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Yuqing Wang, Zhenghao Xu, Tuo Zhao, Molei Tao

Large learning rates, when applied to gradient descent for nonconvex optimization, yield various implicit biases including the edge of stability (Cohen et al., 2021), balancing (Wang et al., 2022), and catapult (Lewkowycz et al., 2020). These phenomena cannot be well explained by classical optimization theory. Though significant theoretical progress has been made in understanding these implicit biases, it remains unclear for which objective functions would they occur. This paper provides an initial step in answering this question, namely that these implicit biases are in fact various tips of the same iceberg. They occur when the objective function of optimization has some good regularity, which, in combination with a provable preference of large learning rate gradient descent for moving toward flatter regions, results in these nontrivial dynamical phenomena. To establish this result, we develop a new global convergence theory under large learning rates, for a family of nonconvex functions without globally Lipschitz continuous gradient, which was typically assumed in existing convergence analysis. A byproduct is the first non-asymptotic convergence rate bound for large-learning-rate gradient descent optimization of nonconvex functions. We also validate our theory with experiments on neural networks, where different losses, activation functions, and batch normalization all can significantly affect regularity and lead to very different training dynamics.

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Zichong Li, Qunzhi Xu, Zhenghao Xu, Yajun Mei, Tuo Zhao, Hongyuan Zha

Spatio-temporal point processes (STPPs) are potent mathematical tools for modeling and predicting events with both temporal and spatial features. Despite their versatility, most existing methods for learning STPPs either assume a restricted form of the spatio-temporal distribution, or suffer from inaccurate approximations of the intractable integral in the likelihood training objective. These issues typically arise from the normalization term of the probability density function. Moreover, current techniques fail to provide uncertainty quantification for model predictions, such as confidence intervals for the predicted event's arrival time and confidence regions for the event's location, which is crucial given the considerable randomness of the data. To tackle these challenges, we introduce SMASH: a Score MAtching-based pSeudolikeliHood estimator for learning marked STPPs with uncertainty quantification. Specifically, our framework adopts a normalization-free objective by estimating the pseudolikelihood of marked STPPs through score-matching and offers uncertainty quantification for the predicted event time, location and mark by computing confidence regions over the generated samples. The superior performance of our proposed framework is demonstrated through extensive experiments in both event prediction and uncertainty quantification.

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Zhenghao Xu, Xiang Ji, Minshuo Chen, Mengdi Wang, Tuo Zhao

Policy-based algorithms equipped with deep neural networks have achieved great success in solving high-dimensional policy optimization problems in reinforcement learning. However, current analyses cannot explain why they are resistant to the curse of dimensionality. In this work, we study the sample complexity of the neural policy mirror descent (NPMD) algorithm with convolutional neural networks (CNN) as function approximators. Motivated by the empirical observation that many high-dimensional environments have state spaces possessing low-dimensional structures, such as those taking images as states, we consider the state space to be a $d$-dimensional manifold embedded in the $D$-dimensional Euclidean space with intrinsic dimension $d\ll D$. We show that in each iteration of NPMD, both the value function and the policy can be well approximated by CNNs. The approximation errors are controlled by the size of the networks, and the smoothness of the previous networks can be inherited. As a result, by properly choosing the network size and hyperparameters, NPMD can find an $\epsilon$-optimal policy with $\widetilde{O}(\epsilon^{-\frac{d}{\alpha}-2})$ samples in expectation, where $\alpha\in(0,1]$ indicates the smoothness of environment. Compared to previous work, our result exhibits that NPMD can leverage the low-dimensional structure of state space to escape from the curse of dimensionality, providing an explanation for the efficacy of deep policy-based algorithms.

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