On a Mystery in Machine Learning
Friday, 7.7.23, 12:00-13:00, Hörsaal II, Albertstr. 23b
In classical regression modelling, the complexity of the model, e.g. measured by the number of parameters, is smaller than the amount of training data. The prediction error exhibits a U-shaped behaviour. The (first) descent is due to decreasing bias, the ascent due to increasing variance. In modern machine learning, often the number of parameters by far exceeds the number of training data points. Intuitively, one could expect that the prediction error explodes with increasing model complexity due to overfitting. Belkin et al. (2019) observed that this is not the case. Instead, the prediction error decreases again when surpassing a certain threshold in model complexity, in some case even below the minimum of the classical, U-shaped regime. A phenomenon the authors denominated as double descent. To understand double descent, we study the simplest setting of linear regression and show that it can be explained by investigating the singular values of the design matrix. Finally, we give an outlook for the non-linear model setting.\n\nBelkin, M.; Hsu, D.; Ma, S.; Mandal, S.: Reconciling modern machine-learning practice and the classical bias–variance trade-off. In: Proceedings of the National Academy of Sciences 116 (2019), jul, Nr. 32, 15849–15854.
A Principal-Agent Framework for Optimal Incentives in Renewable Investments
Friday, 14.7.23, 12:00-13:00, Hörsaal II, Albertstr. 23b
We investigate the optimal regulation of energy production reflecting the long-term goals of the Paris climate agreement.\n\nWe analyze the optimal regulatory incentives to foster the development of non-emissive electricity generation when the demand for power is served either by a monopoly or by two competing agents. The regulator wishes to encourage green investments to limit carbon emissions, while simultaneously reducing intermittency of the total energy production. We find that the regulation of a competitive market is more efficient than the one of the monopoly as measured with the certainty equivalent of the Principal’s value function. This higher efficiency is achieved thanks to a higher degree of freedom of the incentive mechanisms which involves cross-subsidies between firms. A numerical study quantifies the impact of the designed second-best contract in both market structures compared to the business-as-usual scenario. In addition, we expand the monopolistic and competitive setup to a more general class of tractable Principal-Multi-Agent incentives problems when both the drift and the volatility of a multi-dimensional diffusion process can be controlled by the Agents. We follow the resolution methodology of Cvitanić et al. (2018) in an extended linear quadratic setting with exponential utilities and a multi-dimensional state process of Ornstein-Uhlenbeck type. We provide closed-form expression of the second-best contracts. In particular, we show that they are in rebate form involving time-dependent prices of each state variable.