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Life sciences · Preprint
arXiv · September 8, 2026
Raises a question worth testing. It does not answer one.
This preprint presents a theoretical characterization of the time-uniform convergence rate frontier for standard SGD on smooth convex objectives, proving that the rate approaches but never reaches $\sqrt{\log n / n}$ under standard noise assumptions. The work establishes a necessary and sufficient condition—a divergence criterion on a sequence $h$—for achievability of convergence bounds of order $h(n)/\sqrt{n}$. The result is a mathematical contribution to optimization theory that does not directly address clinical, empirical, or practical validation.
Preprint.
Time-uniform convergence rate gets arbitrarily close to $\sqrt{\log n / n}$ but never reaches it. A bound of order $h(n)/\sqrt{n}$ holding uniformly over the problem class is achievable if and only if $\sum_{j=1}^{\infty} \frac{1}{h(2^j)^2} < \infty$. Constructive sufficiency follows from dyadic horizon-free schedule and additive conditional-restart inequality.
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This is a theoretical analysis establishing convergence rate boundaries for SGD, not a clinical or empirical validation study; it raises mathematical questions about algorithm behavior rather than answering applied or experimental ones.
Quoted from the source exactly as published.
Graded across the dimensions that decide whether you should act, each from what the source actually supports. There is no single score, and where a dimension was not assessed it says so.
We study the time-uniform convergence of the raw iterate of standard stochastic gradient descent (SGD) for unconstrained smooth convex objectives. We prove that, under standard noise assumptions, the time-uniform convergence rate gets arbitrarily close to $\sqrt{\log n / n}$ but never reaches it. More specifically, we prove that for every positive, eventually nondecreasing sequence $h$ satisfying $h(n) = o(\sqrt{n})$, a bound of order $h(n)/\sqrt{n}$, holding simultaneously for all $n$ with probability at least $1-α$ and uniformly over the problem class, is achievable if and only if \[ \sum_{j = 1}^{\infty} \frac{1}{h(2^j)^2} < \infty. \] The constructive sufficiency result follows from a dyadic horizon-free schedule together with an additive conditional-restart inequality. The necessity counterpart applies to every deterministic nonnegative schedule and holds even for a one-dimensional analytic smooth convex objective with Gaussian noise.
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