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An Efficient High-dimensional Gradient Estimator for Stochastic Differential Equations

NeurIPS 2024 · Optimization › Optimizers · approach: Randomized gradient estimation · 2 citations (OpenAlex)

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Proposes an unbiased generator gradient estimator for parameter gradients of expected SDE (with jumps) value functions whose cost scales with O(d^2)/O(d^3) simulated SDEs instead of d+dn, and validates it on neural-network-parameterized linear-quadratic control.

Problem

Estimating gradients of path-dependent SDE expectations with respect to a high-dimensional parameter θ (n large, e.g., neural network weights) via pathwise differentiation requires simulating a d+d·n dimensional SDE, which becomes prohibitive as n grows.

Approach

Rewrites the θ-gradient as an expectation involving only space derivatives of the value function (via the PDE/PIDE generator), then estimates those space derivatives by simulating the state, its spatial Jacobian and Hessian; the cost is independent of n and the estimator is unbiased with polynomially growing variance.

Key results

• Generator gradient estimator's runtime stays near-constant as n grows, outperforming pathwise differentiation by more than an order of magnitude for large n (up to n ~ 10^8).
• Averaged standard error ratio (generator gradient / pathwise) is below 1 across all tested n (e.g., 0.971 at n=102, 0.903 at n=21002).
• The generator gradient estimator is unbiased with variance growing at most polynomially in the initial state x (Var ≤ C(|x|+1)^{2m+4}).

Measured effects reported in the paper

Surprising findings

• The estimator also works in some settings violating smoothness assumptions (e.g., ReLU networks, CIR model), supported only by numerical checks.

Limitations

• Requires second-order continuous differentiability and uniform boundedness of space derivatives of coefficients.
• Dimension of the simulated system grows as O(d^3) when volatility depends on θ, which can be costly for large state dimension d.

Experimental setup

Benchmarks: linear-quadratic control with neural-network-parameterized controls

Metrics: average runtime per sample standard error of gradient estimate standard error ratio

Baselines: pathwise differentiation estimator

Builds on: infinitesimal perturbation analysis Feynman-Kac formula stochastic flows (Kunita) likelihood ratio gradient estimation

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