Graph-Enhanced Singular Adaptive Learning (GESAL)

A Novel Framework for Real-Time, Self-Adaptive Large Language Models

Authors: Waseem AlShikh

Date: February 25, 2025

Institution: Writer

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Abstract

The rapid evolution of large language models (LLMs) has transformed natural language processing, yet their static nature limits real-time adaptability to user-specific needs. We introduce Graph-Enhanced Singular Adaptive Learning (GESAL), a framework that enables LLMs to learn dynamically from interactions, storing adaptations in a graph structure. GESAL leverages singular value fine-tuning (SVF), a graph-based memory, and reinforcement learning (RL) to achieve efficient, scalable, and responsive adaptation. This paper details GESAL's theory, implementation, evaluation, and future potential, demonstrating superior performance over traditional methods.

1. Introduction

1.1 Motivation

Large language models (LLMs) excel in general tasks but struggle to adapt to individual users without retraining. GESAL addresses this by enabling real-time learning, leveraging efficient adaptation and structured memory.

1.2 Challenges in Current LLM Adaptation

Full fine-tuning updates all parameters, e.g., $(W\in {\mathbb{R}}^{m\times n}),costing(O(mn))memory.$ PEFT reduces this but lacks dynamic updates. Real-time learning requires stability and memory efficiency, unmet by current methods.

1.3 Contributions of GESAL

• SVF: Adapts weights with $(O(r))parameters,where(r=min(m,n)).$
• Graph Storage: $Scalesas(O(|V|)),where(|V|)isthenumberoftasks.$
• RL: $Optimizes(\pi (y|x))inrealtime.$

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2. Related Work

2.1 Self-Adaptive LLMs

Transformer² (Sun et al., 2025) uses pre-trained experts, not real-time learning.

2.2 Parameter-Efficient Fine-Tuning (PEFT)

$$LoRA(Huetal.,2021)updates(W+\mathrm{\Delta }W),where(\mathrm{\Delta }W=AB),(A\in {\mathbb{R}}^{m\times r}),(B\in {\mathbb{R}}^{r\times n}).GESALusesSVFforfull-rankadaptation.$$

2.3 Graph-Based Learning Systems

Graphs store structured data (Zhang et al., 2024). GESAL adapts this for LLM memory.

2.4 Reinforcement Learning in NLP

$$RLrefinesLLMsvia(J(\theta )=\mathbb{E}[\log \pi (y|x)r])(Williams,1992).GESALappliesthisdynamically.$$

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3. Theoretical Foundations

3.1 Singular Value Decomposition in Neural Networks

For a weight matrix $(W\in {\mathbb{R}}^{m\times n}):$ $$ W = U \Sigma V^T

• $(U\in {\mathbb{R}}^{m\times r}),(V\in {\mathbb{R}}^{n\times r}):orthogonalmatrices.$
• $(\mathrm{\Sigma }\in {\mathbb{R}}^{r\times r}):diagonalsingularvalues.$
• $(r=min(m,n)).$

$$GESALmodifies(\mathrm{\Sigma })with(z\in {\mathbb{R}}^r):$$

$$W^{\prime }=U(\mathrm{\Sigma }\cdot z)V^T$$

This adjusts the magnitude of each singular component, preserving full expressivity with ( O(r) ) parameters.

3.2 Graph Structures for Knowledge Representation

A graph ( G = (V, E) ) organizes adaptations:

• $(V):Nodeswithembeddings(e_v\in {\mathbb{R}}^d)and(z_v).$
• $(E):Edgesbasedoncosinesimilarity,(\text{sim}(e_u,e_v)=\frac{e_u\cdot e_v}{|e_u||e_v|}).$ New nodes form if $$ ( \text{sim} < \theta ), where ( \theta ) is a threshold.$$

3.3 Reinforcement Learning for Real-Time Adaptation

GESAL optimizes: $$ J(z) = \mathbb{E}[\log \pi_z(y|x) r(y)] $$

• ( \pi_z(y|x) ): Policy with adapted weights.
• ( r(y) ): Reward (1 or -2 for positive/negative feedback). Gradient update: $$ \nabla J(z) \approx r(y) \nabla_z \log \pi_z(y|x) $$

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4. GESAL Framework

4.1 Overview

GESAL comprises:

1. SVF Module: Efficient weight adjustment.
2. Graph Storage: Task-specific memory.
3. RL Mechanism: Feedback-driven updates.

4.2 Singular Value Fine-Tuning (SVF) Module

For input ( x \in \mathbb{R}^n ): $$ y = W' x = U (\Sigma \cdot z) V^T x $$

• Precompute ( U, V, \Sigma ) offline.
• Learn ( z ) online.

4.3 Graph-Based Adaptation Storage

Each node ( v \in V ):

• ( e_v ): Average embedding, updated as: $$ e_v' = \frac{\text{count}v \cdot e_v + e{\text{new}}}{\text{count}_v + 1} $$
• ( z_v ): SVF parameters.
• ( R_v ): Set of past responses.

4.4 Real-Time Learning Mechanism

Algorithm:

1. Embed input $(x)as(e_x).$
2. Find $(v=\arg \underset{u\in V}{min}1-\text{sim}(e_x,e_u)).$
3. If $(\text{dist}>\theta )$, create new node.
4. Apply $(z_v),generate(y).$
5. If $(y\in R_v)and(r=-2),regenerate.$
6. Buffer $((x,y,r,v));updatewhenfull.$

4.5 Integration and Workflow

Inference and adaptation alternate, with ( O(1) ) per-input cost and ( O(|V|) ) graph operations.

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5. Implementation Details

5.1 Model Architecture

• Base Model: Llama-3.2-1B (( d = 2048 ), vocab = 128,256).
• SVF Layers: Replace MLP ( W_{\text{c_fc}}, W_{\text{c_proj}} ).

5.2 Graph Node Design

class Node:
    def __init__(self, embedding, z_vectors, count=1):
        self.embedding = embedding  # [2048]
        self.z_vectors = z_vectors  # List of [r]
        self.count = count
        self.past_responses = set()

5.3 SVF Linear Layer

class SVFLinear(nn.Module):
    def __init__(self, original_linear):
        super().__init__()
        U, Sigma, V = torch.svd(original_linear.weight.float())
        self.U = nn.Parameter(U, requires_grad=False)
        self.Sigma = nn.Parameter(Sigma, requires_grad=False)
        self.V = nn.Parameter(V.t(), requires_grad=False)
        self.z = nn.Parameter(torch.ones_like(Sigma), requires_grad=True)

    def forward(self, x):
        Sigma_z = self.Sigma * self.z
        Vx = torch.matmul(self.V, x.T if x.dim() == 2 else x.unsqueeze(-1))
        return torch.matmul(self.U, Sigma_z.unsqueeze(-1) * Vx)

5.4 RL Update Algorithm

For buffer ( B = {(x_i, y_i, r_i, v_i)} ):

logits = model(x_i + " " + y_i)[:, len(x_i):-1, :]
log_probs = F.log_softmax(logits, dim=-1)
loss = -r_i * log_probs.gather(2, targets.unsqueeze(-1)).mean()
if r_i < 0: loss *= 2
loss.backward()

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6. Experimental Evaluation

6.1 Setup and Methodology

• Hardware: 4GB GPU.
• Task: "How many r in [word]?" (e.g., "strawberry").
• Metric: Accuracy after feedback.

6.2 Results and Analysis

Method	Initial Accuracy	Post-5 Feedback	Parameters
Llama-3.2-1B	60%	60%	1B
LoRA	65%	70%	6.8M
GESAL	60%	95%	0.5M

GESAL adapts faster, avoiding repetition via ( R_v ).

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7. Conclusion

GESAL offers a scalable, efficient solution for real-time LLM adaptation, with potential to revolutionize personalized AI.

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8. References

• Hu et al., "LoRA: Low-Rank Adaptation of Large Language Models," 2021.
• Sun et al., "Transformer²: Self-Adaptive LLMs," ICLR 2025.
• Williams, "Simple Statistical Gradient-Following Algorithms," 1992.
• Zhang et al., "Proagent: Building Proactive Cooperative Agents," 2024.

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