Optimal worst-risk minimization in structural equation models with random coefficients
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随机系数结构方程模型中的最优最坏风险最小化
- 研究问题
- 面对多个训练环境及目标变量也可能发生的分布位移，如何构造在限定样本外环境集合中最小化最坏风险的预测器？
- 方法与思路
- 作者在随机系数结构方程模型中限定环境位移强度，推导最坏风险分解，并构造显式估计量及基于二分法的近似估计量。
- 材料陈述的贡献
- 摘要称除参数空间中一个勒贝格零测集外，最坏风险最小化解唯一，且显式估计量与二分法近似估计量均一致。
- 阅读关注点
- 应核对环境位移强度的定义、风险分解条件、唯一解例外集合，以及多项式求解与二分法近似之间的关系。
- 阅读边界
- 这是 arXiv 预印本，同行评审及期刊接收未核实；摘要未给出完整条件，且风险最优预测器未必具有直接因果解释。
原文摘要
The insight that causal parameters are particularly suitable for out-of-sample prediction has sparked a lot development of causal-like predictors. However, the connection with strict causal targets, has limited the development with good risk minimization properties, but without a direct causal interpretation. In this manuscript we derive the optimal out-of-sample risk minimizing predictor of a certain target $Y$ in a non-linear system $(X,Y)$ that has been trained in several within-sample environments. We consider data from an observation environment, and several shifted environments. Each environment corresponds to a structural equation model (SEM), with random coefficients and with its own shift and noise vector, both in $L^2$. Unlike previous approaches, we also allow shifts in the target value. We define a sieve of out-of-sample environments, consisting of all shifts $\tilde{A}$ that are at most $γ$ times as strong as any weighted average of the observed shift vectors. For each $β\in\mathbb{R}^p$ we show that the supremum of the risk functions $R_{\tilde{A}}(β)$ has a worst-risk decomposition into a (positive) non-linear combination of risk functions, depending on $γ$. We then define the set $\mathcal{B}_γ$, as minimizers of this risk. The main result of the paper is that there is a unique minimizer ($|\mathcal{B}_γ|=1$) that can be consistently estimated by an explicit estimator, outside a set of zero Lebesgue measure in the parameter space. A practical obstacle for the initial method of estimation is that it involves the solution of a general degree polynomials. Therefore, we prove that an approximate estimator using the bisection method is also consistent.