Pathwise Differentiable Techniques

• Pathwise differentiability is a framework that extends classical derivatives to entire sample paths and nonsmooth functions, enabling rigorous sensitivity analysis.
• It underpins efficient gradient estimation in variational inference and automatic differentiation by leveraging transport equations and conservative Jacobians.
• The approach supports a range of applications from reflected diffusions in convex domains to optimization in infinite-dimensional statistics and discrete event simulations.

Pathwise differentiability is a foundational concept in stochastic analysis, optimization, and automatic differentiation that generalizes classical derivative notions to functions and stochastic processes potentially lacking smoothness. At its core, pathwise differentiability characterizes the existence and computability of derivatives along entire sample paths or with respect to driving signals/parameters, allowing for rigorous sensitivity analysis, stochastic calculus, and gradient-based optimization in complex or nonsmooth settings. This article presents a comprehensive overview of pathwise differentiable techniques across stochastic processes, transport-based gradient estimation, variational inference, convex analysis, and automatic differentiation, referencing principal developments in the field.

1. Mathematical Foundations and Definitions

Pathwise differentiability (sometimes "path differentiability" in the literature) extends pointwise notions of differentiation to sample paths (in stochastic calculus), or to parameters of functions/processes under perturbations. For locally Lipschitz functions $f:\mathbb{R}^p\to\mathbb{R}^m$, pathwise differentiability is formalized via the existence of a conservative Jacobian $$D: \mathbb{R}^p \rightrightarrows \mathbb{R}^{m\times p}$$, i.e., a set-valued map with closed graph and convex non-empty values, such that for every absolutely continuous path $y:[0,1]\to\mathbb{R}^p$,

$$\frac{d}{dt}f(y(t)) \in D(y(t))\dot{y}(t) \quad\text{for a.e. } t,$$

and the integral representation holds:

$$f(y(1)) - f(y(0)) \in \int_0^1 D(y(t))\dot{y}(t)\,dt.$$

This structure encapsulates Clarke subgradients, convex/concave functions, semialgebraic Lipschitz functions, and differential inclusions (Bolte et al., 2019), and is central to nonsmooth analysis, optimization, and sensitivity theory.

In stochastic analysis, the pathwise derivative of a process (e.g., reflected diffusion) with respect to parameters is constructed as a stochastic process itself that gives sample-path derivatives (see Section 3 below). In semiparametric statistics, pathwise differentiability underpins the expansion of infinite-dimensional parameters and efficient influence function construction (Luedtke et al., 2023).

2. Pathwise Sensitivity in Stochastic Processes

Pathwise differentiability plays a pivotal role in the sensitivity analysis of stochastic processes, particularly in reflected diffusions, SDEs, and random fields.

• Reflected Diffusions in Convex Polyhedral Domains For domains $D=\{x:\langle x,n_i\rangle\ge c_i,\, i=1,\dots,m\}$ and reflected diffusions governed by

$$X_t = x_0 + \int_0^t b(X_s)\,ds + \int_0^t \sigma(X_s)\,dW_s + \sum_{i=1}^m \int_0^t n_i\,dL^i_s,$$

one establishes that, under smoothness, ellipticity, and geometric conditions, the process $X_t$ is pathwise differentiable in its parameters (initial condition, drift, diffusion, reflection directions). The pathwise derivative $J_t$ solves a constrained linear SDE with drift/diffusion coefficients and jumps reflecting the sensitivity to parameter changes:

$$dJ_t = \partial_v b(X_t)\,dt + \partial_v \sigma(X_t)\,dW_t + \text{(reflection terms)},$$

where $J_t$ admits a probabilistic representation for expectation derivatives:

$$\frac{d}{d\epsilon}\Big|_{\epsilon=0}\,\mathbb{E}[f(X^{\theta+\epsilon v}_T)] = \mathbb{E}[\langle\nabla f(X^\theta_T), J_T\rangle],$$

supporting robust sensitivity analysis in stochastic networks, queues, and finance (Lipshutz et al., 2017).

• Taylor Expansion for Random Fields (Dupire Calculus) For path-dependent random fields $u(t,\omega)$ on path space, pathwise derivatives (vertical, horizontal) allow expansion to arbitrary order (forward/backward), generalizing classical Taylor series to pathspace. Remainder terms admit pathwise estimates of the form:

$$|R_{n+1}(t,\omega;\delta t,\delta\omega)| \le C_{n,T}(\omega)|\delta t|^{(n+1+\alpha)/2},$$

which underpins viscosity solution theory for path-dependent PDEs (Buckdahn et al., 2013).

• Pathwise SDE Solutions and Malliavin Calculus Relation For second-order SDEs driven by Hölder-continuous paths, solutions defined via Young integrals are shown to be pathwise differentiable with respect to the driving signal; for fractional Brownian drivers, this coincides with the Malliavin derivative, providing a bridge between deterministic and stochastic calculus (Quer-Sardanyons et al., 2010).

3. Transport Equations and Gradient Estimation in Variational Inference

Pathwise derivatives underpin efficient gradient estimation in variational inference and optimization, supplanting high-variance score-function methods.

• Transport Equation Framework For expectations $\mathbb{E}_{q_\theta}[f(x)]$, the pathwise (reparameterization-trick) gradient derives from the transport (continuity) equation:

$$\partial_\theta q_\theta(x) + \nabla_x\cdot[q_\theta(x) u_\theta(x)] = 0,$$

yielding an unbiased estimator

$$\partial_\theta L = \mathbb{E}_{q_\theta}[u_\theta(x)\cdot\nabla_x f(x)],$$

once a solution $u_\theta(x)$ is chosen (Jankowiak et al., 2018, Jankowiak et al., 2018).

• Distribution-Specific Pathwise Gradients For distributions lacking closed-form reparameterization (Gamma, Beta, Dirichlet), CDF-based and saddlepoint approximations yield accurate, unit-tested pathwise derivative formulas compatible with modern autodiff. For mixtures, multivariate Gaussians, and logistic weights, explicit transport field constructions and null solutions enable adaptive variance reduction (Jankowiak et al., 2018).
• Optimal-Mass-Transport (OMT) Gradients Among all transport fields, curl-free (OMT) solutions minimize kinetic energy and attain lower empirical variance for smooth test functions, demonstrably improving convergence speed and final ELBO in high-dimensional SVI tasks (Jankowiak et al., 2018).
• Mixture Weights and Adaptive Control Variates Divergence-free null solutions provide adaptive control variates, allowing stochastic gradient steps to minimize variance dynamically in mixture models (Jankowiak et al., 2018).

4. Pathwise Differentiability Techniques in Automatic Differentiation and Nonsmooth Optimization

Pathwise differentiable techniques furnish the mathematical foundation for generalized automatic differentiation of nonsmooth or semialgebraic functions:

• Conservative Field Calculus Conservative set-valued fields generalize differentiation to nonsmooth functions by requiring zero circulation on closed loops. A sum, chain, and product rule calculus is developed; Whitney–stratification techniques establish that, over each analytic stratum, conservative fields project to gradients (Bolte et al., 2019).
• Automatic Differentiation The computation graph of any composition of elementary pathwise differentiable functions naturally propagates conservative field elements via forward (directional) or reverse (adjoint) mode. At nondifferentiable nodes (e.g., ReLU, max), one selects a permitted subgradient, ensuring the output remains within the conservative field, and the chain rule applies formally (Bolte et al., 2019). Table: Conservative field computation modes

| Mode | Key Operation | Resulting Object | |--------------|-----------------------------------|-----------------------| | Forward (push)| Directional derivative assembly | Conservative field element| | Reverse (backprop) | Adjoint propagation | Conservative field element|

• SGD Convergence in Nonsmooth Problems The generalized calculus enables robust analysis of stochastic gradient methods for nonsmooth objectives, proving convergence of bulk stochastic iterations to critical points within the conservative field, provided the functions are semialgebraic and the stochastic gradients are unbiased (Bolte et al., 2019).

5. Parametric Monotone Inclusions, Fixed-Point Sensitivity, and Variational Problems

Pathwise differentiability allows for implicit differentiation of solutions to monotone inclusion problems and convex variational models in the nonsmooth regime.

• Strongly Monotone Inclusions For $0\in A_\theta(x)+B_\theta(x)$, with maximal monotone $A_\theta$ and Lipschitz/strongly monotone $B_\theta$, fixed-point maps $H(\theta,x)$ admit contractivity. The solution $x^*(\theta)$ is unique, pathwise differentiable, and its generalized Jacobian is given by the nonsmooth implicit function theorem:

$$J_{x^*}(\theta) = \{(I-V(I-\gamma Z))^{-1}(U-\gamma VW)\},$$

with terms derived from the Clarke Jacobians of primal resolvents and smooth parts (Bolte et al., 2022).

• Automatic Differentiation Compatibility The formal similarity to smooth IFT ensures that implementation within autodiff frameworks (JAX, PyTorch) is direct, and iterative differentiation converges to the implicit Jacobian under the right contractivity conditions (Bolte et al., 2022).
• Fundamental Applications The framework applies to strongly convex minimization, composite primal-dual problems, and min-max optimization, providing explicit formulas for sensitivity analysis and enabling gradient-based learning in equilibrium models (Bolte et al., 2022).

6. Applications in Particle Resampling and Discrete Event Simulation

Recent advances extend pathwise differentiable techniques to resampling methods in sequential Monte Carlo and to differentiable simulation of discrete event systems:

• Diffusion Differentiable Resampling Particle filter resampling uses a continuous diffusion, guided by ensemble score estimations, to transform weighted particles into unweighted samples. Forward and reverse SDEs, coupled with reparameterization and autodiff, yield instant pathwise differentiability, unbiased gradients for stochastic filtering, and provable consistency as particle number increases. Empirical results show statistical performance equal or superior to entropic OT, with efficient computational scaling (Andersson et al., 11 Dec 2025).
• Differentiable Discrete Event Simulation In discrete event simulations (e.g., queueing networks), smoothing strategies (capacity-sharing relaxations, softmin surrogates) ensure that pathwise gradients are computable through autodiff frameworks. Bias and variance scaling favors pathwise estimators, yielding orders-of-magnitude improvements in sample efficiency and gradient quality relative to REINFORCE-type approaches (Che et al., 5 Sep 2024).

7. Role in Infinite-Dimensional Statistics and Efficient Estimation

In infinite-dimensional statistical models, pathwise differentiability characterizes the effect of small distributional perturbations and enables construction of optimal estimators.

• Hilbert-Valued Parameter Estimation Pathwise differentiability for parameters $\nu:\mathcal{P}\to\mathcal{H}$ (in Hilbert spaces) yields a linear local parameter and an adjoint efficient influence operator. Corrected one-step estimators based on the efficient influence function achieve root-$n$ consistency and optimal efficiency. When no such function exists (unbounded operator), regularization yields minimax bias-variance optimal estimators. Exemplar cases include counterfactual density, dose-response functions, conditional average treatment effects, and kernel mean embeddings, all leveraging pathwise differentiable expansions (Luedtke et al., 2023).

• "Pathwise differentiability of reflected diffusions in convex polyhedral domains" (Lipshutz et al., 2017)
• "Pathwise Derivatives Beyond the Reparameterization Trick" (Jankowiak et al., 2018)
• "Pathwise Derivatives for Multivariate Distributions" (Jankowiak et al., 2018)
• "Conservative set valued fields, automatic differentiation, stochastic gradient method and deep learning" (Bolte et al., 2019)
• "Differentiating Nonsmooth Solutions to Parametric Monotone Inclusion Problems" (Bolte et al., 2022)
• "Pathwise Taylor Expansions for Random Fields on Multiple Dimensional Paths" (Buckdahn et al., 2013)
• "Pathwise definition of second order SDEs" (Quer-Sardanyons et al., 2010)
• "Monte Carlo pathwise sensitivities for barrier options" (Gerstner et al., 2018)
• "Path differentiability of ODE flows" (Marx et al., 2022)
• "A Differentiable Partially Observable Generalized Linear Model with Forward-Backward Message Passing" (Li et al., 2 Feb 2024)
• "Differentiable Discrete Event Simulation for Queuing Network Control" (Che et al., 5 Sep 2024)
• "One-Step Estimation of Differentiable Hilbert-Valued Parameters" (Luedtke et al., 2023)
• "Diffusion differentiable resampling" (Andersson et al., 11 Dec 2025)

Pathwise differentiable techniques form the analytic backbone of modern stochastic gradient estimation, robust optimization in nonsmooth settings, infinite-dimensional statistics, and sensitivity analysis for complex models. As these methodologies are further fused with automatic differentiation, measure transport, and structured simulation, they continue to expand the scope and rigor of computational mathematics and machine learning.