Resolving Parametric Misspecification: Joint Schemes for Computation and Learning

本研究的目的是考虑由错误指定的参数复杂的凸优化和变分不等式问题的解决方案。一个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）解决相关的计算问题。不幸的是，随着学习问题的规模和复杂性的增长，这样的框架受到了阻碍，在解决方案的时间显着增加，以获得准确的参数估计。此外，任何参数估计误差都会传播到计算问题，可能会产生破坏性影响。因此，与简单的顺序方法相反，本研究的目标是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与正确参数相对应的计算问题。更一般地说，这些耦合方案将被证明是能够与问题的复杂性，如不确定性和nonsmoothness。重要的是，这种方法将科普在学习问题中使用的观察可能取决于计算过程的情况。本研究将着重于开发全局收敛语句、迭代复杂度结果、参考离线算法的遗憾界限以及探索和开发之间的权衡。如果成功，本研究将在多个层面上产生影响。在基础层面上，这项研究预计将导致新的真正自适应算法，可以同时学习参数和优化相关系统。这些算法将通过解决因错误指定而复杂化的大规模应用问题产生影响，包括投资组合选择问题和网络系统的分布式优化。最后，该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.