Differentiable Optimization

We study neural architectures in which each hidden layer is defined by the stationary state of a dissipative Schrödinger-type dynamics on a learned latent graph. On stable branches, the local stationary problem defines a differentiable implicit graph layer. To learn the graph itself, we optimize over the stratified moduli space of weighted graphs and equip each stratum with a non-degenerate Kähler-Hessian metric that keeps natural-gradient descent and face crossing well posed. We then show that a multilayer stationary network is equivalent to an exact global stationary problem on a supra-graph, and that it admits a penalized global relaxation whose stationary states converge to the exact one as the penalty parameter tends to infinity. Reverse-mode differentiation is recovered as the adjoint of the exact global system, and the penalized adjoint converges to it in the same limit. Finally, under finite-dimensional strong-monotonicity and admissible-lift assumptions, the corresponding represented hypothesis classes coincide among resolvent feed-forward networks, graph-stationary networks, supra-graph stationary systems, and sheaf-based architectures with unitary connection. The resulting structural identifications yield complexity bounds controlled by sparse graph or supra-graph geometry rather than dense ambient connectivity.

Neural networks (NNs) achieve outstanding performance in many domains; however, their decision processes are often opaque and their inference can be computationally expensive in resource-constrained environments. We recently proposed Differentiable Logic Networks (DLNs) to address these issues for tabular classification based on relaxing discrete logic into a differentiable form, thereby enabling gradient-based learning of networks built from binary logic operations. DLNs offer interpretable reasoning and substantially lower inference cost. We extend the DLN framework to supervised tabular regression. We first redesign the final output layer (the SumLayer) to support continuous targets. More critically, we find the original two-phase training procedure used for classification is suboptimal for regression, and thus develop a unified, single-stage optimization procedure. We also demonstrate that temperature annealing of the network's differentiable relaxations is decisive for achieving stable convergence and high accuracy. We evaluate the resulting model on 15 public regression benchmarks, comparing it with modern neural networks and classical regression baselines. Regression DLNs match or exceed baseline accuracy while preserving interpretability and fast inference. Our results show that DLNs are a viable, cost-effective alternative for regression tasks, especially where model transparency and computational efficiency are important.

We study differentially private optimization with matrix-orthogonalized momentum. DP-Muon uses conventional global per-example clipping and one Gaussian gradient release per step; matrix updates and auxiliary updates are post-processing. Our main contribution concerns the additional mean distortion created when fresh Gaussian noise passes through a nonlinear matrix map. Conditioning on the actual adaptive history immediately before the current noise yields an exact Gaussian heat identity. For a smooth Newton-Schulz map, first-order DP-MuonBC reduces this conditional output bias from second to fourth order in the fresh noise scale, and an arbitrary-order extension has bias of order $2K+2$. We prove matrix-block stationarity bounds under global clipping, retain finite-step orthogonalization error explicitly, and give an exact criterion for improvement of the resulting upper bound. A separate inequality exposes the effect of auxiliary Adam updates. GPT-2 experiments on E2E at four privacy targets favor the reported Muon configurations over Adam baselines in test NLL.