Research Shows Mirror Descent Algorithm Exhibits Exponential Sensitivity to Initialization in Non-Quadratic Settings
A new theoretical study demonstrates that Mirror Descent, an optimization algorithm used in reinforcement learning and large language model training, can be exponentially more sensitive to initial conditions than standard Gradient Descent when using non-quadratic regularizers. The finding contrasts with the known stability of Mirror Descent under quadratic regularization and raises concerns about reproducibility in machine learning systems. This matters because Mirror Descent is increasingly used in critical applications like LLM post-training, where initialization often comes from pretrained models.
Researchers have identified a fundamental robustness issue with Mirror Descent (MD), an optimization algorithm that generalizes Gradient Descent beyond Euclidean geometry. While MD with quadratic regularizers (including standard Gradient Descent) is known to be stable for convex objectives, the study shows that non-quadratic regularizers can cause exponential amplification of initialization perturbations. The authors provide a three-dimensional mathematical construction demonstrating that a small initial perturbation of size ε can grow to polylogarithmic or exponential scales after T iterations. For KL-regularized MD on the simplex—a canonical setting in reinforcement learning and LLM alignment—even linear objectives can amplify perturbations exponentially in high-dimensional or near-boundary regimes. The researchers propose adding Bregman regularization toward an anchor point as a stabilization mechanism, though they note that the choice of anchor point significantly affects stability outcomes.
What's missing
The study does not discuss empirical validation of the theoretical bounds on real-world optimization problems, nor does it provide guidance on practical thresholds for when this instability becomes problematic in applied settings. The paper also does not compare computational costs of the proposed stabilization method against the baseline Mirror Descent approach.
What different sources said
- arXiv cs.LGCenter
Mirror Descent Beyond Euclidean Stability: An Exponential Separation in Initialization Sensitivity
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