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KOALA: A Kalman Optimization Algorithm with Loss Adaptivity

Aram Davtyan, Sepehr Sameni, Llukman Cerkezi, Givi Meishvilli, Adam Bielski, Paolo Favaro

TL;DR

KOALA treats iteration-time neural-network loss as a noisy measurement and uses Kalman filtering to estimate model parameters $x$ in a scalable way. It introduces loss-adaptive target risks $\hat{L}^\text{target}_k$, online estimates of measurement and state noise $R_k$ and $Q_k$, and optional momentum dynamics by augmenting the state to include velocities, yielding KOALA-M. The paper provides a convergence analysis for KOALA-V under two target-risk schemes and demonstrates strong empirical performance across vision and NLP benchmarks, with modest memory overhead and automatic adaptation to minibatch noise and learning-rate scheduling. The approach offers a practical, self-tuning alternative to traditional optimizers that preserves linear-time complexity and can integrate with advanced training tricks like Lookahead and SWA, making it suitable for large-scale, high-dimensional neural models.

Abstract

Optimization is often cast as a deterministic problem, where the solution is found through some iterative procedure such as gradient descent. However, when training neural networks the loss function changes over (iteration) time due to the randomized selection of a subset of the samples. This randomization turns the optimization problem into a stochastic one. We propose to consider the loss as a noisy observation with respect to some reference optimum. This interpretation of the loss allows us to adopt Kalman filtering as an optimizer, as its recursive formulation is designed to estimate unknown parameters from noisy measurements. Moreover, we show that the Kalman Filter dynamical model for the evolution of the unknown parameters can be used to capture the gradient dynamics of advanced methods such as Momentum and Adam. We call this stochastic optimization method KOALA, which is short for Kalman Optimization Algorithm with Loss Adaptivity. KOALA is an easy to implement, scalable, and efficient method to train neural networks. We provide convergence analysis and show experimentally that it yields parameter estimates that are on par with or better than existing state of the art optimization algorithms across several neural network architectures and machine learning tasks, such as computer vision and language modeling.

KOALA: A Kalman Optimization Algorithm with Loss Adaptivity

TL;DR

KOALA treats iteration-time neural-network loss as a noisy measurement and uses Kalman filtering to estimate model parameters in a scalable way. It introduces loss-adaptive target risks , online estimates of measurement and state noise and , and optional momentum dynamics by augmenting the state to include velocities, yielding KOALA-M. The paper provides a convergence analysis for KOALA-V under two target-risk schemes and demonstrates strong empirical performance across vision and NLP benchmarks, with modest memory overhead and automatic adaptation to minibatch noise and learning-rate scheduling. The approach offers a practical, self-tuning alternative to traditional optimizers that preserves linear-time complexity and can integrate with advanced training tricks like Lookahead and SWA, making it suitable for large-scale, high-dimensional neural models.

Abstract

Optimization is often cast as a deterministic problem, where the solution is found through some iterative procedure such as gradient descent. However, when training neural networks the loss function changes over (iteration) time due to the randomized selection of a subset of the samples. This randomization turns the optimization problem into a stochastic one. We propose to consider the loss as a noisy observation with respect to some reference optimum. This interpretation of the loss allows us to adopt Kalman filtering as an optimizer, as its recursive formulation is designed to estimate unknown parameters from noisy measurements. Moreover, we show that the Kalman Filter dynamical model for the evolution of the unknown parameters can be used to capture the gradient dynamics of advanced methods such as Momentum and Adam. We call this stochastic optimization method KOALA, which is short for Kalman Optimization Algorithm with Loss Adaptivity. KOALA is an easy to implement, scalable, and efficient method to train neural networks. We provide convergence analysis and show experimentally that it yields parameter estimates that are on par with or better than existing state of the art optimization algorithms across several neural network architectures and machine learning tasks, such as computer vision and language modeling.

Paper Structure

This paper contains 14 sections, 1 theorem, 18 equations, 8 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Prior Work
  3. Risk Minimization through Kalman Filtering
  4. Modeling Noise in Risk Minimization
  5. Risk Minimization as Loss Adaptivity
  6. Kalman Filtering for Stochastic Optimization
  7. Incorporating Momentum Dynamics
  8. Estimation of the Measurement and State Noise
  9. Learning Rate Scheduling
  10. Layer-wise Approximations
  11. Convergence Analysis
  12. Ablations
  13. Experiments
  14. Conclusions

Key Result

Theorem 1

Let $\hat{L}(x)$ be a continuously differentiable function and $\nabla \hat{L}(x)$ be Lipschitz-continuous. Assume that $g \le |\nabla \hat{L}_{k}(x)|\le G$ for all $x$ and $k$, where $g, G > 0$ and $\hat{L}_k(x)$ is the minibatch loss. Let us choose the target risk where $\varepsilon_k$ is any sequence satisfying $\sum_{k=0}^{\infty} \varepsilon_k = \infty$ and $\sum_{k=0}^{\infty} \varepsilon_k

Theorems & Definitions (2)

  • Theorem 1
  • proof