Bayesian Online Natural Gradient (BONG) (2405.19681v2)

Abstract: We propose a novel approach to sequential Bayesian inference based on variational Bayes (VB). The key insight is that, in the online setting, we do not need to add the KL term to regularize to the prior (which comes from the posterior at the previous timestep); instead we can optimize just the expected log-likelihood, performing a single step of natural gradient descent starting at the prior predictive. We prove this method recovers exact Bayesian inference if the model is conjugate. We also show how to compute an efficient deterministic approximation to the VB objective, as well as our simplified objective, when the variational distribution is Gaussian or a sub-family, including the case of a diagonal plus low-rank precision matrix. We show empirically that our method outperforms other online VB methods in the non-conjugate setting, such as online learning for neural networks, especially when controlling for computational costs.

• The paper presents an innovative online variational Bayes update using a single-step natural gradient descent, bypassing the KL term for efficiency.
• The method recovers exact Bayesian inference for conjugate models, outperforming traditional online VB techniques in accuracy and computational cost.
• Empirical evaluations on datasets like MNIST demonstrate superior performance in non-conjugate settings, especially for neural network applications.

Proposal of Bayesian Online Natural Gradient for Sequential Bayesian Inference

The paper proposes a novel approach to sequential Bayesian inference, termed Bayesian Online Natural Gradient (BONG). This approach leverages variational Bayes methods and incorporates several key innovations to improve computational efficiency and inference accuracy in the online learning context.

Key Contributions

1. Innovative Online Variational Bayes (VB) Update:
   • Traditional online VB updates involve adding a Kullback-Leibler (KL) divergence term to regularize to the prior. BONG bypasses this by optimizing only the expected log-likelihood and performing a single step of natural gradient descent starting at the prior predictive.
   • The rationale is that the resultant update implicitly regularizes towards the prior by starting the gradient step at the prior predictive, obviating the need for an explicit KL term.
2. Exact Bayesian Inference for Conjugate Models:
   • The method is proven to recover exact Bayesian inference when the model is conjugate. This is a significant theoretical result, showing that for conjugate exponential family models, the BONG update yields the exact posterior distribution.
3. Empirical Validation:
   • The proposed method is empirically evaluated and shown to outperform other online VB methods, especially in non-conjugate settings and when controlling for computational costs. Specifically, BONG exhibits improved performance in online learning for neural networks.

Computational Advantages

• Efficiency in Online Settings:
  • In online learning scenarios where data arrives sequentially, fast updates are crucial. BONG achieves this by simplifying the update step: a single natural gradient step with a unit learning rate is performed, foregoing the KL term at each time step. This provides substantial computational savings.
• Linearized Approximation for Neural Networks:
  • Recognizing the computational burden posed by neural networks, the paper introduces a linearized approximation to tackle the intractability of expected log-likelihood computations. This approach aligns with closed-form updates akin to an extended Kalman filter (EKF) with Gaussian approximations, efficiently handling high-dimensional parameter spaces.

Experimental Results

The empirical evaluation on datasets like MNIST highlights several critical findings:

• Linearization Benefits:
  • Methods utilizing a linear approximation (hess{) consistently outperform others across various experimental setups, affirming the efficacy of this approximation in complex model architectures like CNNs.
• Natural Gradient vs Standard Gradient Descent:
  • NGD-based methods (BONG, BLR) generally show superior performance over standard gradient descent methods (BOG, BBB), underscoring the importance of accounting for the intrinsic geometry of the variational family.
• Implicit Regularization Advantage:
  • The implicit regularization through single-step NGD in BONG provides better performance compared to methods that explicitly incorporate KL regularization (BLR), suggesting that the implicit regularization approach is beneficial in online settings.
• Variational Family Impact:
  • Full Covariance (FC) and Diagonal plus Low Rank (DLR) Gaussian approximations perform notably better than mean-field approximations, particularly in large models. The choice of variational family and parameterization significantly affects both computational efficiency and inferential accuracy.

Implications and Future Directions

The theoretical and empirical results have several important implications:

• Broader Applicability in Online Learning:
  • The BONG framework is particularly well-suited for applications requiring rapid, sequenced data processing, such as real-time anomaly detection, financial time series forecasting, and adaptive control systems.
• Potential Extensions:
  • Future research could explore the integration of more sophisticated dynamic models for non-stationary data streams and further improvements in approximation techniques for deep learning applications.
• Advanced Regularization Techniques:
  • Incorporating advanced regularization techniques within the BONG framework could yield further improvements, particularly in handling highly non-conjugate or non-linear models encountered in practice.

Conclusion

The paper presents a robust and efficient method for sequential Bayesian inference that significantly streamlines the computational burdens typically associated with online learning. The fusion of variational methods, natural gradient descent, and innovative approximations creates a powerful tool for real-time data processing and offers a fertile ground for future advancements in Bayesian neural network optimization.