This article examines the implicit regularization effect of Stochastic
Gradient Descent (SGD). We consider the case of SGD without replacement, the
variant typically used to optimize large-scale neural networks. We analyze this
algorithm in a more realistic regime than typically considered in theoretical
works on SGD, as, e.g., we allow the product of the learning rate and Hessian
to be $O(1)$. Our core theoretical result is that optimizing with SGD without
replacement is locally equivalent to making an additional step on a novel
regularizer. This implies that the trajectory of SGD without replacement
diverges from both noise-injected GD and SGD with replacement (in which batches
are sampled i.i.d.). Indeed, the two SGDs travel flat regions of the loss
landscape in distinct directions and at different speeds. In expectation, SGD
without replacement may escape saddles significantly faster and present a
smaller variance. Moreover, we find that SGD implicitly regularizes the trace
of the noise covariance in the eigendirections of small and negative Hessian
eigenvalues. This coincides with penalizing a weighted trace of the Fisher
Matrix and the Hessian on several vision tasks, thus encouraging sparsity in
the spectrum of the Hessian of the loss in line with empirical observations
from prior work. We also propose an explanation for why SGD does not train at
the edge of stability (as opposed to GD).

通过分析基于随机梯度下降（SGD）的隐式正则化效应，本文研究了 SGD 无替换的情况，并发现其与添加了一种新型正则化器时的最优化过程等价，导致其在损失函数空间中与注入噪声的 GD 和采样独立同分布的 SGD 有着明显不同的轨迹，并在某些视觉任务中鼓励损失函数海森矩阵的特征值谱中出现稀疏性。

关于无替换的 SGD 轨迹

On the Trajectories of SGD Without Replacement

The global navigation satellite systems (GNSS) play a vital role in transport
systems for accurate and consistent vehicle localization. However, GNSS
observations can be distorted due to multipath effects and non-line-of-sight
(NLOS) receptions in challenging environments such as urban canyons. In such
cases, traditional methods to classify and exclude faulty GNSS observations may
fail, leading to unreliable state estimation and unsafe system operations. This
work proposes a Deep-Learning-based method to detect NLOS receptions and
predict GNSS pseudorange errors by analyzing GNSS observations as a
spatio-temporal modeling problem. Compared to previous works, we construct a
transformer-like attention mechanism to enhance the long short-term memory
(LSTM) networks, improving model performance and generalization. For the
training and evaluation of the proposed network, we used labeled datasets from
the cities of Hong Kong and Aachen. We also introduce a dataset generation
process to label the GNSS observations using lidar maps. In experimental
studies, we compare the proposed network with a deep-learning-based model and
classical machine-learning models. Furthermore, we conduct ablation studies of
our network components and integrate the NLOS detection with data
out-of-distribution in a state estimator. As a result, our network presents
improved precision and recall ratios compared to other models. Additionally, we
show that the proposed method avoids trajectory divergence in real-world
vehicle localization by classifying and excluding NLOS observations.

通过分析全球导航卫星系统 (GNSS) 观测数据，本研究提出了一种基于深度学习的方法来检测非直射路径接收与预测 GNSS 伪距误差，并构建了类似 Transformer 的注意机制来提高模型性能和泛化能力，实验结果表明相比其他模型，该方法在车辆定位中具有更好的准确性和召回率，能够避免轨迹偏离问题。

基于 Transformer 增强 LSTM 网络的 GNSS 观测非 LOS 检测和不确定性预测

Learning-based NLOS Detection and Uncertainty Prediction of GNSS  Observations with Transformer-Enhanced LSTM Network

We consider the problem of learning to perform a task from demonstrations
given by teachers or experts, when some of the experts' demonstrations might be
adversarial and demonstrate an incorrect way to perform the task. We propose a
novel technique that can identify parts of demonstrated trajectories that have
not been significantly modified by the adversary and utilize them for learning,
using temporally extended policies or options. We first define a trajectory
divergence measure based on the spatial and temporal features of demonstrated
trajectories to detect and discard parts of the trajectories that have been
significantly modified by an adversarial expert, and, could degrade the
learner's performance, if used for learning, We then use an options-based
algorithm that partitions trajectories and learns only from the parts of
trajectories that have been determined as admissible. We provide theoretical
results of our technique to show that repairing partial trajectories improves
the sample efficiency of the demonstrations without degrading the learner's
performance. We then evaluate the proposed algorithm for learning to play an
Atari-like, computer-based game called LunarLander in the presence of different
types and degrees of adversarial attacks of demonstrated trajectories. Our
experimental results show that our technique can identify adversarially
modified parts of the demonstrated trajectories and successfully prevent the
learning performance from degrading due to adversarial demonstrations.

提出了一种新技术，可以识别对学习有用的演示轨迹的部分并利用它们进行学习，同时排除受对手修改的轨迹的部分，以提高学习效率，在预示轨迹遭到不同类型和程度的对手攻击的情况下，该算法有效防止了学习表现下降。