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.

翻译：迄今为止，尚无任何用于推理泛化误差的“信息论”框架被证明能够在随机凸优化场景下为梯度下降建立极小极大速率。本研究探讨了通过现有若干信息论框架建立此类速率的可能性：输入输出互信息界、条件互信息界及其变体、PAC-贝叶斯界及其近期的条件变体。我们证明了这些界均无法建立极小极大速率。随后我们考察了梯度方法研究中的常见策略（即通过高斯噪声扰动最终迭代结果，生成带噪的“代理”算法），并证明基于此类代理的分析同样无法建立极小极大速率。我们的结果表明，需要新的思路才能运用信息论技术分析梯度下降方法。