To Clip or not to Clip: the Dynamics of SGD with Gradient Clipping in High-Dimensions

Authors: Noah Marshall, Ke Liang Xiao, Atish Agarwala, Elliot Paquette
Primary area: optimization
Keywords: gradient clipping, high-dimensional probability, stochastic optimization, deep learning theory

Abstract:
The success of modern machine learning is due in part to the adaptive optimization methods that have been developed to deal with the difficulties of training large models over complex datasets. One such method is gradient clipping: a practical procedure with limited theoretical underpinnings. In this work, we study clipping in a least squares problem under streaming SGD. We develop a theoretical analysis of the learning dynamics in the limit of large intrinsic dimension—a model and dataset dependent notion of dimensionality. In this limit we find a deterministic equation that describes the evolution of the loss and demonstrate that this equation predicts the path of clipped SGD on synthetic, CIFAR10, and Wikitext2 data. We show that with Gaussian noise clipping cannot improve SGD performance. Yet, in other noisy settings, clipping can provide benefits with tuning of the clipping threshold. We propose a simple heuristic for near optimal scheduling of the clipping threshold which requires the tuning of only one hyperparameter. We conclude with a discussion about the links between high-dimensional clipping and neural network training.

Decision: accept (poster)
Reviewer scores: 8, 6, 6, 6

Reviews and discussion:
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creation date:
1730208971169

thread_type:
official_review

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creator: reviewer_2wzk

<content>
summary:
This paper provides a theoretical investigation into the effects of gradient clipping in high-dimensional settings. Specifically, the authors analyze how clipping impacts SGD in terms of stability, convergence, and optimization dynamics. They derive a deterministic equivalent for clipped SGD, named Clipped Homogenized SGD, which approximates the risk evolution via a system of Ordinary Differential Equations (ODEs) under high-dimensional assumptions. Key findings indicate that while clipping does not improve performance with Gaussian noise, it can enhance stability and convergence under other noisy conditions. The paper also proposes a heuristic clipping schedule for near-optimal performance with minimal hyperparameter tuning.

strengths:
1. The study provides a new theoretical perspective on gradient clipping, a widely used technique in training deep learning models, by framing it in high-dimensional analysis. This theoretical foundation is novel and helps bridge a gap in understanding the dynamics of SGD under clipping.

2. The paper is well-organized, with a clear presentation of problem setup, definitions, and theoretical results. The analysis is rigorous, covering a range of noise distributions and demonstrating the proposed model’s effectiveness across synthetic and real-world datasets

weaknesses:
1. The analysis is specific to linear regression, which may limit the generalizability of the findings to other types of models. Insights into extending these results or experiments with more complex models, such as neural networks, could enhance the broader relevance of the work.

2. The descent and variance reduction factors play a central role in understanding the stability and effectiveness of clipping. However, the paper lacks a practical interpretation or intuition for these factors, which might make it challenging for practitioners to apply these insights beyond the linear regression context. Further discussion on the role of these factors in practical scenarios, perhaps with illustrative examples, would enhance accessibility and the broader applicability of the results.

questions:
1. Can the authors provide more insights into how sensitive the results are to the hyperparameter choice in the proposed clipping heuristic? Would small variations in the threshold significantly affect the stability or convergence?

2. Could the authors discuss the potential for extending their theoretical findings beyond linear regression to other models, such as generalized linear models? Are there specific challenges or limitations in applying their stability analysis to non-linear models or more complex loss functions?
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creation date:
1730327523549

thread_type:
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creator: reviewer_g87a

<content>
summary:
The paper studies the widely used gradient clipping technique in SGD under quadratic settings. Under specific assumptions, the paper develops a theoretical analysis of the training dynamics by proposing an SDE C-HSGD and proving its connection to clipped SGD. Based on this, the paper shows that when clipping can help convergence and stability, providing insights into understanding the clipping technique.

strengths:
1. To my knowledge, this is a novel perspective looking into clipping, which is interesting and aligns with the common belief that clipping helps when noise is heavy-tailed.

2. The theory seems to be clear and complete.

weaknesses:
1. Although the theory looks promising, I think it's still restrictive due to the settings it applies: 
    - As the author also mentioned in the paper, one of the key benefits of clipping is that it helps SGD converge in generalized smooth settings, which cannot be reflected by quadratics.
    - The theory typically requires the data $x$ to be Gaussian, or close to Gaussian as in Appendix A. This is still a restrictive assumption. 

2. I kindly suggest the authors gather more notation definitions in Section 2, as there are still new notations in the following sections, which may make it inconvenient to follow and refer to when reading.

questions:
1. More intuitive explanations for the connection between C-SGD and S-HSGD are welcomed.

2. As also noted in the weakness part, I think possibly the authors can discuss how the analysis in the Gaussian quadratic settings may have impacts on broader machine learning applications.

3. Do you also take stability into consideration when comparing C-SGD and SGD in Section 5?
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<thread>
creation date:
1730617276138

thread_type:
official_review

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creator: reviewer_dt4j

<content>
summary:
The paper studies the properties of clipped SGD for linear regression with mean squared error loss under specific data and noise distributions. The authors compare the parameters learned from clipped SGD (C-SGD) to the solution of a stochastic differential equation (C-HSGD), demonstrating their close alignment in terms of risk and optimality gap. They also derive an ODE to describe risk evolution. Additionally, learning stability (CSC) and instantaneous risk descent (CCC) are analyzed through descent and variance reduction factors. The authors show that under Gaussian noise, clipped SGD is inferior to unclipped SGD, but under other noise distributions, it improves both performance and stability. Theoretical results are supported by experiments.

strengths:
The paper is very well-written, and the problem it addresses is important for large-scale learning settings.  Particularly, gradient clipping is an important  technique that proves useful in deep learning, which makes it crucial to understand its effects. This paper provides a rigorous characterization of clipped SGD in linear regression, focusing on the descent and variance reduction factors. The authors also propose a learning rate schedule that ensures clipped SGD performs at least as well as standard SGD, offering practical insights for optimization.

weaknesses:
* The problem setting is restricted to linear regression with mean squared loss  and the proposed heuristic seems to be meaningful only when the features follow a Gaussian distribution.

* Calculating the descent and variance reduction factors in more general settings appears challenging, since it involves numerical optimization at every step, which limits the practical applicability of the proposed learning rate scheduling and criteria.

questions:
* In the conclusion, the authors suggest extending the analysis to more complex losses. What are the bottlenecks for such an analysis?
* The heuristic presented in the paper appears to be limited to normally distributed data due to the validity of Eqs. (10a) and (10b). Is it possible to extend this approach to a more general setting, such as sub-Gaussian distributions?
* While the authors used real-world data to validate the C-HSGD approximation, they did not evaluate the heuristics on real-world data (though the setup for Figure 4 is somewhat unclear, please correct me if I am mistaken). Could the authors clarify why this aspect was not explored?

Minor: 
* In Theorem 1, the subscript for $\theta_{sd}$  in the denominator of the definition of $\mathcal{E}$ should include a ceiling (or floor).
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<thread>
creation date:
1730770134165

thread_type:
official_review

<thread_object>
creator: reviewer_cyxq

<content>
summary:
This paper analyzes SGD with gradient clipping by constructing an analogous Stochastic Differential Equation (SDE) and studying its solution. The SDE developed by the authors follows the development in [1] with the addition of two factors $\mu_c$ and $\nu_c$ that capture the effect of gradient clipping. The authors show that their SDE is close to the SGD trajectory and also conduct a stability analysis to obtain a criterion for when clipped SGD is more or less stable than its unclipped version (and allows for larger learning rates). The authors study different noise conditions (Gaussian and Rademacher-type) noise and show that the clipping threshold can be set to obtain larger learning rates. Finally this paper explores the conditions under which clipped SGD can achieve lower risk values than unclipped SGD.

[1] Paquette, C., Paquette, E., Adlam, B., & Pennington, J. (2022). Homogenization of SGD in high-dimensions: Exact dynamics and generalization properties. arXiv preprint arXiv:2205.07069.

strengths:
The theoretical analysis seems strong and the authors provide extensive technical proofs for their theoretical results. While I did not go through all the proofs to verify their correctness, the analysis seems correct. The authors go beyond just showing a correspondence between clipped SGD and the C-HSGD stochastic differential equation to provide some applications of their result to stability analysis and to study when clipping helps in SGD.

weaknesses:
1. The theoretical analysis seems to mostly be an application of previous results obtained in [1]. The new machinery that is used here is Stein's lemma to obtain bounds on the quantities $\mu_c$ and $\nu_c$ that characterize the effects of clipping SGD. This only allows the authors to characterize $\mu_c, \nu_c$ for gaussian data, even though they are able to show the correspondence of the SDE to clipped SGD with non-gaussian data. The technical contribution is arguably only incremental.

2. The condition for testing whether clipped SGD reaches lower risk values (CCC) seems to be hard to evaluate. It involves estimating whether $\mu_c^2 / \nu_c > 1$ anywhere along the trajectory of the SDE. This seems to require solving the SDE to estimate whether clipped SGD is beneficial, as well as estimating the threshold $c$ at which to clip the gradients. This may be harder than simply solving the original least squares optimization problem.

3. While the authors explore how their results extend to the setting of training deep networks, the arguments are made in the regime where deep networks can be approximated well by linear models (the NTK regime). NTK models do not perform as well as deep networks that learn features and that cannot be approximated as linear models.

[1] Paquette, C., Paquette, E., Adlam, B., & Pennington, J. (2022). Homogenization of SGD in high-dimensions: Exact dynamics and generalization properties. arXiv preprint arXiv:2205.07069.

questions:
1. The role of CSC is still unclear to me. Can the authors provide a demonstration of how clipped SGD allows for larger learning rates. None of the figures seem to illustrate how clipped SGD converges faster than unclipped SGD. In case I have missed this please direct me to where I can find this.

2. While there is a figure demonstrating the correspondence of clipped SGD and the SDE with heavy tailed Cauchy noise, there are no experiments comparing unclipped and clipped SGD in that condition. Does clipped SGD converge faster in that situation? Can this be demonstrated in an experiment?
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creation date:
1731702164571

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creator: authors

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title:
Global Response to Reviewers

comment:
We would like to thank all Reviewers for their constructive feedback. We appreciate the opportunity to address concerns raised and clarify our contributions. Below, we summarize and address the main recurring concerns across the reviews:


**Generality of results beyond linear regression**  

   Several reviewers noted that our analysis is limited to linear regression with mean squared error loss, raising concerns about the broader applicability of our findings.

We intentionally focused on this simple setting to provide a solid and rigorous foundation for building intuition about gradient clipping. Understanding optimization under a quadratic loss serves as a baseline to sanity-check more complicated losses.

There is a clear path to analysis on non-quadratic losses; for ordinary SGD, [1] provides a way to extend high-dimensional analysis to a broad family of loss functions including cross-entropy loss; analogous methods should work for clipped SGD. This approach would also allow analysis of some non-linear models, including certain classes of neural networks. The work in [1] is extremely technical; therefore, we chose to establish the theory in the simpler case first, in order to better inform the generalization of the methodology to more complicated losses.

We also believe that our insights are relevant for more realistic non-linear models. Understanding the connections will require a detailed empirical study. We are encouraged by the fact that our key quantities — the mean and variance reduction factors $\mu$ and $\nu$ — have tractable estimators in the full non-linear setting (Appendix C.2). We leave this study for future work.

**Assumptions on the data**  

Some reviewers raised concerns about our data modeling assumptions. The majority of our analysis holds over subgaussian data — a large class which contains many distributions of interest, such as those with finite support, including tokenized NLP data with finite vocabulary.

In our original submission we provided experimental evidence that our results are informative for real data in both vision and NLP contexts (Figure 1) — as expected since those data are subgaussian. Reviewers expressed concern that our heuristics for optimal clipping depended on the Gaussian formulation of the mean reduction factor $\mu$. We have added additional experiments (Appendix C.4) which show that our optimal clipping heuristic is also effective for real, non-Gaussian data. 



**Practical utility of descent and variance reduction factors**  
   
There was also concern about the interpretability and practical utility of the descent ($\mu$) and variance reduction ($\nu$) factors. The interpretation is found in lines 155-161 of our paper and can be summarized as follows:
- The mean reduction factor ($\mu$) measures how much, in expectation, clipping reduces the length of the gradient.
- The variance reduction factor measures how much, in expectation, clipping reduces the variance of the norm-squared of the gradient.
In our setting we also derived simple upper and lower bounds of these factors, with respect to the loss and noise statistics which are easy to calculate and interpret (Equations 10a, 10b). This gave a heuristic for the optimal clipping schedule (Equation 25). Following reviewer comments we have added evidence that the heuristic is relatively _insensitive_ to the constants in the bounds (Appendix C.4).

In our original submission we also included a methodology to compute the reduction factors in non-linear settings using an online estimator (Appendix C.2, ResNet18 and ViT). We neglected to mention that this estimator can track $\mu_{c}$ and $\nu_{c}$ for many $c$ values in parallel at virtually no cost, which improves its utility for choosing a good instantaneous clipping threshold.




_(Revisions in the paper are highlighted in red so they may be seen easily)_


[1] Collins-Woodfin et al., "Hitting the high-dimensional notes: An ode for SGD learning dynamics on GLMs and multi-index models," 2023.
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creator: authors

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comment:
We noticed a minor bug in our code for the heuristic optimal clipping schedule. As a result we have updated Figure 4. This has changed the details of the figure but the overall message remains unaffected.
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title:
Any further questions?

comment:
As the rebuttal period is nearing its conclusion, please let us know if there are any other points we can clarify!
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creation date:
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creator: authors

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title:
Thank you to the reviewers

comment:
We'd like to thank the reviewers for taking the time to review our work.

We appreciate the reviewers' positive feedback and recognition of the strengths and contributions of our work. Comments largely highlighted minor points, and we are pleased that no substantial changes were necessary. 

Our manuscript has been updated with one experiment verifying the effectiveness of our heuristic clipping schedule on real world data, further validating our findings. Thank you for your constructive feedback and for acknowledging the clarity and utility of our analysis of clipped SGD.
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Metareview:
metareview:
The paper studies the properties of clipped SGD for linear regression with mean squared error loss under specific data and noise distributions. The authors have addressed all the reviewers' concerns. All the reviewers are happy to recommend the acceptance of the paper. Please incorporate all the comments and discussions into the final version.