# Limitations of Information-Theoretic Generalization Bounds for Gradient Descent Methods in Stochastic Convex Optimization

arXiv: https://arxiv.org/abs/2212.13556 · HTML: https://mlsanity.com/paper/2212.13556/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- Builds on 44 of its 74 references that are in the map.
- First posted on arXiv 27 Dec 2022.

## Builds on (most cited first, 30 of 44)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Sharpness-Aware Minimization for Efficiently Improving Generalization](https://mlsanity.com/paper/2010.01412/) (2020)
- [Convex Optimization: Algorithms and Complexity](https://mlsanity.com/paper/1405.4980/) (2014)
- [Train faster, generalize better: Stability of stochastic gradient descent](https://mlsanity.com/paper/1509.01240/) (2015)
- [Computing Nonvacuous Generalization Bounds for Deep (Stochastic) Neural Networks with Many More Parameters than Training Data](https://mlsanity.com/paper/1703.11008/) (2017)
- [Online Learning: A Modern Introduction Using Convex Optimization](https://mlsanity.com/paper/1912.13213/) (2019)
- [A PAC-Bayesian Approach to Spectrally-Normalized Margin Bounds for Neural Networks](https://mlsanity.com/paper/1707.09564/) (2017)
- [Information-theoretic analysis of generalization capability of learning algorithms](https://mlsanity.com/paper/1705.07809/) (2017)
- [Stability of Stochastic Gradient Descent on Nonsmooth Convex Losses](https://mlsanity.com/paper/2006.06914/) (2020)
- [Reasoning About Generalization via Conditional Mutual Information](https://mlsanity.com/paper/2001.09122/) (2020)
- [Information-Theoretic Generalization Bounds for SGLD via Data-Dependent Estimates](https://mlsanity.com/paper/1911.02151/) (2019)
- [Tightening Mutual Information Based Bounds on Generalization Error](https://mlsanity.com/paper/1901.04609/) (2019)
- [High probability generalization bounds for uniformly stable algorithms with nearly optimal rate](https://mlsanity.com/paper/1902.10710/) (2019)
- [How much does your data exploration overfit? Controlling bias via information usage](https://mlsanity.com/paper/1511.05219/) (2015)
- [Sharpened Generalization Bounds based on Conditional Mutual Information and an Application to Noisy, Iterative Algorithms](https://mlsanity.com/paper/2004.12983/) (2020)
- [Information-Theoretic Generalization Bounds for Stochastic Gradient Descent](https://mlsanity.com/paper/2102.00931/) (2021)
- [Generalization Error Bounds for Noisy, Iterative Algorithms](https://mlsanity.com/paper/1801.04295/) (2018)
- [Entropy-SGD optimizes the prior of a PAC-Bayes bound: Generalization properties of Entropy-SGD and data-dependent priors](https://mlsanity.com/paper/1712.09376/) (2017)
- [On the role of data in PAC-Bayes bounds](https://mlsanity.com/paper/2006.10929/) (2020)
- [On Generalization Error Bounds of Noisy Gradient Methods for Non-Convex Learning](https://mlsanity.com/paper/1902.00621/) (2019)
- [Generalization Bounds for Uniformly Stable Algorithms](https://mlsanity.com/paper/1812.09859/) (2018)
- [Information-theoretic generalization bounds for black-box learning algorithms](https://mlsanity.com/paper/2110.01584/) (2021)
- [In Defense of Uniform Convergence: Generalization via derandomization with an application to interpolating predictors](https://mlsanity.com/paper/1912.04265/) (2019)
- [Tighter expected generalization error bounds via Wasserstein distance](https://mlsanity.com/paper/2101.09315/) (2021)
- [The intriguing role of module criticality in the generalization of deep networks](https://mlsanity.com/paper/1912.00528/) (2019)
- [SGD Generalizes Better Than GD (And Regularization Doesn't Help)](https://mlsanity.com/paper/2102.01117/) (2021)
- [PAC-Bayes, MAC-Bayes and Conditional Mutual Information: Fast rate bounds that handle general VC classes](https://mlsanity.com/paper/2106.09683/) (2021)
- [On the Generalization of Models Trained with SGD: Information-Theoretic Bounds and Implications](https://mlsanity.com/paper/2110.03128/) (2021)
- [Individually Conditional Individual Mutual Information Bound on Generalization Error](https://mlsanity.com/paper/2012.09922/) (2020)
- [Towards a Unified Information-Theoretic Framework for Generalization](https://mlsanity.com/paper/2111.05275/) (2021)

## Built on by (most cited first, 19 of 19)

- [Tighter Information-Theoretic Generalization Bounds from Supersamples](https://mlsanity.com/paper/2302.02432/) (2023)
- [Information Theoretic Lower Bounds for Information Theoretic Upper Bounds](https://mlsanity.com/paper/2302.04925/) (2023)
- [More PAC-Bayes bounds: From bounded losses, to losses with general tail behaviors, to anytime validity](https://mlsanity.com/paper/2306.12214/) (2023)
- [Information Complexity of Stochastic Convex Optimization: Applications to Generalization and Memorization](https://mlsanity.com/paper/2402.09327/) (2024)
- [Data-dependent Generalization Bounds via Variable-Size Compressibility](https://mlsanity.com/paper/2303.05369/) (2023)
- [Lessons from Generalization Error Analysis of Federated Learning: You May Communicate Less Often!](https://mlsanity.com/paper/2306.05862/) (2023)
- [Sample-Conditioned Hypothesis Stability Sharpens Information-Theoretic Generalization Bounds](https://mlsanity.com/paper/2310.20102/) (2023)
- [Select without Fear: Almost All Mini-Batch Schedules Generalize Optimally](https://mlsanity.com/paper/2305.02247/) (2023)
- [Exactly Tight Information-Theoretic Generalization Error Bound for the Quadratic Gaussian Problem](https://mlsanity.com/paper/2305.00876/) (2023)
- [Generalization Guarantees via Algorithm-dependent Rademacher Complexity](https://mlsanity.com/paper/2307.02501/) (2023)
- [Generalization Bounds via Conditional f-Information](https://mlsanity.com/paper/2410.22887/) (2024)
- [Generalization Guarantees for Representation Learning via Data-Dependent Gaussian Mixture Priors](https://mlsanity.com/paper/2502.15540/) (2025)
- [Leveraging Flatness to Improve Information-Theoretic Generalization Bounds for SGD](https://mlsanity.com/paper/2601.01465/) (2026)
- [Generalization error bounds for iterative learning algorithms with bounded updates](https://mlsanity.com/paper/2309.05077/) (2023)
- [Multiview Representation Learning via Distributed Joint Latent Space Structuring](https://mlsanity.com/paper/2504.18455/) (2025)
- [Information-theoretic Generalization Analysis for VQ-VAEs: A Role of Latent Variables](https://mlsanity.com/paper/2505.19470/) (2025)
- [Tight Generalization Error Bounds for Stochastic Gradient Descent in Non-convex Learning](https://mlsanity.com/paper/2506.18645/) (2025)
- [Generalization Bounds of Stochastic Gradient Descent in Homogeneous Neural Networks](https://mlsanity.com/paper/2602.22936/) (2026)
- [An Accuracy-Information Tradeoff for Loss-Difference Conditional Mutual Information](https://mlsanity.com/paper/2610.09206/) (2026)

## Abstract (arXiv)

To date, no "information-theoretic" frameworks for reasoning about generalization error have been shown to establish minimax rates for gradient descent in the setting of stochastic convex optimization. In this work, we consider the prospect of establishing such rates via several existing information-theoretic frameworks: input-output mutual information bounds, conditional mutual information bounds and variants, PAC-Bayes bounds, and recent conditional variants thereof. We prove that none of these bounds are able to establish minimax rates. We then consider a common tactic employed in studying gradient methods, whereby the final iterate is corrupted by Gaussian noise, producing a noisy "surrogate" algorithm. We prove that minimax rates cannot be established via the analysis of such surrogates. Our results suggest that new ideas are required to analyze gradient descent using information-theoretic techniques.