課後功課答案 11.2. Attention Pooling by Similarity

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1. Parzen windows density estimates are given by . Prove that for binary classification the function , as obtained by Parzen windows is equivalent to Nadaraya--Watson classification.

To prove that the function , as obtained by Parzen windows, is equivalent to the Nadaraya-Watson classification, we need to show that the two expressions are mathematically equivalent.

Parzen Windows Density Estimate:

The Parzen windows density estimate for a data point is given by:

where is the number of data points, and is the kernel function that measures the similarity between and the -th data point .

For binary classification, we have two classes with labels $y \in {-1, 1}$. The Parzen windows density estimates for the two classes are:

where and are the number of data points in the positive and negative classes, respectively.

Nadaraya-Watson Classification:

To show the equivalence, we need to prove that:

Proof:

Substitute the Parzen windows density estimates into the left-hand side:

Multiply the numerator and denominator of the Nadaraya-Watson classification function by , the total number of data points:

Observe that the first term in the product is , which can be rewritten as:

Substituting this expression back into the Nadaraya-Watson classification function, we get:

Comparing the expressions, we can see that , proving the equivalence.

2. Implement stochastic gradient descent to learn a good value for kernel widths in Nadaraya–Watson regression.

We'll use the Gaussian kernel for this example.

nadaraya_watson 函數計算給定查詢點 x 和內核寬度 sigma 的 Nadaraya-Watson 迴歸估計。 sgd_train_sigma 函數執行隨機梯度下降來學習最佳內核寬度。

關鍵步驟是：

將內核寬度 sigma 初始化為起始值（例如 1.0）。

在 SGD 循環的每次迭代中：

對一批資料點 X_batch 和 y_batch 進行取樣。
計算均方誤差相對於核寬度 sigma 的梯度。
使用梯度和學習率更新內核寬度。

返回最大迭代次數後學習到的核心寬度。

請注意，在梯度計算中，我們需要根據核寬度區分 Nadaraya-Watson 估計，這是透過將 True 作為最後一個參數傳遞給 nadaraya_watson 函數來完成的。

此實現透過不直接最小化預測值和目標值之間的均方誤差來避免先前回應中提到的循環依賴問題。相反，它專注於優化內核寬度，這是影響 Nadaraya-Watson 估計的單獨參數。

產生輸入點的二維網格，計算每個網格點的 Nadaraya-Watson 迴歸估計，然後繪製所得迴歸曲面以及輸入資料點。

使用 np.meshgrid 產生輸入點網格。

使用 nadaraya_watson 函數計算每個網格點的迴歸估計。重塑迴歸估計以匹配網格尺寸。

使用 plt.contourf 繪製迴歸曲面，並使用 plt.scatter 繪製輸入資料點。

產生的圖應顯示學習的回歸曲面，並覆蓋輸入資料點。

2.1 What happens if you just use the above estimates to minimize directly? Hint: is part of the terms used to compute .

The issue with directly minimizing is that the target variable is used in the computation of , which leads to a circular dependency.

The Nadaraya-Watson regression estimate is given by:

where is the kernel function, typically a Gaussian kernel:

If you directly minimize with respect to the kernel width , the gradient will be:

The issue is that depends on , which is part of the computation of . This creates a circular dependency, and the resulting gradient update will not lead to a proper optimization of the kernel width.

2.2 Assume that all lie on the unit sphere, i.e., all satisfy . Can you simplify the term in the exponential? Hint: we will later see that this is very closely related to dot product attention.

Absolutely, let's think through this step-by-step:

Assume that all lie on the unit sphere, i.e., for all .

The term can be expanded as:

Since and , we have:

Therefore, the exponential term can be simplified to:

This simplified form of the exponential term is very closely related to dot product attention, where the similarity between and is measured by their dot product . This connection will become more apparent as we explore dot product attention in more detail.

Recall that`mack1982weak` proved that Nadaraya--Watson estimation is consistent. How quickly should you reduce the scale for the attention mechanism as you get more data? Provide some intuition for your answer. Does it depend on the dimensionality of the data? How? 回想一下，mack1982weak證明了Nadaraya-Watson估計是一致的。當您獲得更多數據時，您應該以多快的速度縮小注意力機制的規模？為你的答案提供一些直覺。它取決於資料的維數嗎？如何？

Nadaraya-Watson 估計是一種非參數迴歸技術，用於估計隨機變數的條件期望值。 Mack (1982) 證明了 Nadaraya-Watson 估計在某些條件下是一致的估計。

在註意力機制的背景下，尺度參數控制注意力權重的「柔軟度」。較大的意味著注意力權重更加“分散”，而較小的意味著注意力權重更加“集中”。

當你獲得更多數據時，直覺上你會希望減小尺度以使注意力權重更加集中。這是因為有了更多的數據，您就有更多的資訊可以使用，並且您可以更有選擇性地關注輸入的哪些部分。

作為資料量函數的減少率取決於資料的維度。具體來說：

In low-dimensional settings, you can reduce $\sigma^2$ more quickly as you get more data. This is because in low dimensions, the data becomes more "spread out" as you get more of it, and you can afford to be more selective.

In high-dimensional settings, you should reduce $\sigma^2$ more slowly as you get more data. This is because in high dimensions, the data remains "clumped together" even as you get more of it, and you need to maintain a broader attention distribution to capture relevant information.

The intuition is that in high-dimensional settings, the "curse of dimensionality" means that the data becomes more "sparse" as you add more dimensions. This means that you need to maintain a more "diffuse" attention distribution to capture relevant information, even as you get more data.總之，當你獲得更多數據時，你應該減少注意力機制的尺度的速率取決於數據的維度。在低維設定中，您可以更快地減少，而在高維設定中，您應該更緩慢地減少它以維持更廣泛的注意力分佈。

1. Parzen windows density estimates are given by . Prove that for binary classification the function , as obtained by Parzen windows is equivalent to Nadaraya--Watson classification.

2. Implement stochastic gradient descent to learn a good value for kernel widths in Nadaraya–Watson regression.

2.1 What happens if you just use the above estimates to minimize directly? Hint: is part of the terms used to compute .

2.2 Assume that all lie on the unit sphere, i.e., all satisfy . Can you simplify the term in the exponential? Hint: we will later see that this is very closely related to dot product attention.

tom-ci

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