The "double descent" risk curve was proposed to qualitatively describe the
out-of-sample prediction accuracy of variably-parameterized machine learning
models. This article provides a precise mathematical analysis for the shape of
this curve in two simple data models with the least squares/least norm
predictor. Specifically, it is shown that the risk peaks when the number of
features $p$ is close to the sample size $n$, but also that the risk decreases
towards its minimum as $p$ increases beyond $n$. This behavior is contrasted
with that of "prescient" models that select features in an a priori optimal
order.

本文提出了 “双下降” 风险曲线，用于定性描述可变参数机器学习模型的样本外预测准确性。通过对最小二乘 / 最小规范预测器的两个简单数据模型进行精确的数学分析，显示出风险在特征数 $p$ 接近样本数 $n$ 时达到峰值，但随着 $p$ 超过 $n$ 而逐渐降低至最小值。而这种行为则和需要提前确定最优特征序列的 “先知” 模型有所不同。

弱特征的双下降模型

Two models of double descent for weak features

Using a large-scale Deep Learning approach applied to a high-frequency
database containing billions of electronic market quotes and transactions for
US equities, we uncover nonparametric evidence for the existence of a universal
and stationary price formation mechanism relating the dynamics of supply and
demand for a stock, as revealed through the order book, to subsequent
variations in its market price. We assess the model by testing its
out-of-sample predictions for the direction of price moves given the history of
price and order flow, across a wide range of stocks and time periods. The
universal price formation model is shown to exhibit a remarkably stable
out-of-sample prediction accuracy across time, for a wide range of stocks from
different sectors. Interestingly, these results also hold for stocks which are
not part of the training sample, showing that the relations captured by the
model are universal and not asset-specific.
The universal model --- trained on data from all stocks --- outperforms, in
terms of out-of-sample prediction accuracy, asset-specific linear and nonlinear
models trained on time series of any given stock, showing that the universal
nature of price formation weighs in favour of pooling together financial data
from various stocks, rather than designing asset- or sector-specific models as
commonly done. Standard data normalizations based on volatility, price level or
average spread, or partitioning the training data into sectors or categories
such as large/small tick stocks, do not improve training results. On the other
hand, inclusion of price and order flow history over many past observations is
shown to improve forecasting performance, showing evidence of path-dependence
in price dynamics.

运用深度学习技术，探究供需动态、订单簿及后续价格波动间的非参数普适与稳定关系并在各行业普遍适用；通过金融数据池的模型训练，证明类算法对于多品种数据更具性能优势，而数据归一化、分组等设计并未证明有效。