FOCUS: First Order Concentrated Updating Scheme

Large language models (LLMs) demonstrate remarkable performance, and improving their pre-training process appears to be key to enhancing their capabilities further. Based on the documented success of Adam, learning rate decay, and weight decay, we hypothesize that the pre-training loss landscape features a narrowing valley structure. Through experiments with synthetic loss functions, we discover that when gradient query noise is high relative to the valley's sharpness, Adam's performance falls behind that of Signum because Adam reduces the effective step size too drastically. This observation led us to develop FOCUS, an optimizer that enhances Signum by incorporating attraction toward moving averaged parameters, allowing it to handle noise better while maintaining larger step sizes. In training GPT-2, FOCUS proves to be more stable than Signum and faster than Adam. These results suggest that gradient noise may be an underappreciated limiting factor in LLM training, and FOCUS offers promising solutions.

Physics of Skill Learning

We aim to understand physics of skill learning, i.e., how skills are learned in neural networks during training. We start by observing the Domino effect, i.e., skills are learned sequentially, and notably, some skills kick off learning right after others complete learning, similar to the sequential fall of domino cards. To understand the Domino effect and relevant behaviors of skill learning, we take physicists' approach of abstraction and simplification. We propose three models with varying complexities -- the Geometry model, the Resource model, and the Domino model, trading between reality and simplicity. The Domino effect can be reproduced in the Geometry model, whose resource interpretation inspires the Resource model, which can be further simplified to the Domino model. These models present different levels of abstraction and simplification; each is useful to study some aspects of skill learning. The Geometry model provides interesting insights into neural scaling laws and optimizers; the Resource model sheds light on the learning dynamics of compositional tasks; the Domino model reveals the benefits of modularity. These models are not only conceptually interesting -- e.g., we show how Chinchilla scaling laws can emerge from the Geometry model, but also are useful in practice by inspiring algorithmic development -- e.g., we show how simple algorithmic changes, motivated by these toy models, can speed up the training of deep learning models.

Optimal Control Operator Perspective and a Neural Adaptive Spectral Method

Optimal control problems (OCPs) involve finding a control function for a dynamical system such that a cost functional is optimized. It is central to physical systems in both academia and industry. In this paper, we propose a novel instance-solution control operator perspective, which solves OCPs in a one-shot manner without direct dependence on the explicit expression of dynamics or iterative optimization processes. The control operator is implemented by a new neural operator architecture named Neural Adaptive Spectral Method (NASM), a generalization of classical spectral methods. We theoretically validate the perspective and architecture by presenting the approximation error bounds of NASM for the control operator. Experiments on synthetic environments and a real-world dataset verify the effectiveness and efficiency of our approach, including substantial speedup in running time, and high-quality in- and out-of-distribution generalization.

MarsCode Agent: AI-native Automated Bug Fixing

Recent advances in large language models (LLMs) have shown significant potential to automate various software development tasks, including code completion, test generation, and bug fixing. However, the application of LLMs for automated bug fixing remains challenging due to the complexity and diversity of real-world software systems. In this paper, we introduce MarsCode Agent, a novel framework that leverages LLMs to automatically identify and repair bugs in software code. MarsCode Agent combines the power of LLMs with advanced code analysis techniques to accurately localize faults and generate patches. Our approach follows a systematic process of planning, bug reproduction, fault localization, candidate patch generation, and validation to ensure high-quality bug fixes. We evaluated MarsCode Agent on SWE-bench, a comprehensive benchmark of real-world software projects, and our results show that MarsCode Agent achieves a high success rate in bug fixing compared to most of the existing automated approaches.

Style Transfer: From Stitching to Neural Networks

This article compares two style transfer methods in image processing: the traditional method, which synthesizes new images by stitching together small patches from existing images, and a modern machine learning-based approach that uses a segmentation network to isolate foreground objects and apply style transfer solely to the background. The traditional method excels in creating artistic abstractions but can struggle with seamlessness, whereas the machine learning method preserves the integrity of foreground elements while enhancing the background, offering improved aesthetic quality and computational efficiency. Our study indicates that machine learning-based methods are more suited for real-world applications where detail preservation in foreground elements is essential.

Complex fractal trainability boundary can arise from trivial non-convexity

Training neural networks involves optimizing parameters to minimize a loss function, where the nature of the loss function and the optimization strategy are crucial for effective training. Hyperparameter choices, such as the learning rate in gradient descent (GD), significantly affect the success and speed of convergence. Recent studies indicate that the boundary between bounded and divergent hyperparameters can be fractal, complicating reliable hyperparameter selection. However, the nature of this fractal boundary and methods to avoid it remain unclear. In this study, we focus on GD to investigate the loss landscape properties that might lead to fractal trainability boundaries. We discovered that fractal boundaries can emerge from simple non-convex perturbations, i.e., adding or multiplying cosine type perturbations to quadratic functions. The observed fractal dimensions are influenced by factors like parameter dimension, type of non-convexity, perturbation wavelength, and perturbation amplitude. Our analysis identifies "roughness of perturbation", which measures the gradient's sensitivity to parameter changes, as the factor controlling fractal dimensions of trainability boundaries. We observed a clear transition from non-fractal to fractal trainability boundaries as roughness increases, with the critical roughness causing the perturbed loss function non-convex. Thus, we conclude that fractal trainability boundaries can arise from very simple non-convexity. We anticipate that our findings will enhance the understanding of complex behaviors during neural network training, leading to more consistent and predictable training strategies.

Quantum Langevin Dynamics for Optimization

We initiate the study of utilizing Quantum Langevin Dynamics (QLD) to solve optimization problems, particularly those non-convex objective functions that present substantial obstacles for traditional gradient descent algorithms. Specifically, we examine the dynamics of a system coupled with an infinite heat bath. This interaction induces both random quantum noise and a deterministic damping effect to the system, which nudge the system towards a steady state that hovers near the global minimum of objective functions. We theoretically prove the convergence of QLD in convex landscapes, demonstrating that the average energy of the system can approach zero in the low temperature limit with an exponential decay rate correlated with the evolution time. Numerically, we first show the energy dissipation capability of QLD by retracing its origins to spontaneous emission. Furthermore, we conduct detailed discussion of the impact of each parameter. Finally, based on the observations when comparing QLD with classical Fokker-Plank-Smoluchowski equation, we propose a time-dependent QLD by making temperature and $\hbar$ time-dependent parameters, which can be theoretically proven to converge better than the time-independent case and also outperforms a series of state-of-the-art quantum and classical optimization algorithms in many non-convex landscapes.

On Quantum Speedups for Nonconvex Optimization via Quantum Tunneling Walks

Classical algorithms are often not effective for solving nonconvex optimization problems where local minima are separated by high barriers. In this paper, we explore possible quantum speedups for nonconvex optimization by leveraging the global effect of quantum tunneling. Specifically, we introduce a quantum algorithm termed the quantum tunneling walk (QTW) and apply it to nonconvex problems where local minima are approximately global minima. We show that QTW achieves quantum speedup over classical stochastic gradient descents (SGD) when the barriers between different local minima are high but thin and the minima are flat. Based on this observation, we construct a specific double-well landscape, where classical algorithms cannot efficiently hit one target well knowing the other well but QTW can when given proper initial states near the known well. Finally, we corroborate our findings with numerical experiments.