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计算机代写|机器学习代写Machine Learning代考|COMP7703 The Size of the Validation Set

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计算机代写|机器学习代写Machine Learning代考|The Size of the Validation Set

The choice of the split ratio $\rho \approx m_t / m$ in Algorithm 5 is often based on trial and error. We try out different choices for the split ratio and pick the one resulting in the smallest validation error. It is difficult to make a precise statement on how to choose the split ratio which applies broadly [2]. This difficulty stems from the fact that the optimal choice for $\rho$ depends on the precise statistical properties of the data points.
To obtain a lower bound on the required size of the validation set, we need a probabilistic model for the data points. Let us assume that data points are realizations of i.i.d. random variables with the same probability distribution $p(\mathbf{x}, y)$. Then the validation error $E_v$ (6.6) becomes a realization of a random variable. The expectation (or mean) $\mathbb{E}\left{E_v\right}$ of this $\mathrm{RV}$ is precisely the risk $\mathbb{E}{L((\mathbf{x}, y), \hat{h})}$ of $\hat{h}$ (see (4.1)).
The random validation error $E_v$ fluctuates around its mean. We can quantify this fluctuations using the variance
$$\sigma_{E_v}^2:=\mathbb{E}\left{\left(E_v-\mathbb{E}\left{E_v\right}\right)^2\right} .$$
Note that the validation error is the average of the realizations $L\left(\left(\mathbf{x}^{(i)}, y^{(i)}\right), \hat{h}\right)$ of i.i.d. random variables. The probability distribution of the random variable $L((\mathbf{x}, y), \hat{h})$ is determined by the probability distribution $p(\mathbf{x}, y)$, the choice of loss function and the hypothesis $\hat{h}$. In general, we do not know $p(\mathbf{x}, y)$ and, in turn, also do not know the probability distribution of $L((\mathbf{x}, y), \hat{h})$.

If we know an upper bound $U$ on the variance of the (random) $\operatorname{loss} L\left(\left(\mathbf{x}^{(i)}, y^{(i)}\right), \hat{h}\right)$, we can bound the variance of $E_v$ as
$$\sigma_{E_v}^2 \leq U / m_v .$$

计算机代写|机器学习代写Machine Learning代考|k-Fold Cross Validation

Algorithm 5 uses the most basic form of splitting a given dataset $\mathcal{D}$ into a training and a validation set. Many variations and extensions of this basic splitting approach have been proposed and studied (see [3] and Sect. 6.5). One very popular extension of the single split into training and validation set is known as $k$-fold cross-validation (CV) $[4$, Sec. 7.10]. We summarize $k$-fold CV in Algorithm 6 below.

Figure $6.4$ illustrates the key principle behind $k$-fold CV which is to divide the entire dataset evenly into $k$ subsets which are referred to as folds. The learning (via ERM) and validation of a hypothesis out of a given hypothesis space $\mathcal{H}$ is then repeated $k$ times. During each repetition, we use one fold as the validation set and the remaining $k-1$ folds as a training set. We then average the training and validation error over all repetitions.

The average (over all $k$ folds) validation error delivered by $k$-fold CV is a more robust estimator for the expected loss or risk (4.1) compared to the validation error obtained from a single split. Consider a small dataset and using a single split into training and validation set. We might then be very unlucky and choose data points for the validation set which are outliers and not representative of the overall distribution of the data.

计算机代写|机器学习代写Machine Learning代考|验证集的大小

lleft的分隔符缺失或未被识别

$$\sigma_{E_v}^2 \leq U / m_v$$

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