Linear Feedback Control Systems for Iterative Prompt Optimization in Large Language Models (2501.11979v1)

Abstract: LLMs have revolutionized various applications by generating outputs based on given prompts. However, achieving the desired output requires iterative prompt refinement. This paper presents a novel approach that draws parallels between the iterative prompt optimization process in LLMs and feedback control systems. We iteratively refine the prompt by treating the deviation between the LLM output and the desired result as an error term until the output criteria are met. This process is akin to a feedback control system, where the LLM, despite being non-linear and non-deterministic, is managed using principles from linear feedback control systems. We explore the application of different types of controllers within this framework, providing a mathematical foundation for integrating linear feedback control mechanisms with LLMs.

• The paper introduces a novel framework applying linear feedback control systems, such as PID controllers, to iteratively optimize prompts for Large Language Models by treating output deviation from the desired result as an error.
• It provides a detailed mathematical foundation showing how control signals derived from error feedback can update prompts or influence internal LLM mechanisms like token embeddings and positional encoding.
• The framework is demonstrated with a practical use case involving iterative prompt optimization to generate neural network designs on FPGAs that meet specific resource utilization and timing constraints.

The paper introduces a novel framework for iterative prompt optimization in LLMs using principles from linear feedback control systems. It treats the deviation between the LLM output and the desired result as an error term, iteratively refining the prompt until the output criteria are met.

The key contributions include:

• A novel framework that applies linear feedback control system principles to the iterative prompt optimization process in LLMs.
• A detailed mathematical foundation for integrating PID (Proportional-Integral-Derivative) feedback controller mechanisms with LLMs, demonstrating its functionality with an FPGA (Field Programmable Gate Array) design example.

The paper draws a parallel between iterative prompt optimization and feedback control systems. It manages the LLM using principles from linear feedback control, despite the non-linear and non-deterministic nature of LLMs. The paper explores the application of different types of controllers within this framework, providing a mathematical foundation for integrating linear feedback control mechanisms with LLMs. Traditional prompt optimization methods often rely on heuristic or trial-and-error approaches, which can be inefficient. By leveraging the systematic approach of feedback control systems, this paper aims to provide a more robust and theoretically grounded method for prompt optimization.

A feedback control system automatically adjusts its operation to meet a reference point. The controller processes the error signal and generates a control action to minimize this error, thereby driving the system towards the desired performance.

The PID controller is a widely used feedback control mechanism. The control output $u(t)$ is given by:

$$u(t) = K_p e(t) + K_i \int_{0}^{t} e(\tau) d\tau + K_d \frac{de(t)}{dt}$$

where:

• $e(t)$ is the error signal, $y(t)$: $e(t) = r(t) - \hat{y}(t)$.
  • $r(t)$ is the setpoint
  • $\hat{y}(t)$ is the process variable
• $K_p, K_i, K_d$ are the proportional, integral, and derivative gains, respectively.

The error signal $e(t)$ is fed into the PID controller, which computes the control output $u(t)$ to adjust the process variable $\hat{y}(t)$ to match the setpoint $r(t)$. The term $\beta$ is feedback gain and $\hat{y}(t) = \beta y(t)$.

The performance of a PID controller depends on the proper tuning of its parameters $K_p$, $K_i$, and $K_d$. Various tuning methods, such as the Ziegler-Nichols method and Cohen-Coon method, are used to determine these parameters to achieve the desired system performance.

The feedback loop mechanism employed for iterative prompt optimization using LLMs integrates principles from linear feedback control systems with machine learning techniques employed in LLMs.

The control signal $u(t)$ is calculated as:

$$u(t) = K_p e(t) + K_i \int_{0}^{t} e(\tau) d\tau + K_d \frac{de(t)}{dt}$$

Where $u(t)$ is used to update the prompt $p(t)$:

$p(t+1) = p(t) + u(t)$

The updated prompt $p(t+1)$ is subsequently processed by the LLM to generate the output $\sigma(t)$. The LLM's output is modeled as:

$\sigma(t+1) = f\left(p(t+1)\right)$

The function $f$ represents the LLM. Similarly, the system function is denoted by $\phi$, whose output is given as:

$y(t+1) = \phi\left(\sigma(t+1)\right)$

The output $y(t)$ is fed back into the system through a feedback gain $\beta$, completing the loop.

The paper discusses incorporating LLM properties, including stochasticity, non-determinism, and inherent non-linearity, into the feedback control loop used for iterative prompt optimization. To model the stochastic behavior, the paper introduces a stochastic noise term $\eta(t)$ into the LLM output equation:

$\sigma(t+1) = f\left(p(t+1)\right) + \eta(t)$

To capture the non-linearity, the paper modifies the function $f$ to include a non-linear transformation $g$:

$$\sigma(t+1) = g\left(f\left(p(t+1)\right)\right) + \eta(t)$$

The system output $y(t)$ is influenced by the non-linear and stochastic nature of the LLM. The output equation is modified to include these effects:

$y(t+1) = \phi\left(\sigma(t+1)\right) + \nu(t)$

The paper describes the mechanism by which an LLM processes the input prompt $p(t)$ to generate the output $\sigma(t)$, incorporating the effects of a PID controller. The first step in processing the updated prompt $p(t+1)$ involves tokenization, where the input text is divided into smaller units called tokens. Each token is converted into a high-dimensional vector through an embedding layer. Let $p(t+1)$ be tokenized into $\{p_1, p_2, \ldots, p_n\}$. The embedding process can be represented as:

$$\mathbf{e}_i = \text{Embed}(p_i + u(t)), \quad i = 1, 2, \ldots, n$$

where $\mathbf{e}_i$ is the embedding vector corresponding to the token $p_i$, and the PID controller output $u(t)$ influences the tokenization process by adjusting the prompt.

To incorporate the order of tokens, positional encoding is added to the embedding vectors. This can be mathematically expressed as:

$$\mathbf{e}_i' = \mathbf{e}_i + \mathbf{PE}(i + u(t))$$

where, $\mathbf{PE}(i)$ is the positional encoding vector for the $i$-th position, and $u(t)$ affects the positional encoding by modifying the position indices.

The core of the LLM consists of multiple transformer layers, each comprising self-attention and feed-forward sub-layers. The self-attention mechanism computes a weighted sum of the input embeddings, allowing the model to focus on different parts of the input sequence. The self-attention operation is given by:

$\text{Attention}(\mathbf{Q}(u(t)), \mathbf{K}(u(t)), \mathbf{V}(u(t))) = \text{softmax}\left(\frac{\mathbf{Q}(u(t)) \mathbf{K}(u(t))^T}{\sqrt{d_k}\right) \mathbf{V}(u(t))$

where $\mathbf{Q}(u(t))$, $\mathbf{K}(u(t))$, and $\mathbf{V}(u(t))$ are the query, key, and value matrices derived from the input embeddings influenced by the PID controller output $u(t)$.

Each transformer layer also includes a feed-forward network (FFN) applied to each position separately and identically:

$$\text{FFN}(\mathbf{x}(u(t))) = \text{ReLU}(\mathbf{x}(u(t)) \mathbf{W}_1 + \mathbf{b}_1) \mathbf{W}_2 + \mathbf{b}_2$$

where $\mathbf{W}_1$, $\mathbf{W}_2$, $\mathbf{b}_1$, and $\mathbf{b}_2$ are learnable parameters, and $u(t)$ affects the input $\mathbf{x}$.

After passing through several transformer layers, the final hidden states are used to generate the output tokens. This involves a linear transformation followed by a softmax function to produce a probability distribution over the vocabulary:

$$\mathbf{o}_i = \text{softmax}(\mathbf{h}_i(u(t)) \mathbf{W}_o + \mathbf{b}_o)$$

where $\mathbf{h}_i(u(t))$ is the hidden state of the $i$-th token influenced by $u(t)$, and $\mathbf{W}_o$ and $\mathbf{b}_o$ are the output layer parameters.

The output $\sigma(t+1)$ is then generated by sampling from the probability distribution $\mathbf{o}_i$.

The paper analyzes the impact of the control signal $u(t)$ on the various stages of the LLM processing pipeline. The control signal $u(t)$ is composed of three components: proportional error, integral error, and derivative error, with corresponding gains $K_p$, $K_i$, and $K_d$.

In positional encoding, the proportional component directly adjusts the position indices, ensuring immediate alignment with the current error. The integral component corrects long-term deviations in positional encoding, enhancing the model's ability to maintain context over time. The derivative component smooths positional adjustments, reducing oscillations and ensuring stable positional encoding.

In cases where the LLM remembers the current session, typically through a GUI (Graphical User Interface) interface, the PID controller significantly enhances performance by leveraging the history of prompts and responses.

The paper compares the effectiveness of different controllers—PID, Lead-Lag, LQR (Linear-Quadratic Regulator), and Fuzzy Logic Controller—in the context of LLM prompt optimization. The Lead-Lag controller improves system stability and transient response by adding lead and lag compensations:

$$u(t) = K \left( \frac{T_1 s + 1}{T_2 s + 1} \right) e(t)$$

Where $T_1$ and $T_2$ are the lead and lag time constants, respectively. The LQR controller minimizes a cost function to achieve optimal control:

$J = \int_{0}^{\infty} (x^T Q x + u^T R u) \, dt$

$u(t) = -K x(t)$

Where $Q$ and $R$ are weighting matrices, and $K$ is the gain matrix. The Fuzzy Logic Controller uses fuzzy logic to handle uncertainties and non-linearities:

$u(t) = \text{Fuzzy}(e(t), \frac{de(t)}{dt})$

To illustrate the application of the feedback control approach to LLMs, the paper considers a use case in the domain of neural network implementation on FPGA. The objective is to iteratively refine the prompt ($p(t)$) to generate an optimal design ($y(t)$) that closely matches the desired setpoint ($r(t)$). Specifically, the paper aims to ensure that the utilization of FPGA resources $\lambda_i, i \in$ {LUTs, FFs, DSPs, BRAMs} for the neural network design is less than setpoint $r(t)$ while meeting timing constraints (positive setup/hold slack time).

In the iterative prompt refinement process, the system function $\phi(t)$ represents the relationship between the LLM output $\sigma(t)$ and the system output $y(t)$, which includes various FPGA resources. The system output $y(t)$ can be expressed as a vector of resource utilizations:

$$y(t) = \begin{bmatrix} \lambda_\text{LUTs}(t) & \lambda_\text{FFs}(t) & \lambda_\text{DSPs}(t) & \lambda_\text{BRAMs}(t) & \lambda_\text{Slack}(t) \end{bmatrix}$$