Logarithmic Activation Clocks in Diagonal Linear Networks: When SGD Does and Does Not Follow Lasso

Under a path-monotonicity condition, the from-start average of gradient flow in a diagonal linear network, rather than its raw output, recovers the Lasso regularization path. We ask what survives for stochastic gradient flow (SGF) and finite-step stochastic gradient descent (SGD). Integrating the square-factor log dynamics yields the Lasso KKT system plus coordinatewise Itô variance clocks and a self-normalized martingale. In an exactly solvable least-squares model, fixed SGF noise and a fixed SGD step produce different weighted-Lasso paths; ordinary Lasso therefore need not survive in either regime. In weak noise, only clock mass accumulated before future support motion enters the comparison. Under the stated nonexplosion, localization, and path-variation conditions, the averaged path returns to Lasso without requiring ${\delta }_{\alpha }\log (1/\alpha )\to 0$. For the standard signed product parametrization, an exact rotation produces two nonnegative channels driven by the same minibatch, with distinct finite-step clocks $R(\eta G)$ and $R(-\eta G)$. When the positive and negative Lasso channels have zero downward variation, the stopped, averaged actual-SGD path converges in prediction norm and signed-Lasso objective as $\alpha \downarrow 0$ and ${\eta }_{\alpha }\to 0$, again without an ${\eta }_{\alpha }\log (1/\alpha )\to 0$ condition. For correlated row sampling, our positive results retain explicit tail-control assumptions; a universal off-face excursion bound remains open.