Rough variational principles and applications to adjoint systems

Motivation

• Adjoint sensitivity analysis efficiently computes gradients for many parameters when only a small number of objective functionals are needed.The introduction connects these methods to applications including optimisation, uncertainty quantification, and data assimilation.Source: §1 · p. 2
• FBSDE simulation is typically costly because it iteratively simulates an adapted backward SDE alongside the forward state SDE.The issue arises when the objective is an expectation over the probability space defining the forward SDE.Source: §1 · p. 2

Contribution

• The paper aims to extend the geometric description and structure-preserving discretisation of adjoint systems to rough-path-driven dynamics through a variational characterisation.It builds on stochastic and rough variational principles, deterministic adjoint geometry, and discrete Hamiltonian variational methods.Source: §1 · p. 2
• The continuous framework derives rough Hamilton equations from a Type-II variational principle and establishes conservation laws for rough adjoint sensitivity analysis.It also gives an associated rough Hamilton–Jacobi equation and pathwise conservation laws for sensitivities to initial conditions and parameters.Source: §1 · pp. 2–3
• The discrete framework constructs rough Galerkin integrators equivalent to RSPRK methods, with symplectic flows, discrete conservation laws, and analysed convergence and naturality properties.The paper illustrates RSPRK convergence and the importance of structure preservation for gradient computation in optimisation.Source: §1 · p. 3

2.1 Type-II rough variational principles

• The framework assumes geometric rough paths, for which ordinary calculus rules apply and conservation laws follow closely to their deterministic counterparts.The section identifies Stratonovich lifted Brownian motion as a classic example and notes that non-geometric rough-path variational principles had not appeared in the literature to the authors’ knowledge.Source: §2 · p. 3
• Critical points of the Type-II action with initial position and terminal momentum fixed are exactly solutions of the rough Hamilton equations under the corresponding constrained variations.The result holds in both directions for controlled paths.Source: §2.1 · p. 4
• On a nondegenerate solution branch, the action extremum defines a Type-II generating function, and the rough Hamiltonian flow preserves the canonical symplectic form at the boundary points.The split boundary data are initial position and terminal momentum; the assumed regular branch need not exist for arbitrary boundary data.Source: §2.1 · p. 6
• On nondegenerate branches, the two-parameter Type-II generating function yields a rough Hamilton–Jacobi equation.The result is stated for the associated generating function with one endpoint position or momentum fixed.Source: §2.1 · pp. 7–8
• Linearized variational equations describe perturbations of initial position and terminal momentum, and their solutions preserve the symplectic pairing along the reference solution.The perturbations evolve by pushforward under the rough flow.Source: §2.1 · p. 9
• Under existence of the rough Hamiltonian flow, the Type-II rough variational argument yields a zero-increment identity for Z^r on [t0, t1].Source: §2.1 · p. 10

2.2 Applications to adjoint systems

• The rough adjoint system is a special case of rough Hamilton’s equations and, given a forward solution through t1, has a unique global solution with terminal boundary condition.Source: §2.2 · p. 10
• The rough adjoint system preserves the canonical Liouville 1-form at the endpoints and conserves the pairing of the adjoint with forward variations throughout the interval.Source: §2.2 · pp. 11–12
• Adding running costs through momentum-affine Hamiltonians extends adjoint equations beyond terminal costs and yields a quasi-conservation law for the variational pairing.Source: §2.2 · p. 12

2.3 Adjoint sensitivities.

• For each Brownian rough-path realization, the adjoint gives initial-condition sensitivity δJ=⟨p0, δq0⟩; expectation sensitivities require justified derivative-expectation interchange.Source: §2.3 · p. 14
• With smooth parameter maps and initial conditions independent of parameters, the adjoint yields pathwise and ensemble parameter sensitivities, also recoverable as ϕ0 in an augmented-state formulation.Source: §2.3 · p. 15

3.1 Rough Galerkin discretisation

• The discrete action uses only level-one rough-path increments, and this choice imposes an order barrier on the Runge–Kutta methods constructed by the Galerkin approach.Source: §3.1 · p. 16
• The rough Galerkin method is solvable for sufficiently small time steps and rough-path increments when r = s, the matrix Aiµ is nonsingular, and bi ≠ 0.Under these conditions, its discrete generating function is smooth in qk and pk+1.Source: §3.1 · p. 17
• When solvable, the rough Galerkin discrete generating function defines a flow that preserves the symplectic form between time steps.The discrete flow also has the continuous system’s canonical 1-form conservation law when both Hamiltonians are linear in p.Source: §3.1 · p. 18
• For Hamiltonians affine in p, the rough Galerkin state and linearised integrators satisfy a discrete quasi-conservation law; when L(q) = L(q) = 0, the pairing ⟨pk+1, δqk+1⟩ = ⟨pk, δqk⟩ is conserved.Source: §3.1 · p. 19

3.2 Rough symplectic partitioned Runge–Kutta method (RSPRK)

• Under Proposition 3.1’s solvability conditions, the rough Galerkin method is equivalent to a collocation-type RSPRK method; the linearised variables also admit a coupled RSPRK formulation.Source: §3.2 · pp. 20–21
• RSPRK methods with coefficients satisfying the symplecticity conditions preserve the discrete conservation laws, including the laws for smooth, linear-in-p, and affine-in-p Hamiltonians.Source: §3.2 · p. 22
• Examples include implicit midpoint and adjoint schemes using Heun’s method or standard RK4 for the q dynamics.For affine-in-p adjoint systems, an explicit q scheme gives an explicit p scheme in reverse time.Source: §3.2 · p. 23
• Adjoint and forward Butcher tableaux need not coincide; a symplectic partitioned method built from an order-p Runge–Kutta method need not retain order p, while classical RK4 and its conjugate pair satisfy every order condition through p=3.Source: §3.2 · pp. 23, 25
• Using only rough-path increments imposes an order barrier: under order-p ODE conditions, local error is O(h^((p+1)α)) and global rate is r=min{r0,(p+1)α−1}.Source: §3.2 · pp. 24–25
• For the rough RK4 method (3.24), the global convergence rate is r=min{r0, 4α−1}; it limits to min{r0, 1/3} as α approaches 1/3.Source: §3.2 · p. 25

3.3 Applications to adjoint systems

• For a geometric rough path, forming the rough variational equation commutes with discretisation when its variable uses the tangent lift; for rough Runge–Kutta, this is the perturbation of initial conditions.Source: §3.3 · pp. 26–27
• Forming the rough adjoint commutes with discretisation when the adjoint uses the cotangent lift; rough partitioned Runge–Kutta methods satisfy this when their tableaux obey the symplecticity relations.Source: §3.3 · pp. 27–28

4.1 Kubo Oscillator

• For rough IM on coupled Kubo oscillators, the expected pathwise rate is min{2h −1/2, 3h −1}; observed rates match theory at h = 3/4 and 1/2, while h = 2/5 is truncation-limited.At h = 1/2, the two rate bounds coincide, making the method optimal relative to the Wong–Zakai bound.Source: §4.1 · p. 28, Figure 1 · p. 29
• Strong convergence rates are evaluated across 32 independent fbm realisations, using a ∆t = 2^-16 reference solution and comparing against rates from [HHW18].Figure 2 shows results for h = 3/4, 1/2, and 2/5.Source: §4.1 · p. 29, Figure 2 · p. 30

4.2 Sensitivity analysis

• In the rough Lorenz 96 example (N = 40, K = 2, h = 2/5), tangent vectors grow rapidly and adjoints also grow near t0.The displayed states, tangents, and adjoints are from a single fbm realisation.Source: §4.2 · p. 30, Figure 3 · p. 31
• Figure 4 shows that RSPRK methods achieve near machine precision conservation of ⟨p, δq⟩ for each realisation of the driving fbm, whereas non-symplectic methods develop errors of O(1).Source: Figure 4 · p. 31
• Without symplectic adjoints, BFGS can terminate at iteration 50 (h = 3/4), 25 (h = 1/2), and 10 (h = 2/5) as gradient precision is lost.The gradient-norm criterion cannot be satisfied when gradients are not exact to machine precision.Source: Figure 5 · p. 32, §4.2 · p. 32
• For h = 2/5, empirical-mean gradients are smoother than pathwise gradients, and Quasi–Newton stability is less affected than in the pathwise optimisation example.Figure 6 varies the number of driving-fbm realisations.Source: Figure 6 · p. 32

5 Conclusion and future work

• The Type-II principle with split boundary conditions yields rough Hamilton equations, conservation laws, and a Hamilton–Jacobi equation, then supports pathwise adjoint conservation and quasi-conservation laws and sensitivity gradients.The prescribed endpoints are initial conditions for the state and terminal conditions for the adjoint.Source: §5 · p. 33
• Rough Galerkin discretisation generates symplectic flows with discrete conservation laws; the methods are equivalent to RSPRK methods, and discretisation commutes with forming variational and adjoint equations under tangent and cotangent lifts.The paper identifies the tangent lift for the state scheme and the cotangent lift for the adjoint scheme.Source: §5 · p. 33
• Rough symplectic methods preserve adjoint conservation laws to machine precision and yield more accurate gradients than non-symplectic methods, with accuracy and convergence speed improving in optimisation.The comparison is reported for optimisation with both pathwise and expected loss functions.Source: §5 · p. 33
• Future work includes extending the finite-dimensional framework to rough PDE adjoints and constructing higher-order variational integrators and adaptive time-stepping methods.The stated order barrier is due to excluding higher-order signatures in (3.3).Source: §5 · p. 33
• The framework currently focuses on geometric rough paths; extending it to non-geometric drivers could broaden applications including data assimilation and stochastic optimal control.The paper notes that drivers in those application settings are typically non-geometric.Source: §5 · p. 34

• The calculation considers a controlled path Y against a geometric rough path Z and smooth test functions with prescribed endpoint values.Source: A Calculus of variations of rough path · p. 34
• The proof approximates a piecewise-linear profile by mollified smooth functions, with transitions in intervals of width n^-1 near s and t.Source: A Calculus of variations of rough path · pp. 34–35
• Two terms vanish at rate O(n^-alpha), and continuity of rough integrals extends the resulting equality to the endpoints.Source: A Calculus of variations of rough path · p. 35
• Corollary A.2 treats the two endpoint constraints separately: one fixes ϕ(a) while leaving ϕ(b) free, and the other reverses those roles.Source: A Calculus of variations of rough path · p. 35