Resolving Parametric Misspecification: Joint Schemes for Computation and Learning

本研究的目的是考虑带有错定参数的凸优化和变分不等式问题的解。一种可能朴素的顺序方法将首先(i)通过学习准确地估计问题参数；然后（ii）求解相关的计算问题。不幸的是，随着学习问题在规模和复杂性上的增长，这种框架受到获得准确参数估计的解决时间的显著增加的阻碍。此外，任何参数估计误差都会传播到计算问题中，可能会带来毁灭性的影响。因此，与naïve顺序方法相反，本研究将致力于开发基于梯度的算法，该算法可以同时学习错误指定的参数并解决正确参数对应的计算问题。更一般地说，这些耦合方案将被证明能够解决诸如不确定性和非光滑性等问题的复杂性。重要的是，这种方法将处理在学习问题中使用的观察可能依赖于计算过程的情况。该研究将强调全局收敛声明、迭代复杂性结果、参考离线算法的遗憾边界以及探索和利用之间的权衡。如果成功，这项研究将在几个层面上产生影响。在基础层面上，这项研究有望导致新的真正的自适应算法，既可以学习参数，又可以同时优化相关系统。这些算法将通过解决广泛的大规模应用问题（包括投资组合选择问题和网络系统的分布式优化）而产生影响，这些问题因规范不当而变得复杂。最后，这个项目的教育部分将包括本科研究项目和高中课程模块。

The objective of this research is to consider the solution of convex optimization and variational inequality problems complicated by misspecified parameters. A possibly naive sequential approach would first (i) estimate the problem parameters accurately through learning; and then (ii) solve the associated computational problem. Unfortunately, as the learning problems grow in size and complexity, such a framework is hampered by the significant increase in solution time to obtain accurate parameter estimates. Furthermore, any parameter estimation error propagates to the computational problem, possibly with devastating impact. Accordingly, on contrary to the naïve sequential approach, this research will aim to develop gradient-based algorithms that can simultaneously learn the misspecified parameter and solve the computational problem corresponding to the correct parameter. More generally, these coupled schemes will be shown to be capable of contending with problem intricacies such as uncertainty and nonsmoothness. Importantly, this methodology will cope with situations where the observations used in the learning problem may depend on the computational process. The research will emphasize the development of global convergence statements, iteration complexity results, regret bounds with reference to offline algorithms, and trade-offs between exploration and exploitation.If successful, this research will find impact at several levels. At a fundamental level, this research is expected to lead to new truly adaptive algorithms that can both learn parameters and optimize the associated systems simultaneously. The algorithms will make impact through addressing a wide range of large-scale application problems that are complicated by misspecification, including portfolio selection problems and distributed optimization of networked systems. Finally, the educational component of this project will comprise of undergraduate research projects and high school course modules.