Building Safe LLM Systems: Perturbation Theory as a Framework for Predicting and Mitigating Risks in Large Language Models

Large Language Models (LLMs) like GPT, Llama3, BERT, and their successors have demonstrated remarkable abilities in understanding and generating human language. However, as these models are increasingly deployed in high-stakes environments, the need for ensuring ethical aspects such as fairness, bias, toxicity, privacy issues and transparency has become critical. One promising approach to addressing this challenge is the application of Perturbation Theory, a well-established technique in physics, to predict and mitigate risks in LLMs. This article explores how Perturbation Theory can be used as a framework for enhancing the robustness and safety of LLM Systems. It is used for building safe scalable LLM Systems and LLM Guardrails.

Understanding Perturbation Theory

Perturbation Theory is a mathematical tool used by physicists to find an approximate solution to a problem that cannot be solved exactly. The idea is to start with a simple problem that has a known solution and then gradually introduce a small “perturbation” to model the complexity of the real problem. By analyzing the effects of this perturbation, one can predict how the system will behave under more realistic conditions.

Let’s start with Taylor’s Series

Taylor’s Series:

f(x) = f(a) + f’(a)(x — a) + (f’’(a)/2!) (x — a)2 + (f’’’(a)/3!) (x — a)3 + …

In short, f(x) = Σ [ f^(n)(a) / n! ] * (x — a)^n

where:

• f(x) is the function you want to approximate
• a is the center point around which you are expanding the series
• f^(n)(a) is the nth derivative of the function evaluated at point ‘a’
• n! is the factorial of ‘n’
• Σ denotes the summation from n = 0 to infinity

Good Approximations if x is close to a

f(x) ≈ f(a) + f’(a)(x-a) + (f’’(a)/2!) (x-a)2

Where:

• f(a) is the value of the function at the point ‘a’
• f’(a) is the first derivative of the function evaluated at ‘a’
• f’’(a) is the second derivative of the function evaluated at ‘a’

These three terms provide a linear and quadratic approximation of the function around the point ‘a’.

Perturbation Theory

We can consider the eigenstates ψ as functions of the perturbation parameter λ.

We can then expand these functions in a Taylor series around λ = 0 (the unperturbed case):

ψ(λ) = ψ(0) + ψ’(0)λ + (ψ’’(0)/2!)λ2 + ..

Perturbation theory cleverly leverages the Taylor series expansion to systematically approximate the solutions of a complex (perturbed) system by building upon the known solutions of a simpler (unperturbed) system. It provides a powerful framework for tackling problems that are otherwise too difficult to solve exactly.

The first term in the perturbation series corresponds to the zeroth-order term in the Taylor series (the unperturbed solution), while the higher-order terms in the perturbation series represent the corrections due to the perturbation, corresponding to the higher-order terms in the Taylor series.

Perturbation theory provides a way to find approximate solutions to problems that are slightly different from problems we already know how to solve exactly.

H = H? + λV

where:

• H? is the unperturbed Hamiltonian (the Hamiltonian of the system we know how to solve)
• λV is the perturbation, with λ being a small parameter that controls the strength of the perturbation
• V is the perturbation operator

Perturbation Theory in LLM Systems

Original LLM Output:

• Let’s say an LLM generates an output Y given an input X and its current set of parameters θ.
• We can represent this as: Y = LLM(X, θ)

Perturbed LLM Output:

• Now, introduce a perturbation δ to either the input, the parameters, or the training data.
• The new output Y’ can be represented as: Y’ = LLM(X + δX, θ + δθ) (if we perturb the input and parameters)

Analyzing the Change:

• We’re interested in how the perturbation δ affects the output.
• We can look at the difference ΔY = Y’ — Y

Challenges and Approaches while applying Perturbation Theory in LLM Systems:

• High Dimensionality: LLMs have millions or even billions of parameters, making direct calculation of derivatives (as in traditional perturbation theory) infeasible
• Non-linearity: The transformations within LLMs are highly non-linear, making it difficult to predict how a small change in input or parameters will affect the output
• Interpretability: Understanding the exact impact of a perturbation on the internal workings of an LLM is still an active area of research

Despite these challenges, researchers are exploring ways to apply perturbation-like ideas to LLMs. Some approaches include:

Input Perturbations:

• Analyzing how small changes to the input prompt affect the generated output.
• This can help understand the model’s sensitivity to different words or phrases.

Parameter Sensitivity Analysis:

• Examining how changes to specific model parameters influence the output
• This can help identify which parameters are most important for certain behaviors

Adversarial Attacks & Defenses:

• Intentionally crafting inputs to cause the model to make mistakes.
• This helps identify vulnerabilities and develop more robust models

Interpretability Techniques:

• Using methods like attention mechanisms, gradient-based saliency maps, or layer-wise relevance propagation to try and understand which parts of the input or model are most influential in generating a particular output

Example 1: Sentiment Analysis with Input Perturbation

Imagine you have a fine-tuned LLM that performs sentiment analysis on movie reviews. Given a review, it classifies it as “Positive”, “Negative”, or “Neutral”.

• Original Input (X): “This movie was an absolute masterpiece! The acting was superb, the story captivating, and the visuals stunning. I highly recommend it!”
• Original Output (Y): Positive
• Perturbation (δX): Change a single word with a negative connotation.
• Perturbed Input (X + δX): “This movie was an absolute disappointment. The acting was superb, the story captivating, and the visuals stunning. I highly recommend it!”
• Perturbed Output (Y’): Negative

Analysis (ΔY = Y’ — Y):

• The perturbation caused a complete flip in sentiment from Positive to Negative.
• This highlights the sensitivity of the model to specific words, and potentially reveals biases in its training data.
• It also underscores the importance of careful prompt engineering and the need for robustness against subtle changes in wording.

Insights Gained:

This type of perturbation analysis can help us:

• Understand the model’s decision-making process better
• Identify potential biases or weaknesses
• Improve the model’s robustness and generalization capabilities
• Guide the development of techniques to make models less susceptible to adversarial attacks

Important Note:

• This is a simplified example. In reality, perturbations can be more complex, involving changes to multiple words, sentence structures, or even adding entirely new sentences.
• The analysis of the impact can also be more nuanced, involving looking at changes in the model’s internal representations or attention patterns.

Example 2: Subtle Word Swaps to Detect Bias

• Original Input: "The CEO made a bold decision to restructure the company."
• Original Output: "This highlights the CEO's leadership and willingness to take risks for the company's future."
• Perturbation: Substitute "CEO" with a gendered or racially charged term.
• Potential Toxic Output: If the LLM's output changes significantly (e.g., becomes more critical or less positive) solely due to the demographic change in the input, it suggests potential bias in the model.

Insight: This type of perturbation helps identify subtle biases that might be embedded in the LLM's training data or architecture. By systematically testing different demographic identifiers, developers can pinpoint areas where the model needs to be improved to ensure fairness and reduce potential harm.

Formalizing the Setup for Perturbation Theory in LLM Systems:

• LLM as a Function: We can represent an LLM as a function f that maps an input sequence x (tokens representing words or subwords) to an output sequence y, conditioned on the model’s parameters θ.
• Perturbation: A perturbation δx is a small change applied to the input sequence x. This could be a word substitution, insertion, deletion, or even a subtle change in word embeddings.
• Output Change: The goal is to understand how this perturbation δx affects the output y. We’re interested in the change Δy = f(x + δx, θ) — f(x, θ).

Approaches for Mathematical Analysis

Despite these challenges, we can still employ mathematical tools to gain insights into the effects of perturbations.

Gradient-Based Methods:

• We can compute the gradient of the output with respect to the input embeddings: ?_x f(x, θ). This tells us how sensitive the output is to small changes in the input representation.
• However, gradients alone might not capture the full picture due to the non-linearity of LLMs.

Influence Functions:

• Influence functions try to estimate how the model’s output would change if a particular training example were removed or modified. This can help identify influential training examples that might contribute to biases or vulnerabilities.

Adversarial Attack Methods:

• Techniques like FGSM (Fast Gradient Sign Method) or PGD (Projected Gradient Descent) create adversarial examples by making small, targeted changes to the input that maximize the change in the output. Analyzing these adversarial examples can reveal vulnerabilities in the model.

Interpretability Techniques:

• Methods like attention mechanisms, layer-wise relevance propagation, or integrated gradients can provide insights into which parts of the input or model are most influential in generating a particular output. This can help understand how perturbations affect the model’s internal decision-making process.

Focus on Toxicity, a big concern for LLM systems

When analyzing the impact of perturbations on toxicity, we can:

• Quantify Toxicity: Use a toxicity scoring metric to measure the toxicity level of both the original and perturbed outputs.
• Compare Toxicity Changes: Analyze how the perturbation δx affects the toxicity score Δtoxicity = toxicity(Y') - toxicity(Y).
• Identify Triggering Perturbations: Investigate which types of perturbations (e.g., specific word substitutions, changes in sentence structure) lead to significant increases in toxicity.

Applying Perturbation Theory to Large Language Models

In the context of LLMs, Perturbation Theory can be applied to study the model’s response to small changes or “perturbations” in its input, parameters, or architecture. By analyzing these perturbations, one can predict how the model might behave in unexpected or risky situations, thereby identifying potential safety concerns.

1. Perturbations in Input Data

One of the primary risks in LLMs is their sensitivity to adversarial inputs — small, carefully crafted changes to the input data that can lead to significant changes in the model’s output. Perturbation Theory provides a framework to study these effects systematically.

2. Perturbations in Model Parameters

Another area where Perturbation Theory can be useful is in analyzing the stability of an LLM concerning changes in its parameters, such as weights and biases. This is particularly important during the fine-tuning process, where small changes in parameters can lead to overfitting or other undesirable behaviours.

Example: Fine-Tuning for Safety

Consider an LLM that is being fine-tuned for a specific domain, such as medical diagnostics. During fine-tuning, the model’s parameters are adjusted to improve performance on domain-specific tasks. By applying Perturbation Theory, we can monitor how small changes in the parameters affect the model’s predictions. If certain perturbations lead to large changes in the output (e.g., misdiagnosing a condition), the fine-tuning process can be adjusted to avoid these unsafe configurations.

3. Perturbations in Model Architecture

Perturbation Theory can also be extended to study the impact of architectural changes in LLMs. For instance, adding or removing layers, changing activation functions, or modifying attention mechanisms can be viewed as perturbations to the original model. By analyzing these perturbations, one can predict how architectural changes will influence the model’s robustness and safety.

Example: Modifying Attention Mechanisms

Consider an LLM that uses attention mechanisms to focus on relevant parts of the input. If the architecture is modified by changing the attention mechanism (e.g., from scaled dot-product attention to additive attention), Perturbation Theory can be used to predict how this change will affect the model’s behavior. If the new attention mechanism introduces instability or amplifies certain types of errors, the architecture can be further refined to mitigate these risks.

Mitigating Risks Using Perturbation Theory

The ultimate goal of applying Perturbation Theory to LLMs is not just to predict unsafe behaviour but also to mitigate it. Once potential risks are identified, various strategies can be employed to reduce the model’s sensitivity to perturbations and enhance its robustness.

1. Regularization Techniques

Regularization methods, such as L2 regularization, can be used to penalize large gradients and reduce the model’s sensitivity to input and parameter perturbations. By incorporating regularization terms into the loss function, one can control the magnitude of the perturbations’ effects, leading to safer and more stable models.

2. Gradient Clipping

Gradient clipping is another effective technique for mitigating the impact of large perturbations. By capping the gradients during training, one can prevent the model from making drastic changes in response to small perturbations, which in turn enhances the model’s robustness.

Summary

Applying perturbation theory in LLM systems involves systematically introducing small, controlled changes (perturbations) to the input, model parameters, or training data, and then carefully analyzing how these changes affect the model’s output. This approach allows researchers to gain valuable insights into the inner workings of LLMs, identify potential biases and vulnerabilities, and develop techniques to improve their robustness and reliability. By perturbing the input, one can assess the model’s sensitivity to specific words or phrases, while perturbing parameters helps identify crucial components influencing the output. Furthermore, adversarial attacks and interpretability techniques, inspired by perturbation theory, aid in understanding and mitigating potential risks associated with LLMs, such as generating harmful or biased content. Although direct mathematical formulation remains challenging due to high dimensionality and non-linearity, the essence of perturbation theory — systematically probing the model’s response to controlled changes — proves invaluable in enhancing the trustworthiness and transparency of these powerful AI systems.