Drawing on statistical learning theory, we derive out-of-sample and optimality guarantees about the investment strategy obtained from a regularized portfolio optimization model which attempts to exploit side information about the financial market in order to reach an optimal risk-return tradeoff. This side information might include for instance recent stock returns, volatility indexes, financial news indicators, etc. In particular, we demonstrate that a regularized investment policy that linearly combines this side information in a way that is optimal from the perspective of a random sample set is guaranteed to perform also relatively well (i.e., within a perturbing factor of $O(1/\sqrt{n})$) with respect to the unknown distribution that generated this sample set. We also demonstrate that these performance guarantees are lost in a high-dimensional regime where the size of the side information vector is of an order that is comparable to the sample size. We further extend these results to the case where non-linear investment policies are considered using a kernel operator and show that with radial basis function kernels the performance guarantees become insensitive to how much side information is used. Finally, we illustrate our findings with a set of numerical experiments involving financial data for the NASDAQ composite index.

基于统计学习理论，我们从正则化的投资组合优化模型中获得了关于投资策略的样本外和最优性保证，该模型试图利用有关金融市场的附带信息以达到最佳的风险收益权衡。此辅助信息可能包括例如最近的股票收益，波动率指数，财务新闻指标等。特别是，我们证明了一种正规化的投资政策，该政策以一种从随机样本集的角度来看是最佳的方式线性地组合了此辅助信息。保证也能表现得比较好（即在$Ø（\mathrm{1个}/\sqrt{ñ}）$）关于生成此样本集的未知分布。我们还证明，在辅助信息向量的大小与样本大小可比的数量级的高维方案中，这些性能保证会丢失。我们进一步将这些结果扩展到使用核算子考虑非线性投资策略的情况，并表明利用径向基函数核，性能保证对于使用多少辅助信息变得不敏感。最后，我们用一组涉及纳斯达克综合指数财务数据的数值实验说明了我们的发现。