Stochastic gradient descent for optimal design in ordinary differential equations

背景：许多实验通过一个参数未知的基础模型依赖于时间。这里我们将集中讨论由常微分方程定义的模型。为了获得这些参数真实值的信息，需要在不同的时间点进行观测。直观的是，观察越多，我们获得的信息就越多，然而，由于预算和可用资源的限制，这并不总是可行的。这突出了何时采取有限数量的观测以获得关于模型参数的最多信息的问题。模型的实现是嘈杂的，这使得选择最佳设计点变得困难。优化设计就是对这个问题的解决方案的探索。我们将要使用的统计模型的参数是未知的。我们的目标是通过重要抽样或马尔可夫链蒙特卡罗等推理方法尽可能多地了解这些参数。为了获得推理结果与真实值相比有多好的定量总结，我们使用效用函数。效用函数有多种选择，但最初我们将使用负均方误差。我们的目标是选择一种优化预期效用的设计，即在参数和结果观察的所有可能真值上适当地平均效用。在实践中，这是非常困难的，因为只能对预期效用进行嘈杂的估计，这意味着很难确定最佳时间。为了找到期望效用的最大值，我们将使用随机梯度下降算法。该算法在给定时间估计效用的梯度，使用它来迭代地接近最优时间。项目：该项目最初将着眼于Ryan、Drovandi和Pettitt定义的药代动力学研究模型的变体。我将使用事先建议的重要采样器从参数的后验分布中获得结果。为了区分哪种设计能使效用最大化，我们将实现一个简单的随机梯度下降算法。这将与基本的网格搜索进行比较。在探索了这个简单的例子之后，我可以继续研究寻找最佳设计的更好方法。这些可能包括：*使用与SIR模型等替代模型类似的方法*查看各种效用函数及其执行方式*不仅确定何时进行观察很重要，我们应该确定应该进行多少次观察*是否有更有效的方法实现算法以找到效用函数最大化的时间

Background:Many experiments are dependent on time through an underlying model for which the parameters are unknown. Here we will focus on models defined by ordinary differential equations. In order to gain information for the true value of such parameters, observations need to be made at various time points. It is intuitive that the more observations, the more information we gain, however this is not always feasible due to constraints on budget and resources available. This highlights the problem of when to take the limited number of observations with the aim of attaining the most information about the model parameters. Realisations from the model are noisy, making choosing the best design points difficult to identify. Optimal design is the exploration of a solution to this problem.The statistical models we shall be using have parameters which are unknown. We aim to learn as much about these parameters as possible through inference methods such as importance sampling or Markov Chain Monte Carlo. To gain a quantitative summary on how good the inference results are in comparison to the true values, we use a utility function. There are various choices of utility function but initially we shall use the negative mean square error. Our aim is to choose a design which optimises expected utility i.e. utility averaged appropriately over all possible true values of the parameters and resulting observations.In practice this is incredibly difficult as only noisy estimates of expected utility can be made, meaning it is difficult to identify the optimal times. In order to find the maximum of the expected utility we will use a stochastic gradient descent algorithm. This algorithm estimates the gradient of the utility at given times, using this to iteratively get closer to the optimal times.Project:The project will initially look at a variation of the pharmacokinetic study model defined by Ryan, Drovandi and Pettitt. I shall use an importance sampler with prior proposal to obtain draws from the posterior distribution of the parameters. In order to distinguish which design maximises the utility a simple stochastic gradient descent algorithm will be implemented. This will be compared to a basic grid search.After this simple example has been explored, I could then progress to research better ways of finding the optimal design. These could include:* using similar approach to alternative models such as the SIR model* looking at various utility functions and how they perform* not only is it important to identify when to take observations, we should identify how many observations should be made* are there more efficient ways of implementing an algorithm to find the times which maximises the utility function