Why do similarity matching objectives lead to Hebbian/anti-Hebbian networks? (1703.07914v2)

Abstract: Modeling self-organization of neural networks for unsupervised learning using Hebbian and anti-Hebbian plasticity has a long history in neuroscience. Yet, derivations of single-layer networks with such local learning rules from principled optimization objectives became possible only recently, with the introduction of similarity matching objectives. What explains the success of similarity matching objectives in deriving neural networks with local learning rules? Here, using dimensionality reduction as an example, we introduce several variable substitutions that illuminate the success of similarity matching. We show that the full network objective may be optimized separately for each synapse using local learning rules both in the offline and online settings. We formalize the long-standing intuition of the rivalry between Hebbian and anti-Hebbian rules by formulating a min-max optimization problem. We introduce a novel dimensionality reduction objective using fractional matrix exponents. To illustrate the generality of our approach, we apply it to a novel formulation of dimensionality reduction combined with whitening. We confirm numerically that the networks with learning rules derived from principled objectives perform better than those with heuristic learning rules.

• The paper establishes that similarity matching objectives naturally lead to Hebbian and anti-Hebbian learning rules through variable transformations and a min-max formulation.
• The paper introduces the novel Principal Subspace Whitening objective to enhance dimensionality reduction and demonstrates numerical superiority over heuristic methods.
• The paper derives online algorithms that mirror biological plausibility, bridging computational neuroscience and practical unsupervised learning applications.

Exploration of Similarity Matching in Hebbian/Anti-Hebbian Learning Networks

The paper "Why do similarity matching objectives lead to Hebbian/anti-Hebbian networks?" explores the utilization of similarity matching objectives in deriving neural networks that adhere to local Hebbian and anti-Hebbian learning rules. The inquiry into biologically plausible synaptic learning rules is central to understanding how neural networks self-organize, especially in unsupervised learning scenarios where labeled datasets are not readily available.

Summary and Contributions

The authors provide a rigorous exposition on how similarity matching objectives arise as a natural choice for deriving local Hebbian and anti-Hebbian learning rules in neural networks. The focus is on unsupervised tasks, with dimensionality reduction serving as the primary example. Key contributions of the paper include:

1. Variable Transformations: Through insightful variable substitutions, the paper showcases how the complexity of network optimization can be decomposed into simpler, synapse-specific problems amenable to local learning rules. Locality is crucial in making these rules biologically plausible, meaning a synapse's weight depends only on the neurons it connects.
2. Min-Max Formulation: A novel perspective is introduced by framing the competition between Hebbian and anti-Hebbian rules as a min-max optimization problem, mathematically capturing the notion of adversarial learning.
3. Dimensionality Reduction: The authors not only strengthen the understanding of previously suggested objectives like Principal Subspace Projection (PSP) but also propose a new objective, Principal Subspace Whitening (PSW), incorporating whitening into the dimensionality reduction task.
4. Online Algorithms: The derivation of online algorithms calibrates the theoretical developments with practical applications, as networks must adapt to streaming data typical in neural circuits.
5. Numerical Superiority: The authors provide numerical verification that networks derived from these objectives outperform those developed with heuristic learning algorithms.

Implications

1. Biological Requirement: The derivations have significant implications for neuroscience, as they provide a pathway to constructing computational models that are in closer alignment with biological reality due to their reliance on local weight updates.
2. Theoretical Expansion: The min-max formulation and insights into adversarial interactions offer a unique interpretation that could guide the development of novel learning paradigms, extending beyond purely Hebbian synaptic models.
3. Fractional Matrix Exponents: The exploration of dimensionality reduction using fractional matrix powers provides a theoretical extension that offers a fresh lens through which to view and solve traditional optimization problems.

Future Directions

The paper paves the way for further exploration into a multitude of unsupervised learning tasks beyond dimensionality reduction. The theoretical foundation laid by similarity matching objectives and the derived local learning rules can be extrapolated to more complex, multi-layer networks, potentially enhancing our understanding of deep unsupervised learning architectures.

The practical success and biological plausibility of such algorithms indicate potential future applications in both artificial intelligence systems and the modeling of neural processes. The interplay between feedforward and lateral connections through adversarial min-max dynamics invites further exploration into competitive learning mechanisms in artificial neural networks, with potential applications in areas such as reinforcement learning and adaptive control systems.

In summary, the paper contributes significantly to the field by advancing our understanding of the role similarity matching objectives play in optimizing local learning rules, thereby aligning computational systems closer to biological neural networks. Moreover, the proposed methodological perspective invites further research into multi-dimensional learning tasks, neural adaptation, and the implications for cognitive processes in artificial networks.