### 数学代写|优化算法作业代写optimisation algorithms代考|Overfitting

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础
• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|优化算法作业代写optimisation algorithms代考|The Problem

Definition 5 (Overfitting). Overfitting is the emergence of an overly complicated model (solution candidate) in an optimization process resulting from the effort to provide the best results for as much of the available training data as possible $[64,80,190,202]$.

A model (solution candidate) $m \in X$ created with a finite set of training data is considered to be overfitted if a less complicated, alternative model

$m^{\prime} \in X$ exists which has a smaller error for the set of all possible (maybe even infinitely many), available, or (theoretically) producible data samples. This model $m^{\prime}$ may, however, have a larger error in the training data.

The phenomenon of overfitting is best known and can often be encountered in the field of artificial neural networks or in curve fitting $[124,128,181,191$, 211]. The latter means that we have a set $A$ of $n$ training data samples $\left(x_{i}, y_{i}\right)$ and want to find a function $f$ that represents these samples as well as possible, i.e., $f\left(x_{i}\right)=y_{i} \forall\left(x_{i}, y_{i}\right) \in A$.

There exists exactly one polynomial of the degree $n-1$ that fits to each such training data and goes through all its points. Hence, when only polynomial regression is performed, there is exactly one perfectly fitting function of minimal degree. Nevertheless, there will also be an infinite number of polynomials with a higher degree than $n-1$ that also match the sample data perfectly. Such results would be considered as overfitted.

In Fig. 9, we have sketched this problem. The function $f_{1}(x)=x$ shown in Fig. 9.b has been sampled three times, as sketched in Fig. 9.a. There exists no other polynomial of a degree of two or less that fits to these samples than $f_{1}$. Optimizers, however, could also find overfitted polynomials of a higher degree such as $f_{2}$ which also match the data, as shown in Fig. 9.c. Here, $f_{2}$ plays the role of the overly complicated model $m$ which will perform as good as the simpler model $m^{\prime}$ when tested with the training sets only, but will fail to deliver good results for all other input data.

## 数学代写|优化算法作业代写optimisation algorithms代考|Countermeasures

There exist multiple techniques that can be utilized in order to prevent overfitting to a certain degree. It is most efficient to apply multiple such techniques together in order to achieve best results.

A very simple approach is to restrict the problem space $X$ in a way that only solutions up to a given maximum complexity can be found. In terms of function fitting, this could mean limiting the maximum degree of the polynomials to be tested. Furthermore, the functional objective functions which solely concentrate on the error of the solution candidates should be augmented by penalty terms and non-functional objective functions putting pressure in the direction of small and simple models $[64,116]$.

Large sets of sample data, although slowing down the optimization process, may improve the generalization capabilities of the derived solutions. If arbitrarily many training datasets or training scenarios can be generated, there are two approaches which work against overfitting:

1. The first method is to use a new set of (randomized) scenarios for each evaluation of a solution candidate. The resulting objective values may differ

largely even if the same individual is evaluated twice in a row, introducing incoherence and ruggedness into the fitness landscape.

1. At the beginning of each iteration of the optimizer, a new set of (randomized) scenarios is generated which is used for all individual evaluations during that iteration. This method leads to objective values which can be compared without bias.

In both cases it is helpful to use more than one training sample or scenario per evaluation and to set the resulting objective value to the average (or better median) of the outcomes. Otherwise, the fluctuations of the objective values between the iterations will be very large, making it hard for the optimizers to follow a stable gradient for multiple steps.

Another simple method to prevent overfitting is to limit the runtime of the optimizers [190]. It is commonly assumed that learning processes normally first find relatively general solutions which subsequently begin to overfit because the noise “is learned”, too.

For the same reason, some algorithms allow to decrease the rate at which the solution candidates are modified by time. Such a decay of the learning rate makes overfitting less likely.

If only one finite set of data samples is available for training/optimization, it is common practice to separate it into a set of training data $A_{t}$ and a set of test cases $A_{c}$. During the optimization process, only the training data is used. The resulting solutions are tested with the test cases afterwards. If their behavior is significantly worse when applied to $A_{c}$ than when applied to $A_{t}$, they are probably overfitted.

The same approach can be used to detect when the optimization process should be stopped. The best known solution candidates can be checked with the test cases in each iteration without influencing their objective values which solely depend on the training data. If their performance on the test cases begins to decrease, there are no benefits in letting the optimization process continue any further.

## 数学代写|优化算法作业代写optimisation algorithms代考|The Problem

Oversimplification (also called overgeneralization) is the opposite of overfitting. Whereas overfitting denotes the emergence of overly-complicated solution candidates, oversimplified solutions are not complicated enough. Although they represent the training samples used during the optimization process seemingly well, they are rough overgeneralizations which fail to provide good results for cases not part of the training.

A common cause for oversimplification is sketched in Fig. 11: The training sets only represent a fraction of the set of possible inputs. As this is normally the case, one should always be aware that such an incomplete coverage may fail to represent some of the dependencies and characteristics of the data.

which then may lead to oversimplified solutions. Another possible reason is that ruggedness, deceptiveness, too much neutrality, or high epistasis in the fitness landscape may lead to premature convergence and prevent the optimizer from surpassing a certain quality of the solution candidates. It then cannot completely adapt them even if the training data perfectly represents the sampled process. A third cause is that a problem space which does not include the correct solution was chosen.

Fig. 11.a shows a cubic function. Since it is a polynomial of degree three, four sample points are needed for its unique identification. Maybe not knowing this, only three samples have been provided in Fig. 11.b. By doing so, some vital characteristics of the function are lost. Fig. 11.c depicts a square function – the polynomial of the lowest degree that fits exactly to these samples. Although it is a perfect match, this function does not touch any other point on the original cubic curve and behaves totally differently at the lower parameter area.

However, even if we had included point $P$ in our training data, it would still be possible that the optimization process would yield Fig. 11.c as a result. Having training data that correctly represents the sampled system does not mean that the optimizer is able to find a correct solution with perfect fitness – the other, previously discussed problematic phenomena can prevent it from doing so. Furthermore, if it was not known that the system which was to be modeled by the optimization process can best be represented by a polynomial of the third degree, one could have limited the problem space $\mathbb{X}$ to polynomials of degree two and less. Then, the result would likely again be something like Fig. 11.c, regardless of how many training samples are used.

## 数学代写|优化算法作业代写optimisation algorithms代考|Countermeasures

1. 第一种方法是对候选解决方案的每次评估使用一组新的（随机）场景。产生的客观值可能不同

1. 在优化器的每次迭代开始时，会生成一组新的（随机）场景，用于该迭代期间的所有单独评估。这种方法产生的客观值可以在没有偏差的情况下进行比较。

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## MATLAB代写

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