This commit is contained in:
Marco Realacci 2024-11-02 16:28:37 +01:00
parent be7844b4f3
commit eea09ec9b8
15 changed files with 35749 additions and 63 deletions

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@ -5,5 +5,6 @@
"mathlive-in-editor-mode", "mathlive-in-editor-mode",
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"local-gpt", "local-gpt",
"obsidian-latex-suite" "obsidian-latex-suite",
"companion"
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19
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@ -49,7 +161,7 @@
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@ -40,7 +40,7 @@ as we can see, the system explores more at the beginning, which is good as it wi
- only effective for stationary problems - only effective for stationary problems
- for non-stationary problems we have to use eps-greedy - for non-stationary problems we have to use eps-greedy
### Optimism in the Face of Uncertainty ### Optimism in the Face of Uncertainty - Upper Confidence Bound (UCB)
- ... - ...
- easy problem: - easy problem:
- two arms, one always good and one always bad - two arms, one always good and one always bad
@ -59,17 +59,23 @@ which actions should we peek?
![[Pasted image 20241025090344.png]] ![[Pasted image 20241025090344.png]]
the brackets represent a confidence interval around q*(a). The system is confident that the value lies somewhere in the region. the brackets represent a confidence interval around q*(a). The system is confident that the value lies somewhere in the region.
The problem is that, when a region is large, we don't know where the average value is! So we have to try!
If region is very small, we are very certain! If region is very small, we are very certain!
![[Pasted image 20241025090549.png]] ![[Pasted image 20241025090549.png]]
In this situation we chose Q2 as estimated value is the highest. In this situation we chose Q2 as estimated value is the highest.
![[Pasted image 20241031144640.png]]
But in this case Q1.
#### Action selection #### Action selection
![[Pasted image 20241025090625.png]] ![[Pasted image 20241025090625.png]]
... check slides for formula explaination ... - We will select the action that has the highest estimated value plus the upper-confidence bound exploration term
- The c parameter is a user-specified parameter that controls the amount of exploration
- $N_{t}(a)$ is the number of times an action $a$ has been taken
- to systematically reduce uncertainity, UCB explores more at the beginning - to systematically reduce uncertainity, UCB explores more at the beginning
- UCB's exploration reduces over time, eps-greedy continues to take a random action 10% of the time - UCB's exploration reduces over time, eps-greedy continues to take a random action 10% of the time
### AI generated summary
In the end, we can say that UCB is an effective strategy for balancing exploration and exploitation in multi-armed bandit problems. Unlike ε-greedy methods, which maintain a fixed level of exploration throughout the process, UCB dynamically adjusts its exploration rate based on the uncertainty associated with each action's value estimates. This adaptability makes UCB particularly well-suited for scenarios where initial exploration is crucial to quickly identify high-reward actions but later iterations require more focused exploitation.

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@ -57,7 +57,7 @@ This is a Markov Process but we also have a reward function! We also have a disc
Value function Value function
- The value function v(s) gives the long-term value of (being in) state s - The value function v(s) gives the long-term value of (being in) state s
- The state value function v(s) of an MRP is the expected return starting from state s $𝑉) = 𝔼 [𝐺𝑡 |𝑆𝑡 = 𝑠]$ - The state value function v(s) of an MRP is the expected return starting from state s $𝑉 = 𝔼 [𝐺𝑡 |𝑆𝑡 = 𝑠]$
![[Pasted image 20241030103519.png]] ![[Pasted image 20241030103519.png]]
![[Pasted image 20241030103706.png]] ![[Pasted image 20241030103706.png]]
@ -111,8 +111,8 @@ The state-value function v𝜋(s) of an MDP is the expected return starting from
The action-value function q 𝜋 (s,a) is the expected return starting from state s, taking action a, and then following policy 𝜋 $$q 𝜋(a|s)= 𝔼𝜋 [ Gt | St=s, At=a ]$$ The action-value function q 𝜋 (s,a) is the expected return starting from state s, taking action a, and then following policy 𝜋 $$q 𝜋(a|s)= 𝔼𝜋 [ Gt | St=s, At=a ]$$
![[Pasted image 20241030105022.png]] ![[Pasted image 20241030105022.png]]
- The state-value function can again be decomposed into immediate reward plus discounted value of successor state $$v\pi(s) = E\pi[Rt+1 + v⇡(St+1) | St = s]$$ - The state-value function can again be decomposed into immediate reward plus discounted value of successor state $$v_{\pi}(s) = E_{\pi}[R_{t+1} + v_{\pi}(S_{t+1}) | St = s]$$
- The action-value function can similarly be decomposed $$q\pi(s, a) = E\pi [Rt+1 + q⇡(St+1, At+1) | St = s, At = a]$$ - The action-value function can similarly be decomposed $$q_{\pi}(s, a) = E_{\pi}[R_{t+1} + q_{\pi}(S_{t+1}, A_{t+1}) | St = s, At = a]$$
![[Pasted image 20241030105148.png]]![[Pasted image 20241030105207.png]] ![[Pasted image 20241030105148.png]]![[Pasted image 20241030105207.png]]
![[Pasted image 20241030105216.png]] ![[Pasted image 20241030105216.png]]
putting all together putting all together

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@ -2,44 +2,39 @@ Assistant
Hello, I am your assistant. How can I help you? Hello, I am your assistant. How can I help you?
- - - - - - - - - -
User User
what are UAVs? Bias vs variance
- - - - - - - - - -
Assistant Assistant
Based on our previous conversation and my understanding of the knowledge you've shared with me, I'll respond to your query. **Bias vs Variance**
=====================
**UAVs (Unmanned Aerial Vehicles)** The bias-variance tradeoff is a fundamental concept in machine learning that refers to the balance between two types of errors: **bias** and **variance**.
UAVs, also known as drones or unmanned aircraft systems (UAS), are [[Remote Controlled Aircraft|remote controlled aircraft]] that operate without a human pilot on board. They can be used for various purposes, including: ### Bias
* Surveillance and reconnaissance * The error caused by **bias** measures how far off the predictions of a model are from the true value. It represents the difference between the predicted mean of the model and the correct value.
* Military operations * A high bias indicates that the model is underfitting, meaning it's too simple to capture the underlying patterns in the data.
* Agricultural monitoring
* Environmental monitoring
* Aerial photography and videography
* Package delivery
UAVs typically consist of an airframe, engine, propellers, sensors, and a control system. Some UAVs are controlled remotely by a human pilot, while others can fly autonomously using pre-programmed flight plans or GPS navigation. ### Variance
If you'd like to know more about UAVs, feel free to ask! * The error due to **variance** measures how much the predictions of a model vary for a given point. It reflects the impact of randomness in the data on the prediction results.
* A high variance indicates that the model is overfitting, meaning it's too complex and fits the noise in the training data.
(Note: I couldn't find any specific notes on UAVs in your existing knowledge, so I've provided general information. If you have any additional notes or details, please let me know!) ### Diagrammatic Representation
- - - - -
User The bias-variance tradeoff can be visualized using a **target diagram**, where the center represents a perfect model that predicts the true values.
latex absolute value
- - - - - ### Mathematical Definition
Assistant
It seems that there is no relevant information about "latex absolute value" in your existing knowledge. However, based on my understanding of LaTeX, I can provide you with a brief explanation. The prediction error at point x can be decomposed into bias, variance, and irreducible error:
**Absolute Value in LaTeX** $$\text{Err}(x) = \left(E[\hat{f}(x)] - f(x)\right)^2 + E\left[(\hat{f}(x) - E[\hat{f}(x)])^2\right] + \sigma^2_\epsilon$$
In LaTeX, the absolute value of an expression is denoted by the `\lvert` command for left absolute value and `\rvert` command for right absolute value. The `|` symbol can also be used to represent absolute value. ### Related Concepts
For example: * **Optimism in the Face of Uncertainty - Upper Confidence Bound (UCB)**: Discusses balancing exploration and exploitation in a multi-armed bandit setting, related to bias-variance tradeoff.
* **Gaussian Filtering**: Describes linear filtering using a Gaussian kernel, which can smooth out noise and reduce variance.
* `\lvert x \rvert` represents the absolute value of `x` * **Logistic Regression**: Introduces maximum likelihood estimation (MLE) for logistic regression, related to bias-variance tradeoff in model selection.
* `|x|` is an alternative way to represent absolute value
These concepts can be used to discuss the bias-variance tradeoff in more detail.
If you need more information or have specific questions about LaTeX, feel free to ask!
(Note: I've assumed that your knowledge on LaTeX is limited, so I've provided basic information. If you have any additional notes or details, please let me know!)
- - - - - - - - - -

0
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