Must the dimension and accuracy costs of zeroth-order optimization multiply? Global upper bounds for smooth stochastic nonconvex optimization scale as $O(dε^{-4})$, with both factors separately necessary. We show that their product is unavoidable. We establish an $Ω(dε^{-4})$ lower bound for smooth stochastic nonconvex optimization, yielding an $Ω(dδ^{-1}ε^{-3})$ bound for nonsmooth Goldstein stationarity. Both match the best-known full-space upper bounds, establishing joint minimax optimality in dimension and accuracy. These bounds hold against fully adaptive randomized algorithms with unrestricted queries and outputs, even under arbitrary sample reuse. As a byproduct, we establish noiseless exact-value lower bounds for both smooth and Goldstein stationarity, even against randomized algorithms.
Transformers achieve strong performance across modalities but are bottlenecked by the quadratic cost of attention. Sparse attention can reduce this cost, but effective sparsity must preserve useful long-range communication and map well to GPU hardware. We introduce INTRA--Interleaved Non-contiguous Token spaRse Attention--a hardware-aware static sparse attention design based on a relay-communication view under Computational Query Set (CQS) constraints. Rather than directly covering all token-to-token interactions within each sparse layer, INTRA uses complementary sparse groups to provide short mediated communication paths across layers. Scatter forms relay groups that sample broadly across local neighborhoods, while Gather reconnects these groups in subsequent layers. Under a partition model, this structure guides CQS-compatible sparse patterns with global reachability. Our CQS-aware FlashAttention-style kernel computes structured indices at runtime, loads selected non-contiguous tokens, and packs them densely for computation. On an A100, INTRA achieves a $1.53\times$ speedup in $2K$ FLUX.1-dev generation, with better measured image quality than evaluated CLEAR configurations after LoRA self-distillation. On LLaMA-3.1-8B, it improves long-context prefill efficiency with strong teacher-forced retrieval performance under matched LoRA adaptation. Attention-only benchmarks further show up to $8.4\times$ and $10.4\times$ speedups over MoBA and density-constrained NSA, respectively. Code is available at https://github.com/NovTi/INTRA.