d$^2$p: Structured Soft Attention Is All You Need

Classical dynamic programming algorithms such as Smith-Waterman, edit distance, and constituency parsing solve structured combinatorial problems using hard constraints. Soft relaxations replace hard $\max$ or $\min$ operators with differentiable probabilistic models whose gradients correspond to alignment or parsing marginals, but existing approaches typically treat the DP algorithm itself as fixed, relying on hand-tuned gap penalties, edit costs, span penalties, and temperatures. We show how to learn these parameters directly from data: the marginals these algorithms produce are themselves first-order derivatives, so learning them is inherently a second-order problem. We derive efficient Hessian-vector products and cross-Jacobians for twelve dynamic programming algorithms spanning alignment, edit distance, and parsing; both derivative families admit closed-form covariance expressions under the induced Gibbs distribution. We implement these operators as fused CUDA kernels in d$^2$p, achieving $100$--$20{,}000\times$ speedups over standard PyTorch and making end-to-end parameter learning practical at modern scales. We then demonstrate that learning these parameters is critical in practice. In protein structure alignment, freezing gap penalties collapses performance from $0.74$ to $0.32$ $F_1$ (and to $0.13$ at lower encoder capacity), while jointly learning them recovers biologically meaningful gap regimes and reaches $0.75$ $F_1$ and $0.445$ lDDT, $91$% of the TM-align lDDT ceiling. The same machinery transfers to constituency parsing, where a structured CKY CRF matches dense per-span supervision to within $0.003$ $F_1$ on English with no direct parse-tree structural supervision.