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arXiv 2501.06258cs.LG
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基于预训练神经网络的上下文老虎机优化

Contextual Bandit Optimization with Pre-Trained Neural Networks

Mikhail Terekhov

AI总结:

针对高维奖励下老虎机优化难、小模型神经网络奖励难证次线性悔值的问题,提出E2TC算法利用预训练表示网络权重,经两阶段探索实现高效学习并证次线性悔值,还给出弱训练下误指定线性老虎机的悔值界。

摘要:

老虎机优化是一个难题,尤其是当奖励模型为高维时。当奖励由神经网络建模时,仅在强假设下(通常是网络极宽时)才能证明次线性悔值。本论文研究了在较小模型规模下,预训练如何为该问题提供帮助。我们考虑一种随机上下文老虎机,其奖励由多层神经网络建模:最后一层是线性预测器,之前的层是黑盒神经架构,我们称之为表示网络。我们将预训练建模为提供给学习者的表示网络权重的初始猜测。为利用预训练权重,我们提出了一种名为Explore Twice then Commit(E2TC,两次探索后承诺)的新算法。在两个探索阶段中,该算法首先使用岭回归估计最后一层的权重,然后对所有权重联合执行随机梯度下降。对于局部凸损失函数,我们给出了预训练权重需满足的条件,在此条件下算法可高效学习。我们证明,在这些条件下,当最后一层维度和动作数$K$远小于时间跨度$T$时,E2TC具有次线性悔值。在弱训练场景下,即仅学习最后一层时,问题退化为误指定线性老虎机。我们为该老虎机引入了误指定度量$ε_0$,并据此给出悔值界$O(ε_0\sqrt{d}KT+(KT)^{4 /5})$或$\tilde{O}(ε_0\\

英文摘要:

Bandit optimization is a difficult problem, especially if the reward model is high-dimensional. When rewards are modeled by neural networks, sublinear regret has only been shown under strong assumptions, usually when the network is extremely wide. In this thesis, we investigate how pre-training can help us in the regime of smaller models. We consider a stochastic contextual bandit with the rewards modeled by a multi-layer neural network. The last layer is a linear predictor, and the layers before it are a black box neural architecture, which we call a representation network. We model pre-training as an initial guess of the weights of the representation network provided to the learner. To leverage the pre-trained weights, we introduce a novel algorithm we call Explore Twice then Commit (E2TC). During its two stages of exploration, the algorithm first estimates the last layer's weights using Ridge regression, and then runs Stochastic Gradient Decent jointly on all the weights. For a locally convex loss function, we provide conditions on the pre-trained weights under which the algorithm can learn efficiently. Under these conditions, we show sublinear regret of E2TC when the dimension of the last layer and number of actions $K$ are much smaller than the horizon $T$. In the weak training regime, when only the last layer is learned, the problem reduces to a misspecified linear bandit. We introduce a measure of misspecification $ε_0$ for this bandit and use it to provide bounds $O(ε_0\sqrt{d}KT+(KT)^{4 /5})$ or $\tilde{O}(ε_0\sqrt{d}KT+d^{1 /3}(KT)^{2 /3})$ on the regret, depending on regularization strength. The first of these bounds has a dimension-independent sublinear term, made possible by the stochasticity of contexts. We also run experiments to evaluate the regret of E2TC and sample complexity of its exploration in practice.

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