Can LLMs predict the convergence of Stochastic Gradient Descent? (2408.01736v1)

Abstract: Large-LLMs are notoriously famous for their impressive performance across a wide range of tasks. One surprising example of such impressive performance is a recently identified capacity of LLMs to understand the governing principles of dynamical systems satisfying the Markovian property. In this paper, we seek to explore this direction further by studying the dynamics of stochastic gradient descent in convex and non-convex optimization. By leveraging the theoretical link between the SGD and Markov chains, we show a remarkable zero-shot performance of LLMs in predicting the local minima to which SGD converges for previously unseen starting points. On a more general level, we inquire about the possibility of using LLMs to perform zero-shot randomized trials for larger deep learning models used in practice.

• The paper introduces a novel framework using LLMs to estimate Markov chain transition kernels for forecasting SGD convergence.
• It tokenizes SGD iterates to simulate unseen initialization paths, bridging the gap between theoretical Markov chains and optimization dynamics.
• Experimental results validate the approach by accurately identifying global optima in convex setups and distinguishing local minima in non-convex scenarios.

Can LLMs Predict the Convergence of Stochastic Gradient Descent?

The paper "Can LLMs Predict the Convergence of Stochastic Gradient Descent?" by Oussama Zekri, Abdelhakim Benechehab, and Ievgen Redko explores the intriguing intersection of LLMs and stochastic gradient descent (SGD). The authors investigate whether LLMs can understand and predict the behavior of SGD, particularly in convex and non-convex optimization scenarios, by leveraging the theoretical equivalence between SGD and Markov chains.

Core Contributions

The paper offers several notable contributions:

1. Problem Contextualization: The researchers identify and contextualize the problem of using LLMs to infer the transition probabilities in Markovian dynamical systems, aligning this with the paper of SGD in optimization contexts.
2. Algorithmic Framework: The authors develop a robust algorithmic framework that tokenizes SGD iterates and uses these tokens to fill the transition kernel of the underlying Markov chain. This theoretical link allows for zero-shot prediction of SGD convergence from unseen starting points.
3. Experimental Validation: Preliminary experimental results demonstrate the efficiency of this novel approach. Through multiple trials, the researchers validate that the transition kernel estimation methodology can accurately predict SGD convergence in both convex and non-convex settings.

Methodological Approach

The approach taken to achieve these results involves several key steps:

1. Theoretical Foundations: Building upon foundational studies, the authors draw parallels between SGD and homogeneous Markov chains. Specifically, they reference works by Dieuleveut et al. (2018), who illustrate that with a fixed step size, SGD iterates form a homogeneous Markov chain.
2. Transition Kernel Estimation: By structuring SGD as a transition probability problem of a Markov chain, the authors use LLMs to estimate this kernel. They propose a method to discretize the state space and compute transition matrices for each parameter independently.
3. Forecasting: Using the estimated transition kernels, the LLMs facilitate forecasting by simulating the Markov chain over previously unseen initializations, demonstrating effective convergence predictions.

Numerical and Theoretical Insights

The paper explores both convex and non-convex optimization problems to showcase the model's predictive power:

• Convex Optimization: Simulations show that the transition matrix, derived from LLM predictions, accurately leads to the global optimum when iterated from various starting points.
• Non-Convex Optimization: Here, the model's ability to differentiate between local minima from different initial points is tested, illustrating its adaptability.

Furthermore, the work touches on neural scaling laws in the context of in-context learning (ICL) and highlights the critical role of tokenization. The authors discuss how the choice of tokenizer, for example, Byte Pair Encoding (BPE), influences numerical learning in LLMs, pointing out the complex relationship between token probability distributions and performance.

Implications and Future Directions

The implications of this research are multi-fold:

1. Practical Benefits: The ability to use LLMs for predicting SGD convergence could significantly enhance training efficiency in various machine learning models, particularly in scenarios where computational resources for repeated SGD runs are limited.
2. Theoretical Extensions: Extending this methodology to more complex neural architectures presents an exciting avenue for future research. Additionally, exploring different tokenization strategies could further refine LLMs' capabilities in numerical tasks.
3. Scaling Laws and Generalization: Revisiting ICL neural scaling laws through the lens of Markov chain spectral gaps opens a new theoretical avenue, one that challenges previous understandings and introduces novel perspectives on convergence behavior in dynamical systems.

Conclusion

This work establishes a novel intersection between LLMs and SGD, providing compelling evidence that LLMs can be employed to predict the convergence behavior of SGD through Markov chain transition kernels. The combination of theoretical grounding and practical methodology offers robust validation, suggesting numerous applications and future research potentials in both AI and broader computational fields. The paper paves the way for further explorations into the scalability and adaptability of LLMs in understanding complex dynamical systems and optimization procedures.