随机梯度下降,按照数据生成分布抽取m个样本,通过计算他们梯度的平均值来更新梯度。

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题目: On the Generalization Benefit of Noise in Stochastic Gradient Descent

摘要:

长期以来一直有人认为,在深度神经网络中,小批量随机梯度下降比大批量梯度下降具有更好的泛化能力。但是,最近的论文对此主张提出了质疑,认为这种影响仅是批处理量较大时超优化超参数调整或计算预算不足的结果。在本文中,我们对一系列流行的模型进行了精心设计的实验并进行了严格的超参数扫描,这证明了小批量或中等批量都可以大大胜过测试集上的超大批量。即使两个模型都经过相同数量的迭代训练并且大批量实现较小的训练损失时,也会发生这种情况。我们的结果证实,随机梯度中的噪声可以增强泛化能力。我们研究最佳学习率时间表如何随着epoch budget的增长而变化,并基于SGD动力学的随机微分方程视角为我们的观察提供理论解释。

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Within the context of hybrid quantum-classical optimization, gradient descent based optimizers typically require the evaluation of expectation values with respect to the outcome of parameterized quantum circuits. In this work, we explore the consequences of the prior observation that estimation of these quantities on quantum hardware results in a form of stochastic gradient descent optimization. We formalize this notion, which allows us to show that in many relevant cases, including VQE, QAOA and certain quantum classifiers, estimating expectation values with $k$ measurement outcomes results in optimization algorithms whose convergence properties can be rigorously well understood, for any value of $k$. In fact, even using single measurement outcomes for the estimation of expectation values is sufficient. Moreover, in many settings the required gradients can be expressed as linear combinations of expectation values -- originating, e.g., from a sum over local terms of a Hamiltonian, a parameter shift rule, or a sum over data-set instances -- and we show that in these cases $k$-shot expectation value estimation can be combined with sampling over terms of the linear combination, to obtain "doubly stochastic" gradient descent optimizers. For all algorithms we prove convergence guarantees, providing a framework for the derivation of rigorous optimization results in the context of near-term quantum devices. Additionally, we explore numerically these methods on benchmark VQE, QAOA and quantum-enhanced machine learning tasks and show that treating the stochastic settings as hyper-parameters allows for state-of-the-art results with significantly fewer circuit executions and measurements.

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Within the context of hybrid quantum-classical optimization, gradient descent based optimizers typically require the evaluation of expectation values with respect to the outcome of parameterized quantum circuits. In this work, we explore the consequences of the prior observation that estimation of these quantities on quantum hardware results in a form of stochastic gradient descent optimization. We formalize this notion, which allows us to show that in many relevant cases, including VQE, QAOA and certain quantum classifiers, estimating expectation values with $k$ measurement outcomes results in optimization algorithms whose convergence properties can be rigorously well understood, for any value of $k$. In fact, even using single measurement outcomes for the estimation of expectation values is sufficient. Moreover, in many settings the required gradients can be expressed as linear combinations of expectation values -- originating, e.g., from a sum over local terms of a Hamiltonian, a parameter shift rule, or a sum over data-set instances -- and we show that in these cases $k$-shot expectation value estimation can be combined with sampling over terms of the linear combination, to obtain "doubly stochastic" gradient descent optimizers. For all algorithms we prove convergence guarantees, providing a framework for the derivation of rigorous optimization results in the context of near-term quantum devices. Additionally, we explore numerically these methods on benchmark VQE, QAOA and quantum-enhanced machine learning tasks and show that treating the stochastic settings as hyper-parameters allows for state-of-the-art results with significantly fewer circuit executions and measurements.

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