This is a high-level overview for a self-corrective and self-reflective architecture. Analyse the potential issues, describe them, think about what causes them and what parts of the architecture it affects, and then think of a solution. This architecture is meant to be fully self-sustainable, meaning nothing external is used (e.g. a knowledge base).

LaTeX write-up:
```
\documentclass{article}

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\title{Write-up for Dan Fosing}
\author{Research Institute @ SprykAI}

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\begin{document}
\maketitle

\begin{abstract}
This document presents a architecture for a self-reflective and self-correcting Transformer-based Language Model (LLM). The model is designed to mimic human-like revision processes by allowing dynamic editing of previously generated tokens during text generation. The architecture integrates several components, including a Generator, a Judge module, an Error Detection Module, a Policy Network, Working Memory, and Action Modules.
\end{abstract}

\tableofcontents

\section{Introduction} \label{introduction}

Building language models that can reflect on their outputs and correct errors autonomously is a significant step towards more advanced and reliable artificial intelligence. This document details an architecture that embeds self-reflection and self-correction directly into a Transformer-based LLM's design, enabling dynamic revision of the generated text similar to human editing processes.

Key features of the architecture include:

\begin{itemize}
    \item Dynamic Editing: The model can insert, delete, or replace tokens in the previously generated text.
    \item Error Awareness: An Error Detection Module identifies potential errors at the token level.
    \item Quality Assessment: A Judge Model evaluates the overall quality of the generated text.
    \item Policy Decision Making: A Policy Network decides when to generate new tokens, edit existing ones, or terminate the generation process.
    \item Working Memory: An editable data structure that maintains the current state of the generated text.
\end{itemize}

\section{Architecture Overview} \label{architecture-overview}

The architecture consists of several interconnected components:

\begin{itemize}
    \item Generator Model (G)
    \item Judge Model (J)
    \item Error Detection Module (EDM)
    \item Working Memory (WM)
    \item Policy Network (PN)
    \item Action Modules (AM)
\end{itemize}

\subsection{Component Summary}

\begin{itemize}
    \item Generator Model (G): Generates text sequences based on input prompts.
    \item Judge Model (J): Evaluates the generated sequences and assigns quality scores.
    \item Error Detection Module (EDM): Identifies token-level errors in the generated text.
    \item Working Memory (WM): Stores the generated text and supports editing operations.
    \item Policy Network (PN): Decides the next action based on inputs from the Judge and EDM.
    \item Action Modules (AM): Executes actions such as generating tokens, editing, or terminating the process.
\end{itemize}

\section{Detailed Component Descriptions} \label{detailed-component-descriptions}

\subsection{Generator Model (G)}

\subsubsection{Function}
\begin{itemize}
    \item Generates text sequences based on input prompts and task embeddings.
\end{itemize}

\subsubsection{Architecture}
\begin{itemize}
    \item Based on the Transformer decoder architecture.
\end{itemize}

\subsubsection{Components}
\begin{itemize}
    \item Embedding Layer
    \item Decoder Layers
    \item Output Layer
\end{itemize}

\subsubsection{Input Representation}

\begin{itemize}
    \item Token Embeddings ($E_{\text{token}}$): Embeddings for each token in the vocabulary.
    \item Positional Embeddings ($E_{\text{pos}}$): Positional information for each token.
    \item Task Embeddings ($E_{\text{task}}$): Representations of the task requirements.
\end{itemize}

The input at time $t$ is:

\[
X_t = E_{\text{token}}(s_{t-1}) + E_{\text{pos}}(t-1) + E_{\text{task}}
\]

\subsubsection{Decoder Layers}

Each decoder layer includes:

\begin{itemize}
    \item Masked Multi-Head Self-Attention
    \[
    \text{Attention}(Q, K, V) = \text{softmax}\left( \frac{Q K^T}{\sqrt{d_k}} + M \right) V
    \]
    \begin{itemize}
        \item $Q, K, V$: Linear projections of inputs.
        \item $M$: Mask to prevent attention to future positions.
    \end{itemize}
    \item Feedforward Network (FFN)
    \[
    \text{FFN}(x) = \text{ReLU}(W_1 x + b_1) W_2 + b_2
    \]
\end{itemize}

\subsubsection{Output Layer}

\begin{itemize}
    \item Vocabulary Distribution
    \[
    P(s_t | S_{<t}) = \text{softmax}(W_o h_t + b_o)
    \]
    \begin{itemize}
        \item $h_t$: Hidden state at time $t$.
        \item $W_o, b_o$: Output layer parameters.
    \end{itemize}
\end{itemize}

\subsection{Judge Model (J)}

\subsubsection{Function}
\begin{itemize}
    \item Evaluates the overall quality and correctness of the generated sequence ($S$).
\end{itemize}

\subsubsection{Architecture}
\begin{itemize}
    \item A separate Transformer encoder or feedforward network that processes the generated sequence.
\end{itemize}

\subsubsection{Input to the Judge}

\begin{itemize}
    \item Generated Sequence Representations ($H_S$): Hidden states from the Generator.
    \item Contextual Information ($C$): Task embeddings, input prompts.
\end{itemize}

\subsubsection{Processing and Output}

\begin{itemize}
    \item Sequence Encoding:
    \begin{itemize}
        \item The Judge encodes the sequence using an encoder:
        \[
        H_{\text{judge}} = \text{Encoder}(H_S)
        \]
    \end{itemize}
    \item Aggregation:
    \begin{itemize}
        \item Aggregate the encoded representations:
        \[
        h_{\text{judge}} = \text{Aggregate}(H_{\text{judge}})
        \]
        \item Aggregation methods can include mean pooling, max pooling, or attention-based pooling.
    \end{itemize}
    \item Quality Score ($Q(S)$):
    \[
    Q(S) = \sigma(W_j h_{\text{judge}} + b_j)
    \]
    \begin{itemize}
        \item $\sigma$: Sigmoid activation function.
    \end{itemize}
\end{itemize}

\subsection{Error Detection Module (EDM)}

\subsubsection{Function}
\begin{itemize}
    \item Identifies token-level errors in the generated sequence.
\end{itemize}

\subsubsection{Architecture}
\begin{itemize}
    \item Integrated within the decoder layers as an auxiliary output.
\end{itemize}

\subsubsection{Error Prediction}

For each token ($s_t$), predict the probability of an error:

\[
P_{\text{error}}(t) = \sigma(W_{\text{edm}} h_t + b_{\text{edm}})
\]

\begin{itemize}
    \item $h_t$: Hidden state from the Generator.
    \item $W_{\text{edm}}, b_{\text{edm}}$: Parameters of the EDM.
\end{itemize}

\subsection{Working Memory (WM)}

\subsubsection{Function}
\begin{itemize}
    \item Stores the current generated sequence and supports dynamic editing.
\end{itemize}

\subsubsection{Data Structure}
\begin{itemize}
    \item Implemented using a Rope Data Structure for efficient edits.
\end{itemize}

\subsubsection{Representation}

The working memory ($W$) is:

\[
W = \{ (s_1, h_1), (s_2, h_2), \dots, (s_N, h_N) \}
\]

\begin{itemize}
    \item $s_i$: Token at position $i$.
    \item $h_i$: Corresponding hidden state.
\end{itemize}

\subsubsection{Editing Operations}

\begin{itemize}
    \item Insertion at Position $k$:
    \[
    W' = W_{1:k-1} \oplus (s_{\text{new}}, h_{\text{new}}) \oplus W_{k:N}
    \]
    \item Deletion at Position $k$:
    \[
    W' = W_{1:k-1} \oplus W_{k+1:N}
    \]
    \item Replacement at Position $k$:
    \[
    W' = W_{1:k-1} \oplus (s_{\text{new}}, h_{\text{new}}) \oplus W_{k+1:N}
    \]
\end{itemize}

\subsection{Policy Network (PN)}

\subsubsection{Function}
\begin{itemize}
    \item Decides the next action based on the current state, Judge's feedback, and EDM outputs.
\end{itemize}

\subsubsection{Architecture}
\begin{itemize}
    \item A feedforward network or Transformer decoder that outputs a probability distribution over actions.
\end{itemize}

\subsubsection{Input to the Policy Network}

\begin{itemize}
    \item Current Hidden State ($h_t$)
    \item Judge's Quality Score ($Q(S)$)
    \item Error Probabilities ($P_{\text{error}}(t)$)
\end{itemize}

Combined input vector:

\[
x_{\text{policy}} = [h_t; Q(S); P_{\text{error}}(t)]
\]

\subsubsection{Action Probability Distribution}

\[
\pi(a_t | x_{\text{policy}}) = \text{softmax}(W_{\text{pn}} x_{\text{policy}} + b_{\text{pn}})
\]

\begin{itemize}
    \item $a_t$: Action at time $t$.
    \item Actions include Generate, Edit, Terminate.
\end{itemize}

\subsection{Action Modules (AM)}

\subsubsection{Function}
\begin{itemize}
    \item Executes the action decided by the Policy Network.
\end{itemize}

\subsubsection{Actions}

\begin{itemize}
    \item Generate: Produce the next token.
    \item Edit: Perform insertion, deletion, or replacement in the Working Memory.
    \item Terminate: End the generation process.
\end{itemize}

\subsubsection{Execution of Actions}

\begin{itemize}
    \item Generate:
    \begin{itemize}
        \item Select next token:
        \[
        s_t = \arg\max_s P(s | S_{<t})
        \]
        \item Update Working Memory:
        \[
        W = W \oplus (s_t, h_t)
        \]
    \end{itemize}
    \item Edit:
    \begin{itemize}
        \item Determine edit positions using $P_{\text{error}}(t)$.
        \item Perform edit operations on $W$.
        \item Recompute hidden states $h_i$ for affected positions.
    \end{itemize}
    \item Terminate:
    \begin{itemize}
        \item Finalize output sequence ($S = W$).
    \end{itemize}
\end{itemize}

\section{Data Flow and Operational Mechanisms} \label{data-flow-and-operational-mechanisms}

\subsection{Generation and Self-Correction Loop}

\begin{enumerate}
    \item Initialization:
    \begin{itemize}
        \item Input prompt and task embeddings are processed.
    \end{itemize}
    \item Token Generation:
    \begin{itemize}
        \item Generate token $s_t$ using the Generator.
    \end{itemize}
    \item Error Detection:
    \begin{itemize}
        \item Compute $P_{\text{error}}(t)$ for the generated token.
    \end{itemize}
    \item Judge Evaluation:
    \begin{itemize}
        \item At defined intervals, compute $Q(S)$ for the current sequence.
    \end{itemize}
    \item Policy Decision:
    \begin{itemize}
        \item Use $h_t$, $Q(S)$, and $P_{\text{error}}(t)$ to compute action probabilities $\pi(a_t | x_{\text{policy}})$.
        \item Sample or select the action $a_t$.
    \end{itemize}
    \item Action Execution:
    \begin{itemize}
        \item Perform the action using the Action Modules.
    \end{itemize}
    \item Update Working Memory and Hidden States:
    \begin{itemize}
        \item Update $W$ and recompute $h_i$ as needed.
    \end{itemize}
    \item Loop Continuation:
    \begin{itemize}
        \item Increment $t$ and repeat steps 2-7 until termination.
    \end{itemize}
\end{enumerate}

\subsection{Preventing Endless Correction Cycles}

\begin{itemize}
    \item Confidence Thresholds:
    \begin{itemize}
        \item Corrections are only initiated if $P_{\text{error}}(t) > \theta_{\text{error}}$.
    \end{itemize}
    \item Edit Limits:
    \begin{itemize}
        \item Maximum number of total edits ($E_{\text{max}}$).
        \item Maximum number of edits per position ($e_{\text{max}}(t)$).
    \end{itemize}
    \item Penalty in Reward Function:
    \begin{itemize}
        \item Include an edit penalty ($\lambda_{\text{edit}}$) in the reward function to discourage excessive corrections.
    \end{itemize}
    \item Convergence Criteria:
    \begin{itemize}
        \item Terminate corrections if improvements ($\Delta Q(S)$) fall below a threshold ($\epsilon$).
    \end{itemize}
\end{itemize}

\section{Mathematical Formulations} \label{mathematical-formulations}

\subsection{Loss Functions}

\subsubsection{Generator Loss ($\mathcal{L}_{\text{gen}}$)}

Standard cross-entropy loss:

\[
\mathcal{L}{\text{gen}} = - \sum{t} \log P(s_t | S_{<t})
\]

\subsubsection{Judge Loss ($\mathcal{L}_{\text{judge}}$)}

Binary cross-entropy loss:

\[
\mathcal{L}{\text{judge}} = - \left( y \log Q(S) + (1 - y) \log (1 - Q(S)) \right)
\]

\begin{itemize}
    \item $y$: Ground truth label (1 for correct, 0 for incorrect).
\end{itemize}

\subsubsection{Error Detection Loss ($\mathcal{L}_{\text{edm}}$)}

Weighted binary cross-entropy loss:

\[
\mathcal{L}{\text{edm}} = - \sum_{t} \left( w_{\text{pos}} y_t \log P_{\text{error}}(t) + w_{\text{neg}} (1 - y_t) \log (1 - P_{\text{error}}(t)) \right)
\]

\begin{itemize}
    \item $y_t$: Ground truth error label at position $t$.
\end{itemize}

\subsubsection{Policy Loss ($\mathcal{L}_{\text{policy}}$)}

Using reinforcement learning:

\[
\mathcal{L}{\text{policy}} = - \mathbb{E}{a_t \sim \pi} \left[ R \log \pi(a_t | x_{\text{policy}}) \right]
\]

\begin{itemize}
    \item $R$: Reward function.
\end{itemize}

\subsection{Reinforcement Learning Framework}

\subsubsection{Reward Function}

Include quality score and edit penalties:

\[
R = Q(S) - \lambda_{\text{edit}} E_{\text{count}}
\]

\begin{itemize}
    \item $E_{\text{count}}$: Total number of edits made.
\end{itemize}

\subsubsection{Policy Optimization}

Use policy gradient methods to update the Policy Network parameters.

\section{Training Methodologies} \label{training-methodologies}

\subsection{Pre-Training}

\begin{itemize}
    \item Generator:
    \begin{itemize}
        \item Pre-train on large text corpora using language modeling objectives.
    \end{itemize}
    \item Judge and EDM:
    \begin{itemize}
        \item Pre-train on synthetic data with known errors.
    \end{itemize}
\end{itemize}

\subsection{Synthetic Data Generation}

\begin{itemize}
    \item Introduce artificial errors into sequences to train the Judge and EDM:
    \begin{itemize}
        \item Random token substitutions, deletions, or insertions.
        \item Logical inconsistencies or contradictions.
    \end{itemize}
\end{itemize}

\subsection{Joint Fine-Tuning}

\begin{itemize}
    \item Fine-tune the Generator, Judge, EDM, and Policy Network together using the combined loss function.
    \item Use dynamic loss weighting to balance the contributions of each component.
\end{itemize}

\subsection{Reinforcement Learning with Imitation Learning}

\begin{itemize}
    \item Imitation Learning:
    \begin{itemize}
        \item Initialize the Policy Network using supervised learning with demonstrations.
    \end{itemize}
    \item Reinforcement Learning:
    \begin{itemize}
        \item Fine-tune the Policy Network using the reward function and policy gradient methods.
    \end{itemize}
\end{itemize}

\section{Implementation Details} \label{implementation-details}

\subsection{Data Structures}

\begin{itemize}
    \item Working Memory (WM):
    \begin{itemize}
        \item Implemented using a Rope data structure for efficient editing.
    \end{itemize}
    \item Edit History Tracking:
    \begin{itemize}
        \item Maintain data structures to track the number of edits per position.
    \end{itemize}
\end{itemize}

\subsection{Computational Considerations}

\begin{itemize}
    \item Efficient Hidden State Updates:
    \begin{itemize}
        \item Recompute hidden states starting from the earliest edited position.
        \item Use batching and parallel processing where possible.
    \end{itemize}
    \item Memory Management:
    \begin{itemize}
        \item Optimize memory usage by discarding unnecessary cached computations.
    \end{itemize}
    \item Thresholds and Limits:
    \begin{itemize}
        \item Set appropriate values for $\theta_{\text{error}}$, $E_{\text{max}}$, and $e_{\text{max}}(t)$ based on empirical results.
    \end{itemize}
\end{itemize}

\section{Hyperparameter Tuning and Threshold Settings} \label{hyperparameter-tuning-and-threshold-settings}

\subsection{Key Hyperparameters}

\begin{itemize}
    \item Error Detection Threshold ($\theta_{\text{error}}$): Typically set between 0.5 and 0.9.
    \item Edit Penalty Coefficient ($\lambda_{\text{edit}}$): Adjusted to balance correction benefits and costs.
    \item Maximum Edits ($E_{\text{max}}$): Depends on the average desired length of sequences.
    \item Improvement Threshold ($\epsilon$): Small value (e.g., $0.01$) to determine significant quality improvements.
    \item Loss Weights:
    \begin{itemize}
        \item $\lambda_{\text{judge}}$, $\lambda_{\text{edm}}$, $\lambda_{\text{policy}}$: Tuned to ensure balanced training.
    \end{itemize}
\end{itemize}

\subsection{Tuning Strategies}

\begin{itemize}
    \item Grid Search: Evaluate combinations of hyperparameters on a validation set.
    \item Adaptive Tuning: Adjust hyperparameters during training based on performance metrics.
    \item Cross-Validation: Use cross-validation to assess generalization.
\end{itemize}

\section{Conclusion} \label{conclusion}

This architecture introduces a self-reflective and self-correcting mechanism into a Transformer-based LLM, enabling it to dynamically edit its outputs during generation. By integrating components such as the Generator, Judge, Error Detection Module, Policy Network, and Working Memory, the model can produce higher-quality text outputs without relying on external resources.

Key contributions:

\begin{itemize}
    \item Dynamic Editing: Mimics human-like revision by allowing the model to correct its own errors.
    \item Internal Evaluation: The Judge and EDM provide internal feedback for self-improvement.
    \item Policy-Driven Actions: The Policy Network facilitates decision-making to balance generation and correction.
    \item Preventing Infinite Loops: Strategies are incorporated to prevent endless cycles of self-correction.
\end{itemize}

Researchers can replicate this architecture by following the detailed descriptions, mathematical formulations, training methodologies, and implementation strategies provided.

\section{References} \label{references}

\begin{enumerate}
    \item Vaswani, A., et al. (2017). Attention is All You Need. Advances in Neural Information Processing Systems.
    \item Sutton, R. S., & Barto, A. G. (2018). Reinforcement Learning: An Introduction. MIT Press.
    \item Goodfellow, I., Bengio, Y., & Courville, A. (2016). Deep Learning. MIT Press.
\end{enumerate}

\end{document}
```