Few neural architectures lend themselves to provable learning with gradient
based methods. One popular model is the single-index model, in which labels are
produced by composing an unknown linear projection with a possibly unknown
scalar link function. Learning this model with SGD is relatively
well-understood, whereby the so-called information exponent of the link
function governs a polynomial sample complexity rate. However, extending this
analysis to deeper or more complicated architectures remains challenging.
In this work, we consider single index learning in the setting of symmetric
neural networks. Under analytic assumptions on the activation and maximum
degree assumptions on the link function, we prove that gradient flow recovers
the hidden planted direction, represented as a finitely supported vector in the
feature space of power sum polynomials. We characterize a notion of information
exponent adapted to our setting that controls the efficiency of learning.

在对称神经网络的设置下，通过对激活函数进行分析和对连接函数进行最大度数的假设，我们证明了梯度流可以恢复隐藏的预设方向，该方向在幂和多项式特征空间中表示为一个有限支持的向量，并刻画了适应我们设置的信息指数概念来控制学习的效率。

对称单指标学习

Symmetric Single Index Learning

Collision avoidance systems play a vital role in reducing the number of
vehicle accidents and saving human lives. This paper extends the previous work
using evolutionary neural networks for reactive collision avoidance. We are
proposing a new method we have called symmetric neural networks. The method
improves the model's performance by enforcing constraints between the network
weights which reduces the model optimization search space and hence, learns
more accurate control of the vehicle steering for improved maneuvering. The
training and validation processes are carried out using a simulation
environment - the codebase is publicly available. Extensive experiments are
conducted to analyze the proposed method and evaluate its performance. The
method is tested in several simulated driving scenarios. In addition, we have
analyzed the effect of the rangefinder sensor resolution and noise on the
overall goal of reactive collision avoidance. Finally, we have tested the
generalization of the proposed method. The results are encouraging; the
proposed method has improved the model's learning curve for training scenarios
and generalization to the new test scenarios. Using constrained weights has
significantly improved the number of generations required for the Genetic
Algorithm optimization.

本论文利用对称神经网络建立一种新的对抗性碰撞避免方法，通过在网络权重之间施加约束以减少模型优化搜索空间，提高车辆转向的控制精度，通过在模拟环境中进行训练和验证，分析了提出的方法并评估了其性能，结果表明该方法的学习曲线和一般化性良好。