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计算机代写|机器学习代写machine learning代考|Using and Evaluating Classifiers for Ranking

Often, the goal of training a classifier is not merely to generate exhaustive sets of ‘true’ and ‘false’ instances. For example, if we wanted to identify relevant wehpages in response to a query, or to recommend items that a user is likely to purchase, in practice it may not matter whether we can identify all relevant webpages or products; rather, we might care more about whether we can surface some relevant items among the first page of results returned to a user.
Note that the type of classifiers we have developed so far can straightforwardly be used for ranking. That is, in addition to outputting a predicted label $\left(\delta\left(x_i \cdot \theta>0\right)\right.$ in the case of logistic regression), they can also output confidence scores (i.e., $x_i \cdot \theta$, or $\left.p_\theta\left(y_i=1 \mid x_i\right)\right)$. Thus, in the context of finding relevant webpages or products above, our goal might be to maximize the number of relevant items returned among the few most confident predictions. Furthermore, we might be interested in how the model’s accuracy changes as a function of confidence; for example, even if the model’s accuracy is low overall, is it accurate for the top $1 \%, 5 \%$, or $10 \%$ of most confident predictions?

Note that neither precision nor recall are particularly meaningful if reported in isolation. For instance, it is trivial to achieve a recall of $1.0$ simply by using a classifier that returns ‘true’ for every item (in which case, all relevant documents are returned); such a classifier would of course have low precision. Likewise, a precision close to $1.0$ can often be achieved by returning ‘true’ only for a few items about which we are extremely confident; such a classifier would have low recall.

As such, to evaluate a classifier in terms of precision and recall, we likely want a metric that considers both, or otherwise to place additional constraints on our classifier (as we see below).

The $F_\beta$ score achieves this by taking a weighted average of the two quantities:
$$
F_\beta=\left(1+\beta^2\right) \cdot \frac{\text { precision } \cdot \text { recall }}{\beta^2 \text { precision }+\text { recall }} .
$$
In the case of $\beta=1$ (which is normally called simply the ‘ $F$-score’), Equation (3.27) simply computes the harmonic mean of precision and recall, which is low if either precision or recall is low.

Otherwise, if $\beta \neq 1$, the $F_\beta$ score reflects a situation where one cares about recall over precision by a factor of $\beta .{ }^5$

计算机代写|机器学习代写machine learning代考|Generalization, Overfitting, and Underfitting

So far, when discussing model evaluation in Section $3.3$ (and earlier in sec. 2.2), we have considered training a model to predict labels $y$ from a dataset $X$; we have then evaluated the model by comparing the predictions $f\left(x_i\right)$ to the labels $y_i$. Critically, we’re using the same data to train the model as we’re using to evaluate it.

The risk in doing so is that our model may not generalize well to new data. For example, when fitting a model relating review length to ratings (as in figs. $2.4$ and 2.8), we considered fitting the data with linear, quadratic, and cubic functions. Increasing the degree of the polynomial would continue to lower the errors of the predictor; alternately, we could have modeled review length using a one-hot encoding (so that there was a different predicted value for every length). Such models could fit the data very closely (in terms of their MSE), but it is unclear whether they would capture meaningful trends in the data or simply ‘memorize’ it.

To consider an extreme case, imagine fitting a vector $y$ using only random features. The code below fits a vector of fifty observations using $1,10,25$, and 50 random features, and then prints the $R^2$ coefficient of each model: Here, the $R^2$ coefficients take values of $0.07,0.25,0.35$, and $1.0$ – once we include fifty random features, we can fit the data perfectly. Of course, given that our features were random, this ‘fit’ is not meaningful, and the model has merely discovered random correlations between the observed data and labels.

These arguments point to two issues that need to be addressed when training a model:
(i) We should not evaluate a model on the same data that was used to train it. Rather we should use a held-out dataset (i.e., a test set).
(ii) Features that improve performance on the training data will not necessarily improve performance on the held-out data.

Evaluating a model on held-out data gives us a sense of how well we can expect that model to work ‘in the wild.’ This held-out data, known as a test set, measures how well our model can be expected to generalize to new data.

计算机代写|机器学习代写machine learning代考|COMP4702

机器学习代考


计算机代写|机器学习代写machine learning代考|使用和评估分类器进行排名


通常,训练分类器的目标不仅仅是生成“真”和“假”实例的详尽集合。例如,如果我们想要识别相关的微网页以响应一个查询,或推荐用户可能购买的商品,在实践中,我们是否能够识别所有相关的网页或产品可能并不重要;相反,我们可能更关心是否可以在返回给用户的结果的第一页中显示一些相关的项。注意,我们目前开发的分类器类型可以直接用于排序。也就是说,除了在逻辑回归的情况下输出预测标签$\left(\delta\left(x_i \cdot \theta>0\right)\right.$外,它们还可以输出置信度评分(即$x_i \cdot \theta$或$\left.p_\theta\left(y_i=1 \mid x_i\right)\right)$)。因此,在寻找上面的相关网页或产品的背景下,我们的目标可能是在少数最有把握的预测中最大化返回的相关项目的数量。此外,我们可能会感兴趣的是,模型的准确性如何随着置信度的变化而变化;例如,即使模型的准确性总体上很低,它对最自信的预测的前$1 \%, 5 \%$或$10 \%$是准确的吗?


请注意,如果单独报告,精确度和召回率都不是特别有意义。例如,通过使用为每个项目返回“true”的分类器(在这种情况下,返回所有相关文档)来实现对$1.0$的召回是微不足道的;这样的分类器当然精度很低。同样,接近$1.0$的精度通常可以通过只对少数我们非常有信心的项目返回“true”来实现;这样的分类器会有较低的召回率


因此,为了在精确度和召回率方面评估一个分类器,我们可能需要一个同时考虑这两者的指标,或者在我们的分类器上添加额外的约束(如下所示)


$F_\beta$分数是通过取两个量的加权平均值来实现的:
$$
F_\beta=\left(1+\beta^2\right) \cdot \frac{\text { precision } \cdot \text { recall }}{\beta^2 \text { precision }+\text { recall }} .
$$
对于$\beta=1$(通常被简单地称为’ $F$ -score’),式(3.27)简单地计算了精度和召回率的调和平均值,如果精度或召回率都很低,则该平均值较低


否则,如果$\beta \neq 1$, $F_\beta$分数反映了一种情况,即一个人关心召回比精度的因素$\beta .{ }^5$

计算机代写|机器学习代写machine learning代考|概化,过拟合,和欠拟合

.


到目前为止,在$3.3$节(以及之前的2.2节)讨论模型评估时,我们考虑了训练一个模型来从数据集$X$预测标签$y$;然后,我们通过比较预测$f\left(x_i\right)$和标签$y_i$来评估模型。关键是,我们使用相同的数据来训练模型,正如我们使用相同的数据来评估它


这样做的风险是,我们的模型可能不能很好地推广到新的数据。例如,当拟合一个将评论长度与评级相关的模型时(如图所示)。$2.4$和2.8),我们考虑用线性、二次和三次函数拟合数据。多项式次数的增加将继续降低预测器的误差;另外,我们也可以使用单热编码来建模评审长度(这样每个长度都有不同的预测值)。这样的模型可以非常接近地拟合数据(就其MSE而言),但尚不清楚它们是否能捕捉到数据中有意义的趋势,还是只是“记住”它


为了考虑一个极端的情况,想象只使用随机特征拟合一个向量$y$。下面的代码使用$1,10,25$和50个随机特征来拟合一个由50个观察结果组成的向量,然后打印每个模型的$R^2$系数:这里,$R^2$系数取$0.07,0.25,0.35$和$1.0$的值——一旦我们包含了50个随机特征,我们就可以完美地拟合数据。当然,考虑到我们的特征是随机的,这种“匹配”是没有意义的,模型只是发现了观察到的数据和标签之间的随机相关性


这些论证指出了在训练模型时需要解决的两个问题:(i)我们不应该用用来训练模型的相同数据来评估模型。
(ii)提高训练数据性能的特性并不一定会提高在保留数据上的性能


根据持有数据评估一个模型可以让我们了解到,我们可以预期该模型在“野外”的工作情况如何。这个搁置数据被称为测试集,它衡量我们的模型对新数据的概括程度

计算机代写|机器学习代写machine learning代考

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