Provable Length Generalization in Sequence Prediction via Spectral Filtering (2411.01035v1)

Abstract: We consider the problem of length generalization in sequence prediction. We define a new metric of performance in this setting -- the Asymmetric-Regret -- which measures regret against a benchmark predictor with longer context length than available to the learner. We continue by studying this concept through the lens of the spectral filtering algorithm. We present a gradient-based learning algorithm that provably achieves length generalization for linear dynamical systems. We conclude with proof-of-concept experiments which are consistent with our theory.

• The paper introduces a novel Asymmetric-Regret metric to quantify prediction loss with limited context versus an optimal long-context benchmark.
• The methodology leverages spectral filtering combined with gradient-based learning for Linear Dynamical Systems, achieving robust sublinear regret as sequence length increases.
• The work proposes enhancements, including tensorized spectral filtering and multiple autoregressive components, to improve generalization in neural architectures and LLMs.

Provable Length Generalization in Sequence Prediction via Spectral Filtering

The problem of length generalization in sequence prediction poses significant challenges, especially in machine learning systems where context length—the number of previous tokens required for accurate prediction—can often be limited by computational restrictions. The paper "Provable Length Generalization in Sequence Prediction via Spectral Filtering" addresses this issue by defining a new performance metric termed Asymmetric-Regret. This metric evaluates a learner’s predictive performance when constrained by a shorter context, against a benchmark with access to a longer context.

Central to the paper is the exploration of spectral filtering alongside gradient-based learning algorithms aimed at enhancing length generalization capabilities for Linear Dynamical Systems (LDS). The authors propose a novel asymmetrical benchmark framework that assesses the difference in prediction loss between learners constrained by a limited context length and optimal predictors using extended contexts.

The research underscores the spectral filtering algorithm's robustness, demonstrating its theoretical efficacy in learning linear dynamical systems with substantial context length disparities. The approach extends the theoretical underpinnings of spectral filtering, historically recognized for its robust nature in coping with unobserved hidden states of massive linear systems.

Key numerical results reveal that spectral filtering, when paired with a gradient-based online learning algorithm, achieves sublinear Asymmetric-Regret. This indicates that the gap in performance diminishes with increasing sequence length, maintaining competitive prediction quality even when using significantly abbreviated contexts. Importantly, the research includes proof-of-concept experiments supporting these theoretical claims.

The paper introduces novel variations of the spectral filtering algorithm and theoretical constructs which incorporate multiple autoregressive components. These enhancements are proven to achieve superior length generalization, accommodating a broader spectrum of eigenvalues within marginally-stable LDSs. Moreover, a new tensorized spectral filtering architecture is proposed that augments expressivity, enabling the learning of time-varying linear dynamical systems beyond the capability of standard spectral methods.

The implications of these findings resonate significantly with the challenges faced by LLMs, notably their shortcomings in generalizing across vastly different input lengths than those encountered during training. The empirical results suggest potential improvements in neural architectures and neural LLMs utilizing spectral filtering frameworks, such as the Spectral Transform Unit (STU), which could pave the way for models with intrinsic length generalization capabilities without intricate task-specific modifications.

Future research may focus on scaling these spectral algorithms within deep learning frameworks, further investigating their empirical performance diversity across different tasks. Additionally, understanding the interplay between tensor structure expressivity and its impact on generalization could open new avenues in the development of time-efficient, scalable predictive models.

In conclusion, "Provable Length Generalization in Sequence Prediction via Spectral Filtering" offers a rigorous and promising exploration of length generalization in sequence prediction, emphasizing spectral filtering's potential in bridging efficiency with predictive accuracy in resource-constrained contexts. The work sets a foundation for developing robust predictive systems that generalize effectively beyond their training confines, addressing a crucial limitation in the landscape of modern machine learning architectures.