Research Shows Mirror Descent Algorithm Exhibits Exponential Sensitivity to Initialization in Non-Quadratic Settings
A new arXiv preprint demonstrates that Mirror Descent (MD) optimization algorithms can amplify small initialization errors exponentially faster than standard Gradient Descent when using non-quadratic regularizers. This matters because MD underpins KL-regularized policy optimization widely used in reinforcement learning and large language model post-training. The finding raises reproducibility and reliability concerns for AI systems whose training pipelines rely on these methods.
Researchers have identified a sharp theoretical gap between Mirror Descent and Gradient Descent in terms of sensitivity to initialization. While quadratic-regularized MD—including standard GD and Mahalanobis-geometry variants—is known to be stable for convex smooth objectives, the paper shows that non-quadratic regularizers can cause an initial perturbation of size ε to grow as fast as ε·e^(ΩηT) after T iterations, even when the regularizer is well-conditioned in Euclidean norm. The authors construct a concrete three-dimensional example with a convex, smooth objective and a strongly convex, smooth regularizer that exhibits this exponential amplification. For the practically important case of KL-regularized MD on the probability simplex—directly relevant to language model alignment—even linear objectives can trigger exponential instability in high-dimensional or near-boundary regimes. The paper also proposes a partial remedy: adding a Bregman regularization term anchored at a fixed point (rather than at the initialization itself) can stabilize dynamics while largely preserving optimization guarantees.
What's missing
As a preprint, this work has not yet undergone peer review. The paper's stabilization remedy (Bregman anchoring at a fixed point) is analyzed theoretically but empirical validation on real LLM post-training pipelines is not reported, leaving open how large the instability effect is in practice at typical training scales and hyperparameter regimes.
What different sources said
- arXiv stat.MLCenter
Mirror Descent on Riemannian Manifolds
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