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# CSE510 Deep Reinforcement Learning (Lecture 21)
## Exploration in RL
### Information state search
Uncertainty about state transitions or dynamics
Dynamics prediction error or Information gain for dynamics learning
#### Computational Curiosity
- "The direct goal of curiosity and boredom is to improve the world model."
- "Curiosity Unit": reward is a function of the mismatch between model's current predictions and actuality.
- There is positive reinforcement whenever the system fails to correctly predict the environment.
- Thus the usual credit assignment process ... encourages certain past actions in order to repeat situations similar to the mismatch situation. (planning to make your (internal) world model to fail)
#### Reward Prediction Error
- Add exploration reward bonuses that encourage policies to visit states that will cause the prediction model to fail.
$$
R(s,a,s') = r(s,a,s') + \mathcal{B}(\|T(s,a,\theta)-s'\|)
$$
- where $r(s,a,s')$ is the extrinsic reward, $T(s,a,\theta)$ is the predicted next state, and $\mathcal{B}$ is a bonus function (intrinsic reward bonus).
- Exploration reward bonuses are non-stationary: as the agent interacts with the environment, what is now new and novel, becomes old and known.
[link to the paper](https://arxiv.org/pdf/1507.08750)
</details>
<details>
<summary>Example</summary>
Learning Visual Dynamics
- Exploration reward bonuses $\mathcal{B}(s, a, s') = \|T(s, a; \theta) - s'\|$
- However, trivial solution exists: could get reward by just moving around randomly.
---
- Exploration reward bonuses with autoencoders $\mathcal{B}(s, a, s') = \|T(E(s';\theta),a;\theta)-E(s';\theta)\|$
- But suffer the problems of autoencoding reconstruction loss that has little to do with our task
#### Task Rewards vs. Exploration Rewards
Exploration rewards bonuses:
$$
\mathcal{B}(s, a, s') = \|T(E(s';\theta),a;\theta)-E(s';\theta)\|
$$
Only task rewards:
$$
R(s,a,s') = r(s,a,s')
$$
Task+curiosity rewards:
$$
R^t(s,a,s') = r(s,a,s') + \mathcal{B}^t(s, a, s')
$$
Sparse task + curiosity rewards:
$$
R^t(s,a,s') = r^t(s,a,s') + \mathcal{B}^t(s, a, s')
$$
Only curiosity rewards:
$$
R^c(s,a,s') = \mathcal{B}^c(s, a, s')
$$
#### Extrinsic reward RL is not New
- Itti, L., Baldi, P.F.: Bayesian surprise attracts human attention. In: NIPS’05. pp. 547–554 (2006)
- Schmidhuber, J.: Curious model-building control systems. In: IJCNN’91. vol. 2, pp. 1458–1463 (1991)
- Schmidhuber, J.: Formal theory of creativity, fun, and intrinsic motivation (1990-2010). Autonomous Mental Development, IEEE Trans. on Autonomous Mental Development 2(3), 230–247 (9 2010)
- Singh, S., Barto, A., Chentanez, N.: Intrinsically motivated reinforcement learning. In: NIPS’04 (2004)
- Storck, J., Hochreiter, S., Schmidhuber, J.: Reinforcement driven information acquisition in non-deterministic environments. In: ICANN’95 (1995)
- Sun, Y., Gomez, F.J., Schmidhuber, J.: Planning to be surprised: Optimal Bayesian exploration in dynamic environments (2011), http://arxiv.org/abs/1103.5708
#### Limitation of Prediction Errors
- Agent will be rewarded even though the model cannot improve.
- So it will focus on parts of environment that are inherently unpredictable or stochastic.
- Example: the noisy-TV problem
- The agent is attracted forever in the most noisy states, with unpredictable outcomes.
#### Random Network Distillation
Original idea: Predicting the output of a fixed and randomly initialized neural network on the next state, given the current state and action.
New idea: Predicting the output of a fixed and randomly initialized neural network on the next state, given the **next state itself.**
- The target network is a neural network with fixed, randomized weights, which is never trained.
- The prediction network is trained to predict the target network's output.
> the more you visit the state, the less loss you will have.
### Posterior Sampling
Uncertainty about Q-value functions or policies
Selecting actions according to the probability they are best according to the current model.
#### Exploration with Action Value Information
Count-Based and Curiosity-driven method does not take into
account the action value information
![Action Value Information](https://notenextra.trance-0.com/CSE510/Action_Value_Information.png)
> In this case, the optimal solution is action 1, but we will explore action 3 because it has the highest uncertainty. And it takes long to distinguish action 1 and 2 since they have similar values.
#### Exploration via Posterior Sampling of Q Functions
- Represent a posterior distribution of Q functions, instead of a point estimate.
1. Sample from $P(Q), Q\sim P(Q)$
2. Choose actions according to this $Q$ for one episode $a=\arg\max_{a} Q(s,a)$
3. Update $P(Q)$ based on the sampled $Q$ and collected experience tuples $(s,a,r,s')$
- Then we do not need $\epsilon$-greedy for exploration! Better exploration by representing uncertainty over Q.
- But how can we learn a distribution of Q functions $P(Q)$ if Q function is a deep neural network?
#### Bootstrap Ensemble
- Neural network ensembles: train multiple Q-function approximations each on using different subset of the data
- Computationally expensive
- Neural network ensembles with shared backbone: only the heads are trained with different subset of the data
### Questions
- Why do PG methods implicitly support exploration?
- Is it sufficient? How can we improve its implicit exploration?
- What are limitations of entropy regularization?
- How can we improve exploration for PG methods?
- Intrinsic-motivated bonuses (e.g., RND)
- Explicitly optimize per-state entropy in the return (e.g., SAC)
- Hierarchical RL
- Goal-conditional RL
- What are potentially more effective exploration methods?
- Knowledge-driven
- Model-based exploration

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CSE510_L18: "CSE510 Deep Reinforcement Learning (Lecture 18)",
CSE510_L19: "CSE510 Deep Reinforcement Learning (Lecture 19)",
CSE510_L20: "CSE510 Deep Reinforcement Learning (Lecture 20)",
CSE510_L21: "CSE510 Deep Reinforcement Learning (Lecture 21)",
}

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> [!TIP]
>
> error + known location $\implies$ erasure. $d = 2 \implies$ 1 erasure is correctable.

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# CSE5313 Coding and information theory for data science (Lecture 20)
## Review for Private Information Retrieval
### PIR from replicated databases
For 2 replicated databases, we have the following protocol:
- User has $i \sim U_{m}$.
- User chooses $r_1, r_2 \sim U_{\mathbb{F}_2^m}$.
- Two queries to each server:
- $q_{1, 1} = r_1 + e_i$, $q_{1, 2} = r_2$.
- $q_{2, 1} = r_1$, $q_{2, 2} = r_2 + e_i$.
- Server $j$ responds with $q_{j, 1} c_j^\top$ and $q_{j, 2} c_j^\top$.
- Decoding?
- $q_{1, 1} c_1^\top + q_{2, 1} c_2^\top = r_1 c_1 + c_2 + e_i c_1^\top = r_1 \cdot 0^\top + x_{i, 1} = x_{i, 1}$.
- $q_{1, 2} c_1^\top + q_{2, 2} c_2^\top = r_2 c_1 + c_2 + e_i c_2^\top = x_{i, 2}$.
PIR-rate is $\frac{k}{2k} = \frac{1}{2}$.
### PIR from coded parity-check databases
For 3 coded parity-check databases, we have the following protocol:
- User has $i \sim U_{m}$.
- User chooses $r_1, r_2, r_3 \sim U_{\mathbb{F}_2^m}$.
- Three queries to each server:
- $q_{1, 1} = r_1 + e_i$, $q_{1, 2} = r_2$, $q_{1, 3} = r_3$.
- $q_{2, 1} = r_1$, $q_{2, 2} = r_2 + e_i$, $q_{2, 3} = r_3$.
- $q_{3, 1} = r_1$, $q_{3, 2} = r_2$, $q_{3, 3} = r_3 + e_i$.
- Server $j$ responds with $q_{j, 1} c_j^\top, q_{j, 2} c_j^\top, q_{j, 3} c_j^\top$.
- Decoding?
- $q_{1, 1} c_1^\top + q_{2, 1} c_2^\top + q_{3, 1} c_3^\top = r_1 c_1 + c_2 + c_3 + e_i c_1^\top = r_1 \cdot 0^\top + x_{i, 1} = x_{i, 1}$.
- $q_{1, 2} c_1^\top + q_{2, 2} c_2^\top + q_{3, 2} c_3^\top = r_2 c_1 + c_2 + c_3 + e_i c_2^\top = x_{i, 2}$.
- $q_{1, 3} c_1^\top + q_{2, 3} c_2^\top + q_{3, 3} c_3^\top = r_3 c_1 + c_2 + c_3 + e_i c_3^\top = x_{i, 3}$.
PIR-rate is $\frac{k}{3k} = \frac{1}{3}$.
## Beyond z=1
### Star-product theme
Given $x=(x_1, \ldots, x_j)_{j\in [n]}, y=(y_1, \ldots, y_j)_{j\in [n]}$, over $\mathbb{F}_q$, the star-product is defined as:
$$
x \star y = (x_1 y_1, \ldots, x_n y_n)
$$
Given two linear codes, $C,D\subseteq \mathbb{F}_q^n$, the star-product code is defined as:
$$
C \star D = span_{\mathbb{F}_q} \{x \star y | x \in C, y \in D\}
$$
Singleton bound for star-product:
$$
d_{C \star D} \leq n-\dim C-\dim D+2
$$
### PIR form a database coded with any MDS code and z>1
To generalize the previous scheme to $z > 1$ need to encode multiple $r$'s together.
- As in the ramp scheme.
> Recall from the ramp scheme, we use $r_1, \ldots, r_z \sim U_{\mathbb{F}_q^k}$ as our key vector to avoid occlusion of the servers.
In the star-product scheme:
- Files are coded with an MDS code $C$.
- The multiple $r$'s are coded with an MDS code $D$.
- The scheme is based on the minimum distance of $C \star D$.
To code the data:
- Let $C \subseteq \mathbb{F}_q^n$ be an MDS code of dimension $k$.
- For all $j \in m$, encode file $x_j = x_{j, 1}, \ldots, x_{j, k}$ using $G_C$:
$$
\begin{pmatrix}
x_{1, 1} & x_{1, 2} & \cdots & x_{1, k}\\
x_{2, 1} & x_{2, 2} & \cdots & x_{2, k}\\
\vdots & \vdots & \ddots & \vdots\\
x_{m, 1} & x_{m, 2} & \cdots & x_{m, k}
\end{pmatrix} \cdot G_C = \begin{pmatrix}
c_{1, 1} & c_{1, 2} & \cdots & c_{1, n}\\
c_{2, 1} & c_{2, 2} & \cdots & c_{2, n}\\
\vdots & \vdots & \ddots & \vdots\\
c_{m, 1} & c_{m, 2} & \cdots & c_{m, n}
\end{pmatrix}
$$
- For all $j \in n$, store $c_j = c_{1, j}, c_{2, j}, \ldots, c_{m, j}$ (a column of the above matrix) in server $j$.
Let $r_1, \ldots, r_z \sim U_{\mathbb{F}_q^k}$.
To code the queries:
- Let $D \subseteq \mathbb{F}_q^k$ be an MDS code of dimension $z$.
- Encode the $r_j$'s using $G_D=[g_1^\top, \ldots, g_z^\top]$.
$$
(r_1^\top, \ldots, r_z^\top) \cdot G_D = \begin{pmatrix}
r_{1, 1} & r_{2, 1} & \cdots & r_{z, 1}\\
r_{1, 2} & r_{2, 2} & \cdots & r_{z, 2}\\
\vdots & \vdots & \ddots & \vdots\\
r_{1, m} & r_{2, m} & \cdots & r_{z, m}
\end{pmatrix}
\cdot G_D=\left((r_1^\top,\ldots, r_z^\top)g_1^\top,\ldots, (r_1^\top,\ldots, r_z^\top)g_n^\top \right)
$$
To introduce the "errors in known locations" to the encoded $r_j$'s:
- Let $W \in \{0, 1\}^{m \times n}$ with some $d_{C \star D} - 1$ entries in its $i$-th row equal to 1.
- These are the entries we will retrieve.
For every server $j \in [n]$ send $q_j = r_1^\top, \ldots, r_z^\top g_j^\top + w_j$, where $w_j$ is the $i$-th column of $W$.
- This is similar to ramp scheme, where $w_j$ is the "message".
- Privacy against collusion of $z$ servers.
Response from server: $a_j = q_j c_j^\top$.
Decoding? Let $Q \in \mathbb{F}_q^{m \times n}$ be a matrix whose columns are the $q_j$'s.
$$
Q = \begin{pmatrix}
r_1^\top & \cdots & r_z^\top
\end{pmatrix} \cdot G_D + W
$$
- The user has
$$
\begin{aligned}
q_1 c_1^\top, \ldots, q_n c_n^\top &= \left(\sum_{j \in m} q_{1, j} c_{j, 1}, \ldots, \sum_{j \in m} q_{n, j} c_{j, n}\right) \\
&=\sum_{j \in m} (q_{1,j}c_{j, 1}, \ldots, q_{n,j}c_{j, n}) \\
&=\sum_{j \in m} q^j \star c^j
$$
where $q^j$ is a row of $Q$ and $c^j$ is a codeword in $C$ (an $n, k$ $q$ MDS code).
We have:
- $Q=(r_1^\top, \ldots, r_z^\top) \cdot G_D + W$
- $W\in \{0, 1\}^{m \times n}$ with some $d_{C \star D} - 1$ entries in its $i$-th row equal to 1.
- $(q^j \star c^j)=sum_{j \in m} q^j \star c^j$
- Each $q^j$ is a row of $Q$
- For $j \neq i$, $q^j$ is a codeword in $D$
- $q^i = d^i + w^i$
- Therefore:
$$
\begin{aligned}
\sum_{j \in [m]} q^j \star c^j &= \sum_{j \neq i} (d^j \star c^j) + ((d^i + w^i) \star c^i) \\
&= \sum_{j \neq i} (d^j \star c^j) + w^i \star c^i
&= (\text{codeword in } C \star D )+( \text{noise of Hamming weight } \leq d_{C \star D} - 1)
\end{aligned}
$$
Multiply by $H_{C \star D}$ and get $d_{C \star D} - 1$ elements of $c^i$.
- Recall that $c^i = x_i \cdot G_C$
- Repeat $k^{d_{C \star D} - 1}$ times to obtain $k$ elements of $c^i$.
- Suffices to obtain $x_i$, since $C$ is $n, k$ $q$ MDS code.
PIR-rate:
- = $\frac{k}{# \text{ downloaded elements}} = \frac{k}{\frac{k}{d_{C \star D} - 1} \cdot n} = \frac{d_{C \star D} - 1}{n}$
- Singleton bound for star-product: $d_{C \star D} \leq n - \dim C - \dim D + 2$.
- Achieved with equality if $C$ and $D$ are Reed-Solomon codes.
- PIR-rate = $\frac{n - \dim C - \dim D + 1}{n} = \frac{n - k - z + 1}{n}$.
- Intuition:
- "paying" $k$ for "reconstruction from any $k$".
- "paying" $z$ for "protection against colluding sets of size $z$".
- Capacity unknown! (as of 2022).
- Known for special cases, e.g., $k = 1, z = 1$, certain types of schemes, etc.
### PIR over graphs
Graph-based replication:
- Every file is replicated twice on two separate servers.
- Every two servers have at most one file in common.
- "file" = "granularity" of data, i.e., the smallest information unit shared by any two servers.
A server that stores $(x_{i, j})_{j=1}^d$ receives $(q_{i, j})_{j=1}^d$, and replies with $\sum_{j=1}^d q_{i, j} \cdot x_{i, j}$.
The idea:
- Consider a 2-server replicated PIR and "split" the queries between the servers.
- Sum the responses, unwanted files "cancel out", while $x_i$ does not.
Problem: Collusion.
Solution: Add per server randomness.
Good for any graph, and any $q \geq 3$ (for simplicity assume $2 | q$).
The protocol:
- Choose random $\gamma \in \mathbb{F}_q^n$, $\nu \in \mathbb{F}_q^m$, and $h \in \mathbb{F} \setminus \{0, 1\}$.
- Queries:
- If node $j$ is incident with edge $\ell$, send $q_{j, \ell} = \gamma_j \cdot \nu_\ell$ to node $j$.
- I.e., if server $j$ stores file $\ell$.
- Except one node $j_0$ that stores $x_i$, which gets $q_{j_0, i} = h \cdot \gamma_{j_0} \cdot \nu_i$.
- Server $j$ responds with $a_j = \sum_{j=1}^d q_{j, \ell} \cdot x_{i, \ell}$.
- Where $x_{i, 1}, \ldots, $x_{i, d}$ are the files adjacent with it.
<details>
<summary>Example</summary>
- Consider the following graph.
- $n = 5, m = 7, and i = 3$.
- $q_3 = \gamma_3 \cdot v_2, v_3, v_6$ and $a_3 = x_2 \cdot \gamma_3 v_2 + x_3 \cdot \gamma_3 v_3 + x_6 \cdot \gamma_3 v_6$.
- $q_2 = \gamma_2 \cdot v_1, h v_3, v_4$ and $a_2 = x_1 \cdot \gamma_2 v_1 + x_3 \cdot h \gamma_2 v_3 + x_4 \cdot \gamma_2 v_4$.
![Example of PIR over graphs](https://notenextra.trance-0.com/CSE5313/PIR_over_graphs.png)
</details>
Correctness:
- $\sum_{j=1}^5 \gamma_j^{-1} a_j =( h + 1 )v_3 x_3$
- $h \neq 1, v_3 \neq 0 \implies$ find $x_3$.
Parameters:
- Storage overhead 2 (for any graph).
- Download $n \cdot k$.
- PIR rate 1/n.
Collusion resistance:
1-privacy: Each node sees an entirely random vector.
2-privacy:
- If no edge – as for 1-privacy.
- If edge exists – E.g.,
- $\gamma_3 v_6$ and $\gamma_4 v_6$ are independent.
- $\gamma_3 v_3$ and $h \cdot \gamma_2 v_3$ are independent.
S-privacy:
- Let $S \subseteq n$ (e.g., $S = 2,3,5$), and consider the query matrix of their mutual files:
$$
Q_S = diag(\gamma_3, \gamma_2, \gamma_5) \begin{pmatrix} 1 &\\ h & 1 \\ & 1\end{pmatrix} diag(v_3, v_4)
$$
- It can be shown that $Pr(Q_S)=\frac{1}{(q-1)^4}$, regardless of $i \implies$ perfect privacy.

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CSE5313_L16: "CSE5313 Coding and information theory for data science (Exam Review)",
CSE5313_L17: "CSE5313 Coding and information theory for data science (Lecture 17)",
CSE5313_L18: "CSE5313 Coding and information theory for data science (Lecture 18)",
CSE5313_L19: "CSE5313 Coding and information theory for data science (Exam Review)",
CSE5313_L19: "CSE5313 Coding and information theory for data science (Lecture 19)",
CSE5313_L20: "CSE5313 Coding and information theory for data science (Lecture 20)",
}

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# CSE5519 Advances in Computer Vision (Topic C: 2024 - 2025: Neural Rendering)
## COLMAP-Free 3D Gaussian SplattingLinks to an external site
[link to the paper](https://arxiv.org/pdf/2312.07504)
We propose a novel 3D Gaussian Splatting (3DGS) framework that eliminates the need for COLMAP for camera pose estimation and bundle adjustment.
> [!TIP]
>
> This paper presents a novel 3D Gaussian Splatting framework that eliminates the need for COLMAP for camera pose estimation and bundle adjustment.
>
> Inspired by point map construction, the author uses Gaussian splatting to reconstruct the 3D scene. I wonder how this method might contribute to higher resolution reconstruction or improvements. Can we use the original COLMAP on traditional NeRF methods for comparable results?

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# CSE5519 Advances in Computer Vision (Topic F: 2025: Representation Learning)
## Can Generative Models Improve Self-Supervised Representation Learning?
[link to the paper](https://arxiv.org/pdf/2403.05966)
### Novelty in SSL with Generative Models
- Use generative models to generate synthetic data to train self-supervised representation learning models.
- Use generative augmentation to generate new data from the original data using a generative model. (with gaussian noise, or other data augmentation techniques)
- Using standard augmentation techniques like flipping, cropping, and color jittering with generative techniques can further improve the performance of the self-supervised representation learning models.
> [!TIP]
>
> This paper shows that using generative models to generate synthetic data can improve the performance of self-supervised representation learning models. The key seems to be the use of generative augmentation to generate new data from the original data using a generative model.
>
> However, both representation learning and generative modeling have some hallucinations. I wonder will these kinds of hallucinations would be reinforced, and the bias in the generation model would propagate to the representation learning model in the process of generative augmentation?