ICML 2024 Papers

Ordinary differential equations (ODEs) are widely used to describe dynamical systems in science, but identifying parameters that explain experimental measurements is challenging. In particular, although ODEs are differentiable and would allow for gradient-based parameter optimization, the nonlinear dynamics of ODEs often lead to many local minima and extreme sensitivity to initial conditions. We therefore propose diffusion tempering, a novel regularization technique for probabilistic numerical methods which improves convergence of gradient-based parameter optimization in ODEs. By iteratively reducing a noise parameter of the probabilistic integrator, the proposed method converges more reliably to the true parameters. We demonstrate that our method is effective for dynamical systems of different complexity and show that it obtains reliable parameter estimates for a Hodgkin--Huxley model with a practically relevant number of parameters.

We introduce a novel algorithm for estimating Cumulative Distribution Function (CDF) values under Local Differential Privacy (LDP) by exploiting an unexpected connection between LDP and the current status problem, a classical survival data problem in statistics. This connection leads to the development of tools for constrained isotonic estimation based on binary queries. Through mathematical proofs and extensive numerical testing, we demonstrate that our method achieves uniform and $L_2$ error bounds when estimating the entire CDF curve. By employing increasingly dense grids, the error bound can be improved, exhibiting an asymptotic normal distribution of the proposed estimator. Theoretically, we show that the error bound smoothly changes as the number of grids increases relative to the sample size $n$. Computationally, we demonstrate that our constrained isotonic estimator can be efficiently computed deterministically, eliminating the need for hyperparameters or random optimization.

The vector field of a controlled differential equation (CDE) describes the relationship between a *control* path and the evolution of a *solution* path. Neural CDEs (NCDEs) treat time series data as observations from a control path, parameterise a CDE's vector field using a neural network, and use the solution path as a continuously evolving hidden state. As their formulation makes them robust to irregular sampling rates, NCDEs are a powerful approach for modelling real-world data. Building on neural rough differential equations (NRDEs), we introduce Log-NCDEs, a novel, effective, and efficient method for training NCDEs. The core component of Log-NCDEs is the Log-ODE method, a tool from the study of rough paths for approximating a CDE's solution. Log-NCDEs are shown to outperform NCDEs, NRDEs, the linear recurrent unit, S5, and MAMBA on a range of multivariate time series datasets with up to $50{,}000$ observations.