Probability Bounds and its Analysis (PBA)  Python for Integrated Circuits   An Online Book  

Python for Integrated Circuits http://www.globalsino.com/ICs/  


Chapter/Index: Introduction  A  B  C  D  E  F  G  H  I  J  K  L  M  N  O  P  Q  R  S  T  U  V  W  X  Y  Z  Appendix  
================================================================================= Following is the steps for Proving Probability Bounds Prove for Individual Hypotheses (h):
 [3947a]  Note: Complementary inequality: Original Inequality: For an individual hypothesis ℎ from the hypothesis class , the original inequality asserts:
Complementary Inequality: The complementary inequality asserts that the opposite event occurs with high probability:
In other words, the complementary inequality focuses on the probability of the event not occurring with high probability. This is a common approach in probability analysis. If you're ensuring that an event occurs with high probability, the complementary inequality helps quantify the probability of the opposite event (not occurring) with high probability.  Take a Union Bound Over All h:
To prove this for an individual hypothesis h, we can use concentration inequalities like Hoeffding's inequality or Chernoff bounds, which are often used in statistical learning theory to bound deviations between empirical and true risks. Let's use Hoeffding's inequality as an example: Hoeffding's Inequality states that for any random variables independently drawn from a distribution with range [a, b], and their average , the following holds for any : [3947b] In our case, we can consider as the loss for a single data point when using hypothesis h, and as the empirical risk .Now, we can apply Hoeffding's Inequality with the equation below:  [3947c] Now, we set and based on the range of the loss function: [3947d] We notice that is the same as the one defined in our original inequality, then we simplify the expression within the exponent: [3947e] Substituting this result into Hoeffding's Inequality, and then simplify the righthand side:  [3947f] Now, notice that the righthand side of this inequality is the same as (the desired level of confidence): [3947g] This step is crucial because it demonstrates that the probability of the event on the lefthand side (the deviation between empirical and true risk) is bounded by the desired confidence level .Now, the final conclusion is that for the individual hypothesis h, the probability of the event is indeed bounded by ., and we can observe that the righthand side is equal to , which verifies the desired inequality for the individual hypothesis h in Equation 3947a.Take a Union Bound Over All h: Now, we extend the result to the entire hypothesis class H. Since there are potentially infinitely many hypotheses in H, we consider the worstcase scenario where we have a countably infinite number of hypotheses. To apply a union bound over all hypotheses, we sum the probabilities for each individual hypothesis:  [3947h] By the properties of the union bound, this probability is bounded by:  [3947i] Therefore, we have established the probability bound for the entire hypothesis class H. If you try to express that the absolute difference between the true risk and the empirical risk decreases as (the sample size) increases, you might want to use the Big O notation. The correct way to express this would be: [3947j] wehre,
The expression states that the absolute difference between the true risk and the empirical risk is bounded by a term that decreases as increases. Specifically, it decreases at a rate of . This is a common result in statistical learning theory, indicating that as you collect more data (larger ), your empirical risk estimate becomes a better approximation of the true risk.
Therefore, Equation 3947j summarizes that, as you gather more data, your empirical risk estimate becomes a better and better approximation of the true risk, and the rate of improvement is characterized by. However, please note that the Big O notation is typically used to describe the upper bound of the difference. If you want to describe the convergence behavior more precisely, you may use limit notation, which would be:  [3947k] This notation states that as the sample size ( ) approaches infinity, the absolute difference between the true risk and the empirical risk converges to zero, indicating that the empirical risk becomes an increasingly accurate estimate of the true risk as you collect more data.In statistical modeling and machine learning:
Therefore, when you parameterize a model, you indeed introduce the parameter , which represents the internal configuration of the model. The process of training the model involves finding the optimal values for these parameters to minimize the difference between the model's predictions and the actual data.With Hoeffding inequality, when you have a specific scenario where you are dealing with a set of independent and identically distributed (i.i.d.) random variables that are bounded within the closed interval :
Under these conditions, you can correctly apply Hoeffding's inequality to bound the probability of observing a large deviation between the sample mean and the true mean for random variables bounded within the closed interval . The specified bounds for this scenario would indeed be , meaning that the probability of observing a deviation outside this range is bounded by the properties of Hoeffding's inequality.Based on Hoeffding Inequality at page3973 and page3957, we have,  [3947l] Based on Union Bound, we have,  [3947m] where,
Good even has hypothesis,  [3947n] Now, we extend it for everything in S. With Lipschitzness at page3931, we can have: if l((x, y), θ) is KLipschitzness in θ, then L(θ) and L^(θ) are both KLipschitzness. Based on εcover theory at page3934, we have,  [3947o] In combination of Inequality 3947o and Lipschitzness, we then have,  [3947p]  [3947q] We know,  [3947r] The first and third terms on the righhand side of Equation 3947r are about differences between θ and θ_{0}. Therefore, the values of both terms are less than Kε, and the second term on the righthand side of the equation is less than because θ_{0} is in C. Then,  [3947s] And,  [3947t] Let's set,  [3947u] Then, we get,  [3947v] Based on the discussion above, the logcover size is,  [3947w] K is inside the log, so that it is not sensitive to the choice of K. Assuming (note they are very rough assumptions),  [3947x]  [3947y] Then, There, we have,  [3947aa] Here, generalization error (refer to page3930) is given by.  [3947ab] The first term in the bracket is from the finite hypothesis case, while the second term is discretization error, (see page3929) coming from the KLipschitzness. Since the first term depends on log(1/ϵ) and increases very slowly as ϵ goes to 0, and the second term depends on ϵ, there is trading off between the two. Therefore, sometime, you can ignore the second term when ϵ is close to 0. If you have bigger H (meaning worse bounds), then you need to have more samples. H typically represents the hypothesis space or the complexity of the model you are using. When you have a more complex model (bigger H), it has the potential to fit the training data more closely, but it also increases the risk of overfitting, meaning that the model may not generalize well to unseen data. To illustrate this concept with an equation, you can use the classic biasvariance tradeoff in the context of empirical risk minimization (ERM). The expected prediction error (EPE) of a machine learning model can be decomposed into three components: EPE = Bias^2 + Variance + Irreducible Error  [3947ac]
Now, the relationship between model complexity (H), sample size (N), and the biasvariance tradeoff can be illustrated as follows:
To control the tradeoff between bias and variance, you need more data (larger N). More data helps to stabilize the variance by providing the model with a better understanding of the underlying patterns in the data and reducing its sensitivity to noise. Therefore, when you have a bigger H (worse bounds in terms of overfitting risk), you typically need more samples (larger N) to ensure that the model generalizes well and strikes a balance between bias and variance. This is a fundamental concept in machine learning and statistical learning theory. ============================================ Illustration of the application of Hoeffding's inequality for a set of random variables bounded within the closed interval Code: The equations and inequalities for the various components used in the script are given below: Sample Mean ( ):
 [3947l] True Mean ( ):
 [3947n] Deviation Threshold ( ):
 [3947o] Actual Probability of Deviation:
 [3947p] Here, 1(. is the indicator function, which equals 1 if the condition inside is true and 0 otherwise.Deviation Probability (Actual):
Deviation Probability (Hoeffding Bound):
 [3947q] This is the probability of observing a deviation greater than or equal to between the sample mean and the true mean, as bounded by Hoeffding's inequality.============================================ If is fixed (meaning the hypothesis class is fixed and not changing), and the randomness comes from a dataset that goes into the empirical risk , and if and for all , then the probability bound based on Hoeffding's inequality would be: [3947r]
The sum of these events represents the event that multiple hypotheses ℎ fail to satisfy the condition . We'll discuss the probability of this union and how it relates to the individual probabilities and .Individual Probabilities :
 [3947s]
 [3947t]
Union of Individual Failures _{h} _{h} :
 [3947u]
Sum of the Events :
 [3947v] Now, let's relate these concepts: If you want to find the probability that two or more hypotheses fail (i.e., the sum of the events), it can be expressed as:  [3947w] where, Independence: When hypotheses are independent, individual probabilities P(E_{h}) are not correlated with each other, and the union probability and the sum probability depend on the individual probabilities but are not directly correlated. Correlation: If there is a correlation between the hypotheses, meaning that the failure of one hypothesis affects the failure of others, then the probabilities may be correlated. In such cases, more advanced statistical techniques may be needed to analyze the relationship between the events. The exact correlations and dependencies between these probabilities depend on the specific context and assumptions made about the hypotheses and events being considered. Now, we have:  [3947x] where,: Then, we have:  [3947y] As discussed above, the behavior of in terms of how it scales with (sample size) depends on the specific scenario and assumptions in machine learning and statistical analysis. It can indeed vary between cases where it scales as and cases where it scales as1/n^{1/2}, and there can be other cases as well. These differences are often related to different assumptions and conditions. Let's explore each case and the associated conditions:1. : Assumption: This case typically arises when we assume that the excess risk converges to zero as the sample size ( ) increases. It implies that as you collect more data, the difference between the risk of the learned model ( ) and the risk of the best possible model (ℎ*) decreases at a rate of .Conditions: This assumption is often associated with strong convergence properties of the learning algorithm and may require certain regularity conditions on the data and model class. It's common in scenarios where the law of large numbers or central limit theorems can be applied. 2. : Assumption: This case arises when we assume that the excess risk converges to zero at a slower rate, specifically 1/n^{1/2}, as the sample size ( ) increases.Conditions: This assumption may occur in scenarios where there is more uncertainty in the learning process, and the excess risk decreases more slowly with increasing sample size. It can be related to situations where there is noise in the data, model complexity, or limited information in the data. 3. Other Cases Related to :There can be other cases as well, where the rate of convergence of with respect to is different. For example, you might encounter cases where the excess risk converges as 1/n^{2}or with some other dependence on .Conditions: These cases depend on the specific assumptions and characteristics of the problem, including the choice of learning algorithm, the nature of the data, and the complexity of the model. Therefore, the scaling of with respect to depends on the assumptions and conditions of the specific problem. Stronger assumptions and more favorable conditions can lead to faster convergence (e.g., ), while more challenging scenarios may result in slower convergence (e.g., 1/n^{1/2}). The choice of algorithm, the presence of noise in the data, the model complexity, and other factors all play a role in determining the rate of convergence.


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