Optimizer-Dependent Generalization Bound for Quantum Neural Networks

Authors: Chenghong Zhu, Hongshun Yao, Yingjian Liu, Xin Wang
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)
Keywords: Quantum Machine Learning, Quantum Neural Networks, Stability, Generalization

Abstract:
Quantum neural networks (QNNs) play a pivotal role in addressing complex tasks within quantum machine learning, analogous to classical neural networks in deep learning. Ensuring consistent performance across diverse datasets is crucial for understanding and optimizing QNNs in both classical and quantum machine learning tasks, but remains a challenge as QNN's generalization properties have not been fully explored. In this paper, we investigate the generalization properties of QNNs through the lens of learning algorithm stability, circumventing the need to explore the entire hypothesis space and providing insights into how classical optimizers influence QNN performance. By establishing a connection between QNNs and quantum combs, we examine the general behaviors of QNN models from a quantum information theory perspective. Leveraging the uniform stability of the stochastic gradient descent algorithm, we propose a generalization error bound determined by the number of trainable parameters, data uploading times, dataset dimension, and classical optimizer hyperparameters. Numerical experiments validate this comprehensive understanding of QNNs and align with our theoretical conclusions. As the first exploration into understanding the generalization capability of QNNs from a unified perspective of design and training, our work offers practical insights for applying QNNs in quantum machine learning.

Decision: reject
Reviewer scores: 5, 5, 8, 6, 6

Reviews and discussion:
<thread>
creation date:
1730180662922

thread_type:
official_review

<thread_object>
creator: reviewer_d18t

<content>
summary:
This paper explores the generalization properties of quantum neural networks (QNNs) with data re-uploading strategy. Concretely, they leverages the quantum combs formalism and data re-uploading structure to analyze the stability of QNNs and then give the upper bound of generalization error on such model. They also provide numerical experiments to support the proposed theoretical claims.

strengths:
1. provide rigorous analysis of generalization performance based on stability of SGD and quantum combs, the upper bound gives some insight to select the hyper-parameters of QML model with data re-uploading.
2. give some basic numerical experiments to support the theoretical claims.

weaknesses:
As the theoretical analysis and numerics only focuses on the specific ansatz structure (with data re-uploading strategy), it is unclear that the similar theoretical findings also true for other QML models.

questions:
I have no further questions.
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</thread>
<thread>
creation date:
1730200518708

thread_type:
official_review

<thread_object>
creator: reviewer_2ekn

<content>
summary:
The authors derive a generalization bound of the data re-uploading QNN trained using SGD. They first show the data re-uploading QNNs can be represented by a sequential quantum comb, and study the stability of the data re-uploading QNNs via this representation.

strengths:
Understanding behaviors of quantum machine learning models is important. This paper focuses on the combination of a quantum machine learning setting and SGD, and analyzes the generalization gap based on the stability. The topic is interesting and relevent to the community.

weaknesses:
I have several questions and concerns, which are listed below:

- In eq. (10) implies that if we choose a proper hyper parameters, then the generalization gap becomes small as the iteration of SGD proceeds. In many cases, the generalizaiton gap becomes large as the learning process proceeds in an early stage due to the overfitting. On the other hand, the generalization gap may become small after a sufficiently long learning process due to the benign overfitting. Could you comment on the relationship between the proposed generalization bound and this type of practical situations?

- In line 383, the authors instst "the linear dependence of data dimension and data reuploading time not only offers extra insights into understanding the QNN model’s performance but also provides insights into the design of these models". Is it possible to determine an optimal $L$ based on the tradeoff between the expressivity and generalization? Could you explain more about how we can design the models based on the proposed generalization bound?

questions:
Minor comments:
- In eq. (6), the number of parameters are denoted as $K$, but the expranation of this notation is in Theorem 3.
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</thread>
<thread>
creation date:
1730713323179

thread_type:
official_review

<thread_object>
creator: reviewer_cm7s

<content>
summary:
This paper attempts to estimate the generalization error bound for Quantum Neural Networks (QNNs) in terms of the number of trainable parameters, data uploading times, dataset dimension and classical optimizer hyperparameters by converting the QNN into a quantum comb structure and by leverage the uniform stability property of the stochastic gradient descent algorithm. The authors then validate these bounds by means of numerical experiments with the Breast Cancer, MNIST and Fashion MNIST datasets by tracking the generalization error being estimated as the absolute difference between the train and test error across epochs for different values of data uploading times and learning rates. The analysis suggests that a larger learning rate should be balanced by a smaller learning rate to ensure that each step in the learning is small enough to prevent instability that could arise in more complex models and that the learning rate should be set $\mathcal{O}(1/K)$, where $K$ is the fixed number of learning parameters or the number of  should be set as $O(1/\eta)$ for a fixed learning rate.

strengths:
The main strength of this paper is that it provides useful guidelines to tune the architecture and learning hyperparameters for QNNs based on the general quantum comb architecture. Each quantum comb is equivalent to a causal QNN. The Choi operator of the quantum comb is the Choi operator of the corresponding causal QNN. Corollary 4 is a useful result that ties all the relevant parameters including number of trainable parameters, data uploading times, dataset dimension and classical optimizer hyperparameters to provide a generalization error bound with SGD-based optimization.

weaknesses:
The main weakness of the paper is the slightly limited experimental evaluation. The choice of datasets are not clearly motivated and only a limited number of values for data uploading times $L$ and the learning rate $\eta$ are tried out, so it is difficult to assess from the plots where we move from stable regime to the unstable regime (see the plot for Fashion MNIST in Figure 1 for an example). Also, the error bars are very large for experiments that vary the learning rate.

questions:
1. Why did you choose the datasets you chose for the experimental evaluation? It would be good to have a table that shows the datasets chosen with the number of training examples.
2. Have you considered the complexity of the learning task (eg. lack of easy separability between the classes) or the number of features in your data as affecting your generalization bounds? How does it factor into your bound provided in Corollary 4 or is it independent of that?
3. For the Fashion MNIST dataset experiment, you may wish to try more values of $L$ for Figure 1 and $\eta$ for Figure 2 because there is rather ubrupt transition from stability to unstability from $L = 2$ to $L = 8$ and from $\eta = 0.01$ to $\eta = 0.05$.
4. The error bounds are rather large for the experiments that vary the learning rate (Figure 2), making the conclusions you are drawing somewhat untenable, you may wish to run more trials to reduce the size of your error bars. I understand you reran 5 times but that maybe too little for some of the experiments. Why did you not do this?
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</thread>
<thread>
creation date:
1730715820855

thread_type:
official_review

<thread_object>
creator: reviewer_x6gc

<content>
summary:
This paper investigates the generalization properties of quantum neural networks (QNNs) through the lens of learning algorithm stability. The authors establish a connection between quantum combs—a framework for organizing variable subcircuits—and data re-uploading, a technique for iteratively encoding data within quantum circuits. They derive an upper bound on the generalization error, relating it to the number of re-uploading layers, dataset size, parameter count, learning rate, and optimization steps. Experimental results are provided to support the theoretical findings.

strengths:
1. This study initiates the attempt to establish the optimization-dependent generalization error bound of quantum neural networks by analyzing the stability of optimizing quantum neural networks (QNNs) with SGD.
2. The derived generalization bound offers practical insights for selecting hyperparameters, such as recommending a smaller learning rate when using a large number of parameters, which could be beneficial for optimizing QNNs.
3. Numerical experiments are conducted, providing empirical support for the theoretical results and suggesting that the bounds are valid under various settings.

weaknesses:
1. While this paper presents a useful framework for optimization-dependent generalization error bounds, its novelty is somewhat limited in both theoretical derivation and implications. Specifically, two of the three main implications have already been observed in Ref. [1], and the third can be derived from Ref. [2], which discusses model stability under different convexity conditions—a result that may be applied to quantum settings as well.
2. Although it is interesting to establish the connection between the quantum comb and the QNNs with data-reuploading structure, the discussion of this connection seems to be separated from the derivation of the generalization error bound. I mean by this the derivation of the generalization error bound does not employ any necessary properties of the quantum comb. This raises questions about the added value of quantum combs in the context of this study.

[1]. Caro, Matthias C., et al. "Encoding-dependent generalization bounds for parametrized quantum circuits." Quantum 5 (2021): 582.
[2]. Hardt, Moritz, Ben Recht, and Yoram Singer. "Train faster, generalize better: Stability of stochastic gradient descent." International conference on machine learning. PMLR, 2016.

questions:
Could you clarify the role of quantum combs in the derivation of the generalization bounds, specifically whether they are essential to the analysis or if the same results could be obtained without them?
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</thread>
<thread>
creation date:
1730848787426

thread_type:
official_review

<thread_object>
creator: reviewer_893s

<content>
summary:
In this paper, the authors provide generalization bounds using the _stability_ approach for a family of quantum neural networks (QNNs) using data-reuploading strategy and stochastic gradient descent (SGD) as an optimizer by conceptualizing QNNs as a special form of quantum combs. The generalization bound number of trainable parameters, number of times data is reuploaded, dataset dimension, and classical optimizer-dependent parameters, providing guidelines for designing and training QNNs. Numerical experiments to support theoretical claims have been conducted on three classical datasets; e.g. MNIST, Breast Cancer, and FMNIST.

strengths:
This work makes an interesting connection of QNNs to quantum combs to leverage the rich theoretical framework of the latter to analyze the dynamics of QNNs, although a brief connection of quantum combs to QNNs using data re-uploading strategy was mention in [1].

Provides theoretical guidelines on designing and training near-term quantum machine learning (QML) models--a fundamental question in QML.

[1] Mo, Yin, et al., "Parameterized quantum comb and simpler circuits for reversing unknown qubit-unitary operations", arXiv:2403.03761

weaknesses:
- Limited novelty in theoretical analysis: In my opinion, the paper heavily borrows proof strategies and techniques from [2] which studies SGD for smooth, Lipschitz and convex problems for deep neural networks. The only novelty is the connection of data reuploading QNNs to quantum combs and then using well-established lemmas from the rich theory of quantum combs.

- The generalization bounds based on stability especially that of [2] becomes too loose and vacuous as training progresses, even in the practical regime of the training time where deep neural networks are known to generalize well [3]. I believe that as the generalization bound essentially uses the same proof techniques and the technical setting is the same, your bound might also undergo the similar phenomenon. This is a limitation which should be highlighted in the work. 

- The work also lacks analysis on the robustness of the generalization bound in presence of standard quantum noise models such as depolarizing noise, state preparation and measurement error (SPAM), thermal relaxation, readout error etc. It would be beneficial if the authors can provide analysis of their generalization bounds for one of the above noise models, e.g. for depolarization noise.

**Minor Issues**:

- Line 161: the definition of CX gate is incorrect.

- Line 164: A more of a pedantic take, apologies if the authors do not agree with me on this: as QNN is a variational quantum algorithm (VQA), I think they comprise of 3 things: ansatz, observable and the classical optimizer. I say this because the design of observables also is related to the presense of barren plateaus. For instance, [4] proves that when you have local observables there is a possibility of absence of barren plateaus (BPs) while for global observables, they prove the presence of BPs.

- In Fig. 2 caption and in the surrounding text, the authors mention the use angle encoding as a classical data encoding scheme, however, this is not correctly depicted in Fig. 2 as there are two RY gates! Please correct this to avoid any confusion.

- Please add reproducibility statement according to the author guidelines of ICLR.


[2] Hardt, Moritz et al., "Train faster, generalize better: Stability of stochastic
gradient descent", ICML (2016)

[3] Kawaguchi et al., "Generalization in Deep Learning", CUP (2022)

[4] Cerezo, Marco et al., "Cost Function Dependent Barren Plateaus in Shallow Parametrized Quantum Circuits", Nat. Comms. (2021)

questions:
- Can your proof techniques be used for proving generalization bounds for data re-uploading QNNs where input-scaling parameters are also trainable? That is, the output function of a data re-uploading QNN is $f(\mathbf{\theta}, \mathbf{\lambda}, \mathbf{x}, \mathcal{M})$, where $\mathbf{\lambda}$ are input-scaling parameters used when data is being re-uploaded and are trainable, e.g. data encoding operation is $U(\mathbf{x}, \mathbf{\lambda})$ instead of $U(\mathbf{x})$. In your current analysis, $\mathbf{\lambda} = \mathbf{1}$ and is not trainable. I ask this because the function $f(\mathbf{\theta}, \mathbf{\lambda}, \mathbf{x}, \mathcal{M})$ is more general and recovers your defintion of $f(\mathbf{\theta}, \mathbf{x}, \mathcal{M})$ and can in principle do a bit more than your data reuploading model.

- Does your definition of stability hold also for memorization algorithm[5]--a learning algorithm which always outputs the function $f(\mathbf{x}_i, \mathbf{\theta}_S, \mathcal{M}) = -1$, except that it alters $f$ in a small number of locations to fit the training set?

- Can you provide any numerical evidence on the how tight is your analysis of generalization bound?

- What properties (or hyperparameters of QNNs) and operations (regularization, weight decay, gradient clipping etc.) induce stability?

- DNNs with random initialization tend to behave like random Gaussian processes (in limit of many neurons)[6] which allows for their characterization of their trainability and generalization. However, this is not the case for randomly initialized QNNs as they instead generally form Wishart processes[7] which give rise to barren plateau phenomenon and exponentially many local poor minima. That being said, does your generalization bound result help to mitigate these issues and if yes, how?

- Why is there a significant difference between the generalization error reported in Fig. 3 with settings [L=8 (purple or violet) with $\eta=0.01$] and Fig. 4 with settings [L=8 with $\eta=0.01$ (red)]. The generalization error should be more or less similar which is not the case if I look at the plots. Can you please explain this phenomenon?

- In the numerical experiments to support your theoretical claims, why do you use parametrized quantum circuits (PQCs) that are not universal and classically simulable? This completely defeats the purpose of using QML, no? Because at the end, we want a quantum model (specified by a PQC) which is expressive (for the domain), trainable, noise robust and can not be classically simulable--holy grail of QML.



[5] Chatterjee, Satrajit, "Learning and Memorization", ICML (2018)

[6] Lee et al., "Deep neural networks as Gaussian processes", ICLR (2018)

[7] Anschuetz, Eric, "A Unified Theory of Quantum Neural Network Loss Landscapes", arXiv:2408.11901
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Metareview:
metareview:
The paper explores the generalization performance of QNNs with a data re-uploading structure through the framework of algorithmic stability. The authors derive generalization error bounds using the uniform stability of the stochastic gradient descent algorithm and attempt to establish connections between QNNs and quantum combs. Simulation results provide partial validation of the theoretical findings. The submission received mixed reviews with varying scores and confidence levels. While reviewers acknowledged the significance of the topic and its potential insights, concerns about the technical contributions and the relevance of quantum combs limited the paper's overall impact.