Probabilistic symmetries and invariant neural networks (1901.06082v2)

Abstract: Treating neural network inputs and outputs as random variables, we characterize the structure of neural networks that can be used to model data that are invariant or equivariant under the action of a compact group. Much recent research has been devoted to encoding invariance under symmetry transformations into neural network architectures, in an effort to improve the performance of deep neural networks in data-scarce, non-i.i.d., or unsupervised settings. By considering group invariance from the perspective of probabilistic symmetry, we establish a link between functional and probabilistic symmetry, and obtain generative functional representations of probability distributions that are invariant or equivariant under the action of a compact group. Our representations completely characterize the structure of neural networks that can be used to model such distributions and yield a general program for constructing invariant stochastic or deterministic neural networks. We demonstrate that examples from the recent literature are special cases, and develop the details of the general program for exchangeable sequences and arrays.

• The paper establishes a probabilistic framework for invariant and equivariant neural networks by linking maximal invariants with outsourced randomness.
• It derives functional representations for invariant and equivariant conditional distributions that guide the construction of deterministic neural network architectures.
• The results highlight the practical benefits of embedding symmetry principles to enhance model generalization, robustness, and training efficiency.

Probabilistic Symmetries and Invariant Neural Networks

The paper "Probabilistic Symmetries and Invariant Neural Networks" by Benjamin Bloem-Reddy and Yee Whye Teh addresses the quantitative characterization of neural networks that model data invariant under the action of a compact group, focusing on group invariance from a probabilistic perspective. The work bridges the relationship between probabilistic symmetry, particularly invariant and equivariant conditional distributions, and deterministic functional forms within neural network architectures, extending the scope and applicability of symmetry in model design.

The primary result is a characterization of invariant and equivariant neural network structures through the concept of probabilistic symmetry, which is exemplified in the general theory of exchangeability. Building on the long history of using symmetry in statistical and machine learning models—epitomized by de Finetti's theorem on exchangeable sequences—the authors establish a framework for constructing neural network architectures by encoding invariance under group actions and exploiting the sufficiency of maximal invariant statistics.

Key Results

1. Invariant Conditional Distributions: The authors establish a functional representation of invariant conditional distributions, showing that for a random variable $X$ invariant under a group $G$, the conditional distribution of an output $Y$ is $G$-invariant if there exists a function $f \colon [0,1] \times S \to Y$, expressed by $Y = f(\eta, M(X))$, where $M$ is a maximal invariant and $\eta$ is outsourced randomness independent of $X$.
2. Equivariant Conditional Distributions: Similarly, they provide conditions for equivariant representations. A conditional distribution is $G$-equivariant if there exists an equivariant function $Y = f(\eta, X)$ such that transformations on $X$ predictably transform $Y$. Central to this result is the introduction of a representative equivariant, which aligns the process of reaching equilibrium positions across transformations.
3. Connection to Neural Network Architectures: These probabilistic notions of symmetry align closely with the design of architectures for invariant and equivariant neural networks, extending existing paradigms such as convolutional networks. The paper details how sufficiency and adequacy relationships typically found in statistics, such as the role of empirical measures in exchangeable arrays, manifest in functionally equivalent deterministic networks.
4. Practical Implications: The derived theoretical representations imply substantial structure in neural network architecture design, emphasizing stochastic network variants that preserve statistical sufficiency, providing simpler proofs and enhanced capacity for generalization in invariant tasks. Notably, the authors identify that the employment of stochastic elements within function sharing architectures may yield benefits in training and model robustness.
5. Generalization to Other Data Structures: The paper hints at the applicability of these results beyond standard neural networks to cover broader examples of exchangeable random structures beyond sequences and arrays, suggesting potential applications in other graph-based and time-series domains.

Implications for Future Research

The results hold significance for advancing the generalization capability of neural networks in semi-supervised and data-scarse applications. These insights prompt further inquiry into the computational complexity of learning symmetric functions within larger invariance frameworks and the viability of translating these principles to high-dimensional and hierarchical models.

Looking forward, exploring universal approximation capabilities for stochastic functions with symmetric properties and extending the ability of probabilistic networks to handle complex polymerization of invariances may bolster compatibility with real-world phenomena where latent invariant structures are prevalent. Additionally, the rigorous statistical grounding of neural architectures via probabilistic symmetry may guide future efforts in model distillation and transfer learning scenarios.

Overall, this paper serves as a foundation for the marriage of probabilistic symmetry concepts with practical machine learning architecture design, empowering the industry to better embed domain-specific structural knowledge within deep learning models, thereby enhancing their interpretability and performance in complex environments.