## 3 Proposed Algorithms
### Variance Feedback
** McDiarmid's Test for Drift Detection  
The experiments in MDDM [16] demonstrated that under identical conditions, [PERSON]'s inequality exhibits lower detection latency compared to [PERSON]'s inequality in drift detection tasks. Therefore, we believe that the [PERSON] test is also applicable in situations where \(\sigma^{2}>\sigma_{c}^{2}\). The [PERSON] test is based on [PERSON]'s inequality. **Theorem 3**. : _[PERSON]'s inequality--Let \(X_{1}\), \(X_{2}\),..., \(X_{n}\) be \(n\) independent random variables. If there exists a function mapping \(f\), such that for \(X_{1}\),...,\(X_{n}\), it holds that for any \(i\) and \(X_{i}\ ^{\prime}\in R\), there is:_  
Figure 5: Workflow of Variance feedback (Including weight generation, mean generation, and statistical test selection). \[\left|f(X_{1},\ldots,X_{i},\ldots,X_{n})-f(X_{1},\ldots,X^{\prime}_{i},\ldots,X_{ n})\right|\leq c_{i} \tag{16}\]  
This implies that by substituting any value for \(X_{i}\) to modify the function \(f\), at most \(c_{i}\) changes. Thus, for all \(\varepsilon_{M}>0\), we have:  
\[\Pr(E[f]-f\geq\varepsilon_{M})\leq\exp\left(-\frac{2\varepsilon_{M}^{2}}{ \sum_{i=1}^{n}c_{i}^{2}}\right) \tag{17}\]  
The significance threshold derived from the McDiarmid inequality can be expressed as Equation (18). \[\varepsilon_{M}\ =\ \sqrt{\frac{\sum_{i=1}^{n}c_{i}^{2}}{2}\ln\frac{1}{\delta}} \tag{18}\]  
\[c_{i}\ =\ \frac{w_{i}}{\sum_{i=1}^{n}w_{i}} \tag{19}\]  
where \(w_{i}\) represents the weight of data instances. **Statistical Test 3. ** Kolmogorov's Test for Drift Detection  
Literature [20] derived the [PERSON]'s inequality and obtained the partial sum Kolmogorov inequality under Bernoulli distribution. This inequality can be proven to generate significance thresholds more compact than the Hoeffding bound. In the following, this research will introduce this inequality. For convenience, this form of the [PERSON] inequality will be referred to as the Kolmogorov inequality. **Theorem 4. **_[PERSON]'s inequality--For a sequence of independent Bernoulli random variables \(X_{1}\), \(X_{2\ldots}\) with \(E(X_{i})=p_{i}\), which may not necessarily have the same distribution. For \(\forall\varepsilon<1\), \(\overline{p}_{k}+\varepsilon\)\(>1/2\) or \(\overline{p}_{k}<1/2\). Then_  
\[\Pr\left(\sup_{k\geq n}(\overline{X}_{k}-\overline{p}_{k})>\varepsilon_{K} \right)\leq\exp\left(-n\left(2\varepsilon_{K}^{2}+\frac{1}{3}\varepsilon_{K}^ {4}e^{\frac{2}{4}}\right)\right) \tag{20}\]  
where \(\overline{X}_{k}\ =\ (1/k)\sum_{i=1}^{k}X_{i}\) and \(\overline{p}_{k}\ =\ (1/k)\sum_{i=1}^{k}p_{i}\). Research findings indicate that when \(k=n\), this inequality can be transformed into a statistical testing method applicable to drift detection, termed Kolmogorov's test, as shown in Equation (21). \[\Pr(p_{max}-p>\varepsilon_{K})\leq\exp\left(-n\left(2\varepsilon_{K}^{2}+ \frac{1}{3}\varepsilon_{K}^{4}e^{\frac{2}{4}}\right)\right) \tag{21}\]  
It is easy to prove that the significance threshold \(\varepsilon_{K}\) generated by the Kolmogorov test is smaller than the threshold \(\varepsilon_{H}\) of the Hoeffding test. Therefore, under the condition where \(\sigma^{2}>\sigma_{c}^{2}\), it also theoretically possesses faster drift detection speed. All three statistical testing methods mentioned above are applicable to \(\sigma^{2}>\sigma_{c}^{2}\). In the experimental section (Section 4), we will employ each of these three statistical testing methods for drift detection to validate the effectiveness of the variance feedback strategy proposed in this paper.