Matrix-Free Differentiable CBF Safety Layer

1. Integration point: use a nominal policy $u_{\mathrm{nom}}=\pi_\phi(z)$ in an RL actor, imitation policy, or differentiable model-predictive controller. Construct CBF inequalities in affine-control form $A(z)u+s=b(z)$ from $\dot h_j(z,u)+\alpha_j h_j(z)\ge0$, append slack $s\ge0$, and output the filtered control $u$ from the splitting layer. Keep the filter active during both training and inference. 2. Forward pass: initialize $y^0=(u_{\mathrm{nom}},s^0)$ with $s^0=\operatorname{softplus}(b-Au_{\mathrm{nom}})$. For $k=0,\ldots,K-1$, compute $v^k=P_{C_1}(y^k)$, compute $w^k=P_{C_2}(v^k-\zeta\widetilde Qv^k)$ using $P_{C_2}(x)=x-M^\top(MM^\top)^\dagger(Mx-b)$, and apply the DYS fixed-point update. Check the primal residual $\|Ay+s-b\|_2$ and nonnegativity of $s$ after every iteration. 3. Backward pass: do not materialize $dT/dy$. Given incoming vector $r$, apply the four factors in the displayed Jacobian in reverse order: mask the slack coordinates by $\mathbf{1}[s_i>0]$, scale control coordinates by $(1-\zeta Q_u^\top Q_u)$, and apply the affine-set projector using matrix-vector products with $M$ and $M^\top$ plus conjugate gradients or a sparse factorization of $MM^\top$. Accumulate gradients through $A(z)$, $b(z)$, and $u_{\mathrm{nom}}$. At kink points $s_i=0$, use the deterministic subgradient mask $\mathbf{1}[s_i>0]$ and test a straight-through alternative. 4. Quantities from the mathematics are the orthant mask, projection formula, and matrix-free Jacobian factorization. Estimate $MM^\top$ solves numerically and tune $\zeta\in(0,1)$ and iteration count $K$ by residual and gradient tradeoff. First experiment: train a 20-agent planar point-mass or quadrotor actor on collision avoidance and goal reaching, comparing this layer with an unrolled generic differentiable QP solver and a penalty-only actor at equal FLOPs. Measure safety violations, return, wall-clock training time, peak memory, and loss descent. Scale from 4 to 64 agents; success is near-zero violations with lower memory and comparable return when the dense KKT baseline becomes impractical.