We study exact verification of ReLU networks in an adversarial smoothed model. Every network weight and bias is independently perturbed by Gaussian noise, clipped to $[-2,2]$, and rounded to the exact dyadic grid determined by the input bit complexity. We show that, under the standard assumption $\mathrm{NP}\not\subseteq\mathrm{BPP}$, there is no sound and complete verifier whose expected running time is polynomial in network size, bit complexity, and inverse noise level for every base instance. The conclusion already holds at the fixed noise level $σ_\star=2^{-11}$ for one-hidden-layer networks over a unit box, with hidden fan-in at most three and base coefficients in $[-1,1]$. The proof combines an exact gap embedding with a quantitative robustness argument. For every E3SAT formula $Φ$ with $m$ clauses, a four-ReLU-per-clause construction satisfies $\max_{x\in[0,1]^n} g_Φ(x)=(m-\operatorname{unsat}(Φ))/3$, and coordinatewise threshold rounding never decreases the objective. A weighted parameter-sensitivity inequality and Gaussian concentration then show that a verification gap linear in $m$ survives the aggregate perturbation of all coefficients with probability at least $1-e^{-m/8}$. The proof includes clipping, exact dyadic rounding, output-layer perturbations, polynomial-bit sampling of the rounded Gaussian law, and the conversion from expected smoothed running time to a BPP algorithm. Computational checks test the exact identity and illustrate the different scaling of extensive and constant gaps; they are diagnostics rather than evidence for the complexity theorem. The result concerns worst-case base networks in the stated absolute-noise model, but it shows that parameter nondegeneracy alone does not yield a universal smoothed-polynomial guarantee for exact verification.
We study the problem of efficient online proportional sampling from a high-dimensional domain under a $σ$-smoothed adversary, where the sampling distribution is induced by a dynamically evolving weight function defined over a sequence of piecewise-structured partitions. This setting captures a broad range of applications, including principal-agent games (e.g., pricing and contract design), and algorithm configuration and parameter tuning. The central challenge is maintaining an efficient data structure as the induced partition grows increasingly complex over time -- naively, the number of subregions can grow as $O(t^d)$ by round $t$ in $d$ dimensions. We design a data structure that supports efficient updates and proportional sampling while avoiding the cost of explicitly maintaining this exponential growth, where the discontinuities are structured from axis-parallel hyperplanes. Under a $σ$-smoothed adaptive adversary, we prove a tight $O(\sqrt{σT})$ bound on the depth of our data structure, and an $O(\log T)$ bound under a random-order adversary -- to our knowledge, the first such results for this class of problems. We apply this framework to online learning with piecewise-structured rewards, obtaining efficient no-regret algorithms under both full-information and bandit feedback, with provable sublinear regret guarantees.