Deep Empirical Risk Minimization in finance: looking into the future

Analysis of the Generalization Error: Empirical Risk Minimization over Deep Artificial Neural Networks Overcomes the Curse of Dimensionality in the Numerical Approximation of Black-Scholes Partial Differential Equations

The development of new classification and regression algorithms based on empirical risk minimization (ERM) over deep neural network hypothesis classes, coined deep learning, revolutionized the area of artificial intelligence, machine learning, and data analysis. In particular, these methods have been applied to the numerical solution of high-dimensional partial differential equations with great success. Recent simulations indicate that deep learning-based algorithms are capable of overcoming the curse of dimensionality for the numerical solution of Kolmogorov equations, which are widely used i…

On optimality of empirical risk minimization in linear aggregation

In the first part of this paper, we show that the small-ball condition, recently introduced by Mendelson (2015), may behave poorly for important classes of localized functions such as wavelets, piecewise polynomials or trigonometric polynomials, in particular leading to suboptimal estimates of the rate of convergence of ERM for the linear aggregation problem. In a second part, we recover optimal rates of covergence for the excess risk of ERM when the dictionary is made of trigonometric functions. Considering the bounded case, we derive the concentration of the excess risk around a single point…

基于样本的优化确定等价风险最小化：算法、收敛速率及应用

Optimized Certainty Equivalent Risk Minimization Using Samples: Algorithms, Convergence Rates, and Applications

本文研究优化确定等价（OCE）风险的优化问题，推导其梯度表达式、估计量界及随机梯度算法收敛速率，并通过实验验证其在投资组合优化等任务中的应用。

Non-convex online learning via algorithmic equivalence

We study an algorithmic equivalence technique between non-convex gradient descent and convex mirror descent. We start by looking at a harder problem of regret minimization in online non-convex optimization. We show that under certain geometric and smoothness conditions, online gradient descent applied to non-convex functions is an approximation of online mirror descent applied to convex functions under reparameterization. In continuous time, the gradient flow with this reparameterization was shown to be exactly equivalent to continuous-time mirror descent by Amid and Warmuth 2020, but theory f…

理解深度学习中的双下降现象

Understanding the Double Descent Phenomenon in Deep Learning

本教程解释深度学习中的双下降现象，涵盖其机制、归纳偏置的作用，并通过线性模型示例及近期工作提供多角度理解。

General risk measures for robust machine learning

A wide array of machine learning problems are formulated as the minimization of the expectation of a convex loss function on some parameter space. Since the probability distribution of the data of interest is usually unknown, it is is often estimated from training sets, which may lead to poor out-of-sample performance. In this work, we bring new insights in this problem by using the framework which has been developed in quantitative finance for risk measures. We show that the original min-max problem can be recast as a convex minimization problem under suitable assumptions. We discuss several …