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# 计算机代写|机器学习代写Machine Learning代考|COMP4702 ERM for Bayes Classifiers

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## 计算机代写|机器学习代写Machine Learning代考|ERM for Bayes Classifiers

The family of ML methods referred to as Bayes estimator uses the $0 / 1$ loss (2.9) to measuring the quality of a classifier $h$. The resulting empirical risk minimization is
\begin{aligned} \hat{h} &=\underset{h \in \mathcal{H}}{\operatorname{argmin}}(1 / m) \sum_{i=1}^m L\left(\left(\mathbf{x}^{(i)}, y^{(i)}\right), h\right) \ & \stackrel{(2.9)}{=} \underset{h \in \mathcal{H}}{\operatorname{argmin}}(1 / m) \sum_{i=1}^m \mathcal{I}\left(h\left(\mathbf{x}^{(i)}\right) \neq y^{(i)}\right) . \end{aligned}
The objective function in this optimization problem is non-differentiable and nonconvex (see Fig. 4.2). This prevents us from using gradient-based optimization methods (see Chap. 5) to solve (4.16).

We will now approach the empirical risk minimization (4.16) via a different route by interpreting the datapoints $\left(\mathbf{x}^{(i)}, y^{(i)}\right)$ as realizations of i.i.d. RVs with the common probability distribution $p(\mathbf{x}, y)$.

As discussed in Sect. 2.3, the empirical risk obtained using 0/1 loss approximates the error probability $p(\hat{y} \neq y)$ with the predicted label $\hat{y}=1$ for $h(\mathbf{x})>0$ and $\hat{y}=$ $-1$ otherwise (see (2.10)). Thus, we can approximate the empirical risk minimization (4.16) as
$$\hat{h}^{(2.10)} \underset{h \in \mathcal{H}}{\operatorname{argmin}} p(\hat{y} \neq y) .$$

## 计算机代写|机器学习代写Machine Learning代考|Training and Inference Periods

Some ML methods repeat the cycle in Figure 1 in a highly irregular fashion. Consider a large image collection which we use to learn a hypothesis about how cat images look like. It might be reasonable to adjust the hypothesis by fitting a model to the image collection. This fitting or training amounts to repeating the cycle in Figure 1 during some specific time period (the “training time”) for a large number.

After the training period, we only apply the hypothesis to predict the labels of new images. This second phase is also known as inference time and might be much longer compared to the training time. Ideally, we would like to only have a very short training period to learn a good hypothesis and then only use the hypothesis for inference.

## 计算机代写|机器学习代写Machine Learning代考|ERM for Bayes Classifiers

$$\hat{h}=\underset{h \in \mathcal{H}}{\operatorname{argmin}}(1 / m) \sum_{i=1}^m L\left(\left(\mathbf{x}^{(i)}, y^{(i)}\right), h\right) \quad \stackrel{(2.9)}{=} \underset{h \in \mathcal{H}}{\operatorname{argmin}}(1 / m) \sum_{i=1}^m \mathcal{I}\left(h\left(\mathbf{x}^{(i)}\right) \neq y^{(i)}\right) .$$

$$\hat{h}^{(2.10)} \underset{h \in \mathcal{H}}{\operatorname{argmin}} p(\hat{y} \neq y) .$$

## MATLAB代写

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