 W (Ww X softmax (X:WQx (cuss))
RD(F)
Ex
sup w VICLaj EJ
n
-Ec
sup
(ww, o (Wy. X softmax (X.W
|| Bro n
<B <.
sup [W Bwe
Wo (Wi X softmax (X, (X)
Bu BWe
Ee sup
Go (Wy x softmax (X, ||
1=1
2 2B Bw Lo Ee
G softmax (X ||
sup Wy.
1=1
2B Bwe
Bwy Lo
Be sup
e softmax (x |I
n
Wax
2B Bw Lo
Be
sup Wo
x |2, (X, W
Wx
QK (ICLs) ||
(Use |Pa|| S 1| ||=)|
n
1=1
2B Bwo BweLB
sup ||softmax
WK Bell
Σ
4 x
12,00
n
Woo
1=1
4B Bwo BwyLo Lo
sup ioo
Σ
xiX
1|2,00
(Use Lemma B.1)
n
4B Bw
Bw Lio
sup max ||l| W
*css W2x
[E|] |Eel
xx,7
1,,00
n
Woo
4 BB BW BW Log
sup max [ || [E ioWk
xx 1|2,00
1=1
AB Bwo Bwy BWox Lo
max x #
C(Css)
Eell Σ
xx, 1,,00
i,
AB. Bw Bw, Bwax La La Eel
G
1|2,00
n
4 Bw BWv BWax Lu R2
max Ee
Gajil
1/2
4 4B Bw Bwy Bwox Lo R²
max
Exl
c
(Use Jensen's Inequality)
AB Bw Bw, Bwax La R22
max
Σ
(Use Lemma A.3)
n
4B Bw BW Bwax R
Vn
27


Reviews:
[{"review_id": "xEcHZu4GTO", "reviewer": "Reviewer_eMeP", "summary": "This work focuses on the generalization analysis of the controllable learning scenario. It establishes a unified theoretical framework and develops a novel vector-contraction inequality for controllable learning based on the Lipschitz continuity of loss functions. \nThe authors first formalize the function class of controllable learning framework, and derive a general Rademacher complexity upper bound for the loss function space based on the Lipschitz continuity and the Rademacher complexity of the controllable learning function class. Based on the general analysis, the authors give the Rademacher complexities and the generalization analyses for Hamming loss and ranking loss that is used in controllable learning. \nFor some specific learning methods, i.e., embedding-based methods and hypernetwork-based methods, the authors give the analyses of the empirical Rademacher complexities respectively. In addition, they also give the Rademacher complexity of the class of feedforward neural networks, graph neural networks and transformers.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your constructive comments and active interest in helping us improve the quality of the paper.\n\nThe following are our responses to the Questions:\n\n**1. Response to Weakness.**\n\nWe 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.\n\n**2. Response to Suggestion.**\n\nWe provide two real-world examples of controllable learning scenarios as follows. \n\n1. 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.\n\n2. 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.\n\nThese 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.\n\n**3. Response to Question.**\n\nFrom 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.", "claims_and_evidence": "This paper mainly considers the generalization ability of the controllable learning scenario. The authors use the empirical Rademacher complexity as the fundamental tool for generalization analysis, which is quite standard. I went through the proofs of the main theorems in this paper, and I think that the proofs are convincing and correct.", "methods_and_evaluation_criteria": "The proposed theoretical results make sense for the problem. This paper studies the Hamming loss and ranking loss for controllable learning, which is widely adopted for this problem. The authors study the generalization ability of these two loss functions based on the Rademacher complexity, which is also a standard tool that is suitable for this problem.", "theoretical_claims": "I have roughly checked the correctness of two main theorems, and I think the proofs are correct. For Theorems 4.5 and 4.7, the proofs basically follow the standard Rademacher analysis for multi-label learning in [Zhang and Zhang, 2021], which is published in ICML 2024. I think the proofs of Theorems 4.5 and 4.7 are correct. For theorems in section 5, which give the Rademacher complexity upper bound for some specific models, e.g., feedforward neural networks, graph neural networks and transformers. The proofs basically follow the upper-bound of the operator norm of each component in deep learning and the Lipschitz continuity of activation functions, which is also quite standard, as in [Neyshabur et al., 2015; Bartlett et al., 2017; Trauger and Tewari, 2024].", "experimental_designs_or_analyses": "This paper focuses on theoretical understanding, and hence no experiment is involved.", "relation_to_broader_scientific_literature": "I think that the key contributions of this paper are important. Since many algorithms for controllable learning is proposed, this paper first establishes a unified framework to analyze the generalization ability of controllable learning algorithms. It gives a formal definition of the function class of controllable learning, and apply the empirical Rademacher complexity to derive the generalization bound, which verifies the effectiveness of existing controllable learning algorithms theoretically.", "essential_references_not_discussed": "I don’t think that the authors miss any important related works. The authors cite most theoretical works of the generalization analysis of machine learning algorithms. Most of the cited works are state-of-the-art results in this area.", "other_strengths_and_weaknesses": "The detailed strengths of this work are as follows:\n1.\tThis paper is well-written and easy to follow. The problems are clearly stated and the key contributions make sense. \n2.\tThe originality of this paper is good. The authors combine the theoretical tool that is used to analyze the generalization ability of multi-label learning to the controllable learning scenario. This work gives the theoretical understanding of existing controllable learning algorithms, and hence have relatively significant impact on this area.\n\nDespite the strengths above, this work has the following weaknesses:\n1. The formulation of controllable learning in Section 3 is a little bit confusing. It could be better if some examples were provided to further explain the function class in Eqn. (2).", "other_comments_or_suggestions": "The clarity of this paper would be further improved if the authors were to give some real-world examples of the controllable learning scenario.", "questions_for_authors": "What is the relationship between controllable learning and multi-label learning? It seems that, from Section 3, controllable learning is a special application of multi-label learning. Can the theoretical results generalize to general multi-label learning scenario?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "B5WtQBzcME", "reviewer": "Reviewer_Gt3V", "summary": "This paper analyzes the generalization of controllable learning methods to understand controllability in trustworthy machine learning better. It establishes a unified and practical framework for the theoretical study of controllable learning methods and proposes a novel vector-contraction inequality that can derive tight generalization bounds for them. In addition, it derives generalization bounds for two typical controllable learning methods: embedding-based and hypernetwork methods. These results reveal the impact of different manipulation methods based on inputs and control functions.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your constructive comments and active interest in helping us improve the quality of the paper.\n\nThe following are our responses to the Questions:\n\n**1. Response to the Weakness and Suggestion.**\n\nAll 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.\n\n**2. Response to the Question.**\n\nAlthough 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.", "claims_and_evidence": "This paper mainly proposes two claims: \n\n1. The generalization analysis of controllable learning needs to address two key challenges, i.e., establishing the relationship between generalization bound and controllability and reducing the dependency of generalization bounds on the number of task targets.\n\n2. Different manipulation methods based on the input and control function will lead to significant differences in the constant $A$ of the generalization bounds in Theorem 4.5 and 4.7.\n\nThe theoretical results in this paper provide clear and convincing evidence for these two claims.", "methods_and_evaluation_criteria": "Although specific evaluation metrics are not available to measure the generalization performance of controllable learning methods, the two metrics analyzed in this paper are commonly used in practice.", "theoretical_claims": "I have checked most of the theoretical proofs, and the proofs' correctness supports the claims made in the main paper.", "experimental_designs_or_analyses": "This is a purely theoretical paper. All the analyses in the Remarks in this paper are reasonable and valid. These analyses are well structured and well supported the claims of this paper.", "relation_to_broader_scientific_literature": "Although existing controllable learning methods have demonstrated empirical success, theoretical research on controllable learning methods is completely unexplored. This paper lays the foundation for theoretically understanding the success of controllable learning methods from the perspective of generalization.", "essential_references_not_discussed": "This paper adequately discusses the related works.", "other_strengths_and_weaknesses": "Strengths:\n\n1) This paper provides a theoretical foundation for understanding the generalization performance of controllable learning methods and for further analyzing a wider range of controllable learning methods. The proposed unified theoretical framework covers two typical controllable learning methods well, and the derived generalization bound has a weak dependency on the number of task targets, which is consistent with the practical situation, that is, it provides a theoretical guarantee that proper controllable learning methods can handle multiple task targets well.\n\n2) This paper proposes a novel vector-contraction inequality specific to controllable learning. This inequality can induce a tight generalization bound with a logarithmic dependency on the number of task targets. It decouples the associations between multiple task targets, consistent with the potentially significant differences between task targets in practice and the losses corresponding to each task target in the weighted Hamming loss.\n\n3) This paper provides generalization analysis for two typical controllable learning methods. The modular decomposition of theoretical results on specific controllable learning methods enhances readability and provides theoretical tools and preliminary results for further analysis in the future. It also provides an interface for subsequent expansion of modular theoretical results on more models. The theoretical techniques of deep models in this paper may be of independent interest.\n\n4) This paper is well written and structured and easy to follow. The motivations and contributions are stated clearly. Each theoretical result is explained and analyzed in detail, making the theoretical results easy to understand. In addition, the generalization bounds of specific controllable learning methods are consistent with the selection of models in practice, which explains the empirical success of existing controllable learning methods.\n\nWeaknesses:\n\nThe theoretical tools and intermediate steps involved in the proofs of theoretical results in the main paper are vague. Although detailed proofs are provided in the appendix, the lack of theoretical details in the main paper may make the theoretical contributions not better presented.", "other_comments_or_suggestions": "I suggest providing proof sketches in the main paper to show the proof process and key theoretical techniques.", "questions_for_authors": "Although specific evaluation metrics for the generalization performance of controllable learning are currently lacking, the question is whether the theoretical results obtained from the two losses involved in this paper are valid without loss of generality, that is, whether the theoretical results obtained from the relevant assumptions can be extended to potentially more evaluation metrics that will appear in the future.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "VMEJJKrxuK", "reviewer": "Reviewer_Rc3E", "summary": "This paper investigates the theory bounds of controllable learning methods, highlighting the need for a deeper understanding of controllable learning methods from a generalization perspective. This paper first gives a formal definition of the general function classes of controllable learning, then develops a novel vector-contraction inequality and derives tight generalization bounds for the general function classes of controllable learning. In addition, this paper analyzes two typical controllable learning methods and derives corresponding generalization bounds for specific methods. These theoretical results reveal the impact of different manipulation functions and control functions on the generalization bounds.\n\n## update after rebuttal", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your constructive comments and active interest in helping us improve the quality of the paper.\n\nThe following are our responses to the Questions:\n\n**1. Response to the Weakness 1.**\n\nThe 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.\n\n**2. Response to the Weakness 2 and Question 1.**\n\nControllable 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.\n\n**3. Response to the Question 2.**\n\nThe 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.\n\n[1] Maksim Lapin, Matthias Hein, Bernt Schiele. \"Top-k Multiclass SVM\", NIPS 2015.\n\n[2] Yunwen Lei, Ürün Dogan, Ding-Xuan Zhou, Marius Kloft. \"Data-dependent generalization bounds for multi-class classification\", IEEE TIT 2019. \n\n[3] Yi-Fan Zhang, Min-Ling Zhang. \"Generalization Analysis for Multi-Label Learning\", ICML 2024.\n\n[4] Shaojie Li, Yong Liu. \"Sharper Generalization Bounds for Clustering\", ICML 2021.", "claims_and_evidence": "This paper investigates the generalization of controllable learning methods for the first time, establishes a theoretical framework for controllable learning, and derives a series of tight generalization bounds. These theoretical results provide strong evidence for the claim that different manipulation and control functions will affect the constants in the generalization bounds.", "methods_and_evaluation_criteria": "Yes. The proposed theoretical analysis method can produce tight theory bounds.", "theoretical_claims": "Yes. I have reviewed the proofs and the theoretical claims are correct.", "experimental_designs_or_analyses": "Yes. The analyses in this paper are complete and all the analyses are valid.", "relation_to_broader_scientific_literature": "Previous research on controllability mainly focused on methods in information retrieval and recommender systems, and the related research lacked theoretical analysis. This work is an important step towards theoretical understanding of the controllability of machine learning from a generalization perspective, and it also explains why controllable learning methods have good generalization properties.", "essential_references_not_discussed": "All the essential related works have been adequately discussed.", "other_strengths_and_weaknesses": "Strengths:\n\n1. This paper is the first work to theoretically analyze controllability. It not only provides a basic framework for the theoretical analysis of controllable learning methods, but also provides some general analysis methods and theoretical tools applicable to typical controllable learning methods, which is a key contribution to a deeper understanding of controllability in machine learning.\n\n2. This paper introduces many valuable theoretical results and techniques, among which the proposed novel vector-contraction inequality may have potential applications in a wider range of problem settings. In addition, the capacity-based theoretical results on deep neural networks can also provide deeper insights into understanding the role of these models in controllable learning methods.\n\n3. The theoretical results in this paper are all rigorously proved, and a lot of analysis and explanations make the theoretical results easy to understand. The assumptions corresponding to the theoretical results are reasonable and consistent with the practical situations, and the argumentation process is logically coherent.\n\nWeaknesses:\n\n1. I noticed that the theoretical results in Section 5 are specific to the weighted Hamming loss, but the multi-target bipartite ranking loss is also analyzed in Section 4. Can the theoretical results in Section 5 be directly applied to the multi-target bipartite ranking loss?\n\n2. Although the theoretical results presented in this paper are important, there seems to be a lack of discussion on how to use these theoretical results to guide the design of controllable learning methods in real scenarios.", "other_comments_or_suggestions": "Typos: Line 421, right column, \"explanation the models\" --> \"explanation for the models\".", "questions_for_authors": "1. Can these theoretical results provide effective guidance for model settings of different modules of controllable learning methods in practical situations?\n\n2. The assumptions corresponding to the proposed vector-contraction inequality seem to be general. Can this lemma be applied to theoretical analysis of other problem settings? Or can it be applied to other loss functions?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Hwd7mO4y3H", "reviewer": "Reviewer_9TGj", "summary": "The paper establishes a statistical generalization theory for controllable learning, which enables machine learning models to dynamically adapt to task requirements during testing. This adaptability is crucial in many real-world applications of decision systems, such as recommender systems and information retrieval, where models need to accommodate diverse user preferences and changing objectives. Despite its empirical success, the generalization performance of controllable learning methods has not been comprehensively analyzed.\n\nTo fill this gap, the authors propose a unified theoretical framework for controllable learning and introduce a vector-contraction inequality, which yields the tightest known generalization bounds. These bounds are largely independent of the number of task targets, except for logarithmic factors. The key mathematical technique involves the development of a novel vector-contraction inequality. The paper then derives generalization bounds for two common controllable learning methods: embedding-based methods, which adjust model inputs, and hypernetwork-based methods, which generate task-specific model parameters. Additionally, the authors present modular theoretical results for frequently used control and prediction functions, offering broad theoretical guarantees for controllable learning methods. The findings enhance the understanding of controllability in machine learning and provide new insights for further research.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your constructive comments and active interest in helping us improve the quality of the paper.\n\nThe following are our responses to the Questions:\n\n**1. Response to Suggestions.**\n\nThe difficulty of theoretical analysis for controllable learning lies in two aspects. \n\nFirst, 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.\n\nSecond, 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: \n\n$$\\mathbb{E}\\left[\\sup_{\\boldsymbol{f} \\in \\mathcal{F}} \\frac{1}{n} \\sum_{i=1}^n \\sum_{j=1}^c \\epsilon_{ij} f_j\\left(\\boldsymbol{x}_{i}\\right)\\right]  $$\n\n$$\\leq  c \\max_j \\mathbb{E}\\left[\\sup_{f_j} \\frac{1}{n} \\sum_{i=1}^n  \\epsilon_{ij} f_j\\left(\\boldsymbol{x}_{i}\\right) \\right].$$ \n\nThe 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.", "claims_and_evidence": "I think the claims are supported by theoretical results and there is no concern in my opinion regarding the theoretical results in this paper.", "methods_and_evaluation_criteria": "N/A. This work is entirely theoretical, which provides a new statistical/mathematical theory for the generalization of controllable machine learning.", "theoretical_claims": "Yes. I read the proof of Lemma 4.4 (the contraction inequality) in Appendix A.3.2, which is claimed to be the main technical tool for theoretical analysis. The proof looks valid. I also look at the proof (sketch) for Theorem 4.5 and 4.7.\n\nI did not check all the details of the proof of the other lemmas in the appendix though.", "experimental_designs_or_analyses": "N/A", "relation_to_broader_scientific_literature": "This paper falls into the greater scientific area of statistical/mathematical theory for generalization of machine learning models and methods. To this end, this paper first develops a new mathematical theorem which is a vector contraction inequality for Rademacher complexity (i.e. Lemma 4.4). It appears to me that this contribution is of independent interest and might be applied to the theoretical analysis of controllable learning by other works. Then, based on this tool, a unified theory on controllable learning is derived in Section 4, which is another potentially extendable theoretical technique.", "essential_references_not_discussed": "This paper describes two major areas of empirical works in controllable machine learning: the embedding based method where the specific tasks are mapped into embeddings, and hypernetwork based method which generates new model parameters on given task requirements.", "other_strengths_and_weaknesses": "Strength:\n\n**1** Clarity: This paper is clearly written, effectively conveying its main ideas. The structure is well-organized, with general theoretical results presented in Section 4 and their application to specific machine learning frameworks in Section 5. Overall, the paper is easy to follow.\n\n**2** Originality: The paper claims to be the first theoretical work on the generalization of controllable learning.\n\nWeakness:\n\n**1** Technical challenge and significance: It is not immediately clear what major technical challenges were encountered in developing the theory in this work. While the paper presents the first generalization theory for controllable learning, this alone does not sufficiently demonstrate that it tackled and resolved a significant technical difficulty.", "other_comments_or_suggestions": "I believe this paper makes a valid contribution to generalization theory. However, there is room for improvement. \n\nI would suggest including a discussion on the major technical challenges in the theoretical analysis. This would clarify what has hindered previous works from establishing a generalization theory for controllable learning using conventional analytical tools. For instance, the discussion could briefly explain why the number of tasks poses difficulties for traditional analysis methods. While the discussion does not need to be long, it should explicitly highlight the obstacles that have made developing a generalization theory for controllable learning challenging and how this paper overcomes them. This would help reinforce the paper's contribution, which is particularly important for a work that claims to be the first theoretical study of this learning method.", "questions_for_authors": "Please see my suggestion regarding major technical novelty in the above session **Comments or Suggestions**.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper provides generalization bounds for "controllable learning", an extension of the multi-label learning setting where at in addition to the sample, the learner also receives a task requirement $z$ at test time. The proofs mostly rely on the construction of auxiliary datasets involving every (input, class) pair, similarly to [MC,MC1,MC2,MC3,ZZ].  Overall, the paper is **quite comprehensive**, tackling several architectures and loss functions, including transformers and even pairwise losses. The original version attempted to tackle GNNs, though in my opinion the results did not make sense.

Reviewers eMeP, Gt3V and Rc3E were very positive about the paper, all giving a score of 4. Reviewer 9TGj gave a score of 3 which was later updated to 4. Although reviewers eMeP, Gt3V and Rc3E claim to have thoroughly read the supplementary, they missed out on several obvious issues in the original submission: **firstly**, there was a minor error in the log factors in Lemma A.5, which was inherited from the previous literature. This has been **fixed** by relying on the correct version from [CRL]. 
**Secondly,** the original submission provided a "Rademacher complexity bound" of a node based graph neural network. Although the authors' answer was originally evasive, after further discussion, we agreed that the bound couldn't be used to prove any excess risk or generalization bound for any well defined learning setting, and the results were **removed (solved)**. **Lastly**, the results for the pairwise learning setting also involved an undefined learning setting, imprecise proofs and a final result which cannot be used directly to prove generalization bounds. This has since been more or less **resolved** thanks to continuous discussion with the authors. 

**Novelty**:  While I don't share the enthusiasm of the reviewers as I personally feel the novelty is limited (compared to [MC,MC1,MC2,MC3]) by the high standards of ICML, the paper still provides original results in multiple scenarios and covers a relatively large scope. 



**Correctness:** I thank the authors for their exceptional efforts fixing the paper during the discussion phase with me. They have **managed to address all of my concerns** in their revised version. Therefore, I am happy to recommend **acceptance**.  
Whilst I believe the three major correctness issues have been solved and the proofs are now correct at a high level, I *strongly advise the authors to thoroughly double check the paper for any remaining inaccuracies* before the camera ready deadline. From the revisions performed during the revision phase, I believe the authors are capable of it. 


----------------------


[MC] Y. Lei, Ü. Dogan, D.-X. Zhou, and M. Kloft, "Data-Dependent Generalization Bounds for Multi-Class Classification", IEEE Transactions on Information Theory 2019. See arXiv https://arxiv.org/pdf/1706.09814

[MC1] L. Wu, A. Ledent, Y. Lei, and M. Kloft, "Fine-grained Generalization Analysis of Vector-Valued Learning", AAAI 2021.

[MC2] W. Mustafa, Y. Lei, A. Ledent, and M. Kloft, "Fine-grained Generalization Analysis of Structured Output Prediction", IJCAI 2021.

[MC3] Hieu et al. Generalization Analysis for Deep Contrastive Representation Learning, AAAI 2025

[ZZ] Zhang and Zhang, Generalization analysis for multi-label learning. ICML 2024.

[CRL] Yunwen Lei, Tianbao Yang, Yiming Ying, Ding-Xuan Zhou, “Generalization Analysis for Contrastive Representation Learning”, ICML 2023.
Decision: Accept (poster)