参数空间对称性：深度学习理论的统一几何框架

Core Insights - The article discusses the evolution of deep learning models from millions to billions of parameters, highlighting the lack of systematic understanding of their effectiveness [2] - A key focus is on the concept of parameter space symmetry, which refers to the existence of multiple parameter configurations that yield the same model function, complicating optimization and generalization analysis [4][6] Group 1: Parameter Space Symmetry - Parameter space symmetry allows different parameter combinations to produce identical outputs, exemplified by the interchange of neurons in hidden layers [4][6] - This symmetry is mathematically defined as transformations that keep the loss function invariant, forming a group that defines equivalent orbits in parameter space [6] Group 2: Types of Symmetry - In addition to discrete symmetries, most neural network architectures exhibit continuous symmetries, such as scaling and linear transformations, which maintain function invariance [8] - Complex architectures like Transformers combine various symmetries from their components, including multi-head attention mechanisms [8] Group 3: Impact on Loss Landscape - Symmetry creates a complex yet structured optimization space, where continuous symmetries can stretch isolated minima into flat manifolds, affecting the interpretation of generalization metrics [10] - Observed phenomena like "mode connectivity," where independently trained models can connect through low-loss paths, are partially attributed to continuous symmetries [10] Group 4: Optimization Methods - The presence of symmetry leads to the phenomenon of "equal loss, different gradients," suggesting new algorithmic possibilities for optimization methods that seek better gradient points within equivalent orbits [15][19] - Some optimization strategies leverage symmetry as a degree of freedom, while others aim to reduce it as redundancy, indicating its importance in algorithm design [19] Group 5: Learning Dynamics - Continuous symmetries correspond to conserved quantities, which remain constant during training, revealing insights into the stability of the training process and the implicit bias of optimization [21][23] - The structure of parameter space symmetry influences the statistical distribution of learning trajectories and outcomes [23] Group 6: Connections Across Spaces - Parameter space symmetry is interconnected with data space and internal representation space, where model parameters often reflect the symmetry present in the data distribution [27][28] - Emerging directions like Weight Space Learning utilize symmetry as a new data structure, facilitating the analysis and generation of model properties [28][29] Group 7: Future Directions - The widespread existence of parameter space symmetry offers a new mathematical language for deep learning, linking complex behaviors of models with established tools from group theory and geometry [30] - This perspective is influencing various practical fields, from optimization acceleration to model fusion and new model design, transforming theoretical concepts into actionable algorithmic principles [30]