insight - Natural Language Processing - # Non-identifiability and Inductive Biases in Autoregressive Language Models

Core Concepts

Autoregressive language models can exhibit desirable qualities, such as zero-shot rule extrapolation, in-context learning, and data-efficient fine-tuning, that are not solely a consequence of good statistical generalization. These properties arise due to the non-identifiability and inductive biases of these models, which require separate theoretical explanation beyond the study of overparametrized models in the interpolation regime.

Abstract

The paper argues that the current focus on statistical generalization in deep learning theory is insufficient for understanding the success of large autoregressive language models (LLMs). LLMs often operate in the "saturation regime", where they achieve near-optimal training and test loss, but this does not guarantee the presence of desired model properties, such as zero-shot rule extrapolation, in-context learning, and data-efficient fine-tuning.
The authors highlight that autoregressive probabilistic models are inherently non-identifiable, meaning that models with equivalent test loss can exhibit markedly different behaviors. They support this position with mathematical examples and empirical observations, illustrating the practical relevance of non-identifiability through three case studies:
The non-identifiability of zero-shot rule extrapolation: LLMs can extrapolate rules on out-of-distribution (OOD) prompts, but this behavior is not implied by minimal test loss and arises due to inductive biases.
The approximate non-identifiability of in-context learning (ICL): ICL can be ε-non-identifiable, meaning that models within a small KL divergence of each other can differ in their ICL abilities, despite having the same test loss.
The non-identifiability of fine-tunability: Functionally equivalent models can have different parametrizations, leading to different fine-tuning and transfer performance, due to parameter-dependent inductive biases.
The authors argue that to understand LLMs, we need to move beyond the interpolation regime and study them in the saturation regime, focusing on better generalization measures, computational language modeling, and inductive biases that enable desirable model properties.

Stats

"LLMs are trained on massive datasets, these models achieve both low training and test loss; thus, they generalize in the statistical sense."
"Multiple models with perfect generalization may exist and may behave differently."
"Trained Transformers have better-than-chance ability to extrapolate rules on OOD prompts in a zero-shot manner."
"For a mixture of HMMs pre-training distribution p, the LLM is an in-context learner in the limit of infinite examples in the prompt."
"Functionally equivalent models can have different parametrizations, leading to different fine-tuning and transfer performance."

Quotes

"Understanding LLMs Requires More Than Statistical Generalization"
"AR probabilistic models are inherently non-identifiable: models zero or near-zero KL divergence apart—thus, equivalent test loss—can exhibit markedly different behaviors."
"OOD rule extrapolation is non-identifiable, LLMs still extrapolate due to inductive biases"
"When the pre-training distribution is a mixture of Hidden Markov Models (HMMs) as in Xie et al. (2022), we demonstrate that ICL is ε−non-identifiable, even with infinite data, due to the insensitivity of the KL."
"Functionally equivalent models can have different parametrizations. These are indistinguishable by the test loss but may differ after fine-tuning in downstream performance, as inductive biases in different parametrizations affect gradient dynamics."

Deeper Inquiries

In order to develop generalization measures that better capture the properties of natural languages like compositionality, systematicity, and symbolic reasoning, we need to move beyond traditional statistical generalization frameworks. One approach could be to adapt existing frameworks, such as PAC-Bayes, to incorporate these language-specific properties. For example, we can define metrics that assess a model's ability to understand and generate compositional structures, follow systematic rules, and reason symbolically.
To capture compositionality, we can design evaluation metrics that test a model's ability to combine basic linguistic units to form complex structures. This could involve assessing how well a model can generate novel phrases or sentences by combining known words and grammatical rules. Systematicity can be evaluated by testing whether a model can generalize patterns and rules to new instances systematically. Symbolic reasoning can be assessed by evaluating a model's ability to manipulate symbolic representations and perform tasks that require symbolic manipulation.
Additionally, we can explore the use of formal languages, such as Probabilistic Context-Free Grammars (PCFGs), as testbeds for studying these language-specific properties in LLMs. By defining formal languages with controlled complexity and structure, we can evaluate how well LLMs capture compositionality, systematicity, and symbolic reasoning in a well-controlled setting. Computational models of language, such as Finite State Machines (FSMs) and Recurrent Neural Networks (RNNs), can be used to simulate language processing tasks and study how LLMs behave in these controlled environments.
By combining insights from computational linguistics, theoretical computer science, and formal language theory, we can develop new evaluation metrics and experimental setups that better capture the unique properties of natural languages and provide deeper insights into the behavior of LLMs in relation to compositionality, systematicity, and symbolic reasoning.

Formal languages and computational models of language can serve as powerful tools for studying the behavior of Large Language Models (LLMs) in well-controlled settings and gaining insights into their inductive biases. By defining formal languages with specific grammatical rules and structures, such as Probabilistic Context-Free Grammars (PCFGs), we can create controlled environments to evaluate how LLMs process and generate language.
One approach is to use formal languages to design language tasks that require specific linguistic properties, such as compositionality, systematicity, and symbolic reasoning. By presenting LLMs with tasks based on these properties, we can observe how well they perform and generalize in these controlled settings. Computational models, such as Finite State Machines (FSMs) and Recurrent Neural Networks (RNNs), can be used to simulate language processing tasks and compare the behavior of LLMs with these established models.
Additionally, leveraging formal languages allows us to define clear evaluation metrics for assessing the performance of LLMs in terms of language-specific properties. By quantifying how well LLMs capture compositionality, systematicity, and symbolic reasoning in formal language tasks, we can gain insights into their inductive biases and how these biases influence their language processing capabilities.
Overall, formal languages and computational models provide a structured framework for studying LLM behavior in controlled settings, enabling researchers to isolate specific linguistic properties and analyze how inductive biases shape the learning and reasoning processes of these models.

In addition to the qualitative model properties discussed in the paper, other relevant properties for understanding the success of Large Language Models (LLMs) could include robustness, interpretability, and transferability. Robustness refers to a model's ability to maintain performance in the face of adversarial inputs or noisy data. Interpretability relates to how easily humans can understand and interpret the decisions made by the model. Transferability assesses how well a model can apply knowledge learned from one task to another task or domain.
To characterize these qualitative model properties, tools from algorithmic information theory and the analysis of neural network architectures can be employed. For robustness, metrics like adversarial robustness and generalization to noisy data can be quantified using algorithmic information theory to measure the complexity of adversarial examples or noisy inputs. Analysis of neural network architectures can reveal how model structures contribute to robustness or vulnerability to adversarial attacks.
Interpretability can be assessed by analyzing the internal representations of LLMs and quantifying the complexity of these representations using tools from algorithmic information theory. By examining the information content of model activations or attention patterns, researchers can gain insights into how interpretable the model's decision-making process is. Transferability can be studied by analyzing the transfer learning performance of LLMs across different tasks and domains, and algorithmic information theory can help quantify the complexity of knowledge transfer between tasks.
Overall, by considering a broader range of qualitative model properties and leveraging tools from algorithmic information theory and neural network analysis, researchers can gain a more comprehensive understanding of the factors contributing to the success of LLMs and develop insights into how these models learn, reason, and generalize across various tasks and domains.

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