Plug-In Classification of Drift Functions in Diffusion Processes Using Neural Networks

Technical Summary: Plug-In Classification of Drift Functions in Diffusion Processes Using Neural Networks

Problem Formulation
This paper addresses the supervised multiclass classification problem for time-homogeneous diffusion processes observed at discrete time points. The underlying process $X = (X_t)_{t \in [0,T]}$ satisfies the stochastic differential equation (SDE):
$$dX_t = b_Y(X_t) dt + \sigma(X_t) dB_t$$
where $Y \in \{1, \dots, K\}$ is a discrete class label, $b_Y$ is the class-specific drift function, and $\sigma$ is a diffusion coefficient assumed to be identical across all classes. The objective is to construct a classifier $g_{D_N}$ that predicts the label $Y$ given a discretely observed trajectory $\bar{Z} = (Z_{t_0}, \dots, Z_{t_M})$. The study assumes the drift functions are distinct across classes while the diffusion coefficient remains constant, a setting where the class information is encoded solely in the drift.

Methodology: Neural Network-Based Plug-In Classifier
The proposed approach is a "plug-in" classifier that explicitly exploits the diffusion structure of the data, contrasting with "end-to-end" methods that treat the entire trajectory as a raw input vector. The methodology proceeds in two main steps:

1. Bayes Characterization and Discretization:
   The authors first derive a multidimensional characterization of the Bayes classifier. Under standard regularity assumptions (Lipschitz continuity, Novikov's condition), the optimal classifier assigns the label $k$ that maximizes a functional $\pi^*_k(X)$. This functional is defined via a softmax transformation of terms involving the drift $b_k$ and the diffusion coefficient $\sigma$:
   $$F^*_k(X) = \int_0^T b_k(X_s)^\top (\sigma \sigma^\top)^{-1}(X_s) dX_s - \frac{1}{2} \int_0^T |\sigma^{-1}(X_s) b_k(X_s)|^2 ds$$
   Since trajectories are observed at discrete times $t_m = m\Delta$, the continuous functional $F^*_k$ is approximated by a Riemann-sum version $\bar{F}_k$.

2. Drift Estimation via Neural Networks:
   Since the true drift functions $b_k$ are unknown, they are estimated nonparametrically using feedforward neural networks (specifically sparse ReLU networks). For each class $k$, the drift is estimated by minimizing an empirical loss based on the discrete increments of the training trajectories belonging to that class. The estimated drifts $\hat{b}_k$ are then plugged into the discretized functional $\bar{F}_k$ to compute estimated class probabilities $\hat{\pi}_k$, and the final classifier predicts the class with the highest probability.

Key Contributions
The paper makes four primary theoretical and numerical contributions:

1. Multidimensional Bayes Characterization: The authors extend the Bayes classifier characterization from one-dimensional settings (Denis et al. [2020]) to the multidimensional case (Proposition 2.4), providing the theoretical foundation for the plug-in approach in higher dimensions.
2. Risk Decomposition: Theorem 2.5 establishes a decomposition of the excess misclassification risk. It bounds the gap between the proposed classifier and the Bayes classifier in terms of the time discretization step $\Delta$ and the $L^2$-error of the drift estimation.
3. Convergence Rates: Theorem 2.7 derives explicit convergence rates for the excess risk when neural networks are used for drift estimation. The rates depend on the sample size $N$, the time step $\Delta$, and the smoothness of the drift functions (characterized by compositional structure).
4. Structural Advantage: The analysis highlights that exploiting the diffusion structure allows the classifier to learn from all observed increments (effectively $N \times M$ data points), whereas direct trajectory-based classifiers treat the whole path as a single sample in a high-dimensional space.

Results and Numerical Experiments
The theoretical findings are supported by two numerical experiments:

• Example 1 (Double-Layer Potential Drifts): In a three-class problem with fluctuating drifts, the proposed NN-based plug-in classifier significantly outperforms B-spline-based methods (Denis et al. [2024]) in one dimension, particularly for large sample sizes where B-splines saturate. Crucially, the method remains computationally tractable and effective in high dimensions ($d \in \{2, 5, 10, 50\}$). The empirical convergence rates align with the theoretical $N^{-1/2}$ rate (up to logarithmic factors), demonstrating that the dimension affects the constants but not the exponent when drift functions admit a compositional structure. Furthermore, the method substantially outperforms direct trajectory-based classifiers (FNNs, RNNs, TCNs, Transformers) that ignore the SDE structure.
• Example 2 (Denis et al. [2024] Benchmark): Revisiting a standard benchmark, the proposed method achieves classification risk comparable to the B-spline-based classifier and the Bayes benchmark, validating its performance in established settings.

Significance and Claims
The paper claims that the proposed plug-in classifier offers a robust alternative to direct trajectory-based learning for diffusion processes. By explicitly modeling the underlying SDE structure, the method avoids the "curse of dimensionality" associated with treating the entire trajectory as a high-dimensional input vector. The theoretical analysis confirms that the convergence rate is determined by the drift estimation accuracy rather than the dimension of the observation space (under compositional assumptions).

The authors acknowledge limitations, noting that the constants in the risk bounds grow with the dimension $d$ due to the application of Burkholder–Davis–Gundy inequalities and Gronwall's lemma. Consequently, while the convergence rate (exponent) remains favorable, larger sample sizes may be required in high-dimensional settings to observe significant risk reduction. Additionally, the paper notes that classification is intrinsically more challenging than drift estimation, as the excess risk is bounded by 1; if the drift estimator is inaccurate, the classifier may perform near-randomly.

Overall, the work extends the scope of diffusion process classification from one-dimensional to multidimensional settings and provides a theoretical justification for using structure-aware plug-in classifiers over generic deep learning architectures for this specific class of problems.