Sharp Root-Exponential Rates for Smooth Convex Optimization in a Serial Finite-Bit Model

We establish the sharp convergence rate of deterministic smooth convex optimization under simultaneous limits on bit computation and live memory, providing a concrete benchmark for first-order algorithm design with finite resources. In fixed dimension $d$, we consider a finite-bit model in which useful numerical replies must be fully consumed in sequence. Each round includes one oracle call and permits at most $2^{12}d$ bit actions; live storage is bounded by $$\begin{gathered} 2^{20} \\ lbraceA+d[p+ \\ lceil \\ lo{g}_2(d+1) \\ rceil] \\ rbrace \end{gathered}$$ bits, where $A$ is the input bit length and $p$ is the current numerical precision scale. We construct a single uniform algorithm with increasing precision whose reported point $z_n$ satisfies $$\begin{gathered} F(z_n)-F^{ \\ ast} \\ le{C}_0 \\ exp(-c_d \\ sqrtn-n_0) \end{gathered}$$ at every sufficiently late round, uniformly over admissible rounding errors. The algorithm requires no prior knowledge of the smoothness constant, minimizer distance, target accuracy, or horizon. Conversely, for every deterministic program obeying the bit-work budget and reply protocol, even with unrestricted memory, we construct one fixed smooth convex objective whose run under a fixed nearest-grid oracle satisfies $$\begin{gathered} F(z_{n_j})-F^{ \\ ast} \\ gec \\ exp(-C \\ sqrt{n}_j) \end{gathered}$$ along an unbounded subsequence of rounds. Together, these bounds establish $$\begin{gathered} exp[- \\ Theta( \\ sqrtn)] \end{gathered}$$ as the sharp asymptotic order of eventual every-round guarantees in this protocol. With $O_d(1)$ bit actions per round and serial reply consumption, the same root-exponential order holds in total charged bit work.