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On the Generalization and Approximation Capacities of Neural Controlled Differential Equations
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Gives the first statistical analysis of Neural CDEs: a Lipschitz-based generalization bound depending on sampling gap, plus a decomposition of approximation and discretization (sampling) bias via continuity of the CDE flow, validated on synthetic data.
Problem
Theoretical understanding of how NCDEs generalize and approximate on irregularly sampled time series, and how sampling coarseness affects prediction risk.
Approach
Combines controlled differential equation theory (continuity of the CDE flow, Gronwall) with Lipschitz-based Rademacher complexity bounds to derive a generalization bound that depends on the sampling gap |D|, and decomposes total risk into worst-case, discretization bias, and approximation bias terms for a well-specified CDE model.
Key results
- Generalization bound scales with sqrt of model complexity (depth q, hidden dim p) and grows with the sampling discretization |D| through M^D_Theta.
- Discretization bias is proven to be linear in the mesh size |D|, vanishing at linear speed for equidistant grids.
- On fBM classification (H=0.4 vs 0.6), generalization error correlates with sampling gap |D| and average maximal path variation.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Train a shallow NCDE on the teacher-student task | NCDE (q=1) · Teacher-student setup | outcome interpolation | Helped |
| Train a shallow NCDE on the teacher-student task | NCDE (q=1) · Teacher-student setup | latent-state interpolation | Helped |
| Increase the sampling gap by randomly downsampling paths | NCDE · Fractional Brownian motion rough-versus-smooth classification | binary cross entropy loss | Hurt |
| Increase average maximal path variation between sampled points | NCDE · Fractional Brownian motion rough-versus-smooth classification | binary cross entropy loss | Hurt |
Limitations
- Assumes parameters lie in a bounded set Theta and that the loss is Lipschitz and outcomes bounded.
- Bounds grow exponentially with depth q when B_A L_sigma > 1.
- Approximation bias left as a decomposition; applying neural-net approximation rates is left for future work.
Experimental setup
Datasets: synthetic teacher-student CDE data fractional Brownian motion sample paths (H=0.4 vs H=0.6)
Metrics: generalization error binary cross entropy parameter norm evolution correlation with sampling gap
Builds on: Neural Controlled Differential Equations Neural ODE Lipschitz-based generalization bounds
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