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arXiv Oct 29, 2026

Gradient descent's implicit bias is no longer set by initialization alone: step size tilts the limit toward a perturbed weighted L1 minimizer

The training limit of deep linear diagonal networks had been characterized only for gradient flow, the continuous version with vanishing step size, where the implicit bias is fixed entirely by initialization. This analysis rewrites discrete gradient descent and stochastic gradient descent themselves as perturbed mirror gradient descent, proves convergence of the iterates for any number of layers, and characterizes the limit as a convex minimization problem parameterized by a perturbed initialization. The perturbation shrinks linearly with step size, so the limit corresponds to weighted L1 minimization whose weights depend on initialization, step size and the whole iterate trajectory.

Ludwig-Maximilians-Universität München

Wiebke Bartolomaeus（First author）

Implicit Bias Linear Diagonal Networks Gradient Descent Stochastic Gradient Descent Mirror Gradient Descent

BLS refines task vectors at test time, letting zero-shot policies surpass OLS and online adaptation baselines on OGBench, DMC, and HumEnv

The authors propose BLS, an offline zero-shot task-inference method for Behavioral Foundation Models that starts from an ordinary-least-squares (OLS) task vector and refines it with a soft-margin contrastive loss plus a trust-region loss to reduce both reward reconstruction error and successor-measure mismatch; they prove a suboptimality gap upper bound characterized by successor-measure and reward residuals, and empirically BLS outperforms OLS, ZOL, ReLA, and LoLA on average across five feature representations on OGBench (proprioceptive and pixel observations), 20 DMC tasks, and 45 HumEnv humanoid control tasks, with negligible inference overhead.