Continuous-Time Proximal Gradient Descent

• Continuous-time proximal gradient descent is the continuous analogue of the forward–backward splitting method, transforming discrete proximal updates to an ODE framework.
• It ensures monotonic decrease of the objective function and, under proximal PL conditions, achieves exponential convergence without a discrete step-size restriction.
• The framework generalizes to complex settings including Wasserstein-space optimization and infinite-dimensional stochastic control, broadening its application to LASSO and feedback policy algorithms.

Continuous-time proximal gradient descent is the continuous-time counterpart of forward–backward splitting for composite optimization. In its basic Euclidean form, it concerns objectives $F(x)=f(x)+g(x)$ with $f$ continuously differentiable and $g$ convex, closed, and proper, and it appears both as the forward–backward Euler interpretation of

$$x_{n+1}=\operatorname{prox}_{\gamma H}\bigl(x_n-\gamma \nabla F(x_n)\bigr)$$

and as the ODE

$$\dot x=-x+\operatorname{prox}_{\alpha g}\bigl(x-\alpha \nabla f(x)\bigr),$$

whose equilibria satisfy $0\in \nabla f(x)+\partial g(x)$ (Salim et al., 2020, Gokhale et al., 2024). The same structural idea extends to optimization over probability measures in Wasserstein space, to Hilbert spaces of controls and feedback policies, and to saddle systems where proximal augmented Lagrangian iterations become implicit gradient descent–ascent schemes (Reisinger et al., 2022, Liu et al., 10 Jun 2026).

1. Euclidean composite dynamics and forward–backward structure

In Euclidean space, the underlying composite problem is to minimize a proper lower semicontinuous convex function $G$, or more specifically a decomposition $G=F+H$ with a smooth part and a nonsmooth convex regularizer. For a proper l.s.c. convex $G:\mathbb R^d\to(-\infty,+\infty]$, the Euclidean subdifferential is

$$\partial G(x)=\{y: G(x)+\langle y,z-x\rangle \le G(z),\ \forall z\},$$

and the associated gradient flow is the differential inclusion

$f$0

This flow admits the Evolution Variational Inequality characterization

$f$1

while the proximal operator is

$f$2

The classical proximal point scheme is therefore a backward Euler discretization of the gradient flow, and the forward–backward iteration

$f$3

is the discrete proximal gradient method (Salim et al., 2020).

A second Euclidean formulation treats proximal gradient descent directly as a continuous-time dynamical system. With

$f$4

the proximal gradient dynamics are

$f$5

This system is the continuous-time analogue of the discrete-time proximal gradient method

$f$6

obtained by viewing the vector field as “the proximal gradient update minus the current point” (Gokhale et al., 2024). In this formulation, fixed points of the proximal mapping coincide with stationary points of the composite problem, because

$f$7

These two viewpoints are compatible rather than competing. One emphasizes forward–backward Euler as a discretization of a composite flow, while the other studies an explicit ODE whose vector field is itself built from the proximal map. Together they supply the canonical Euclidean vocabulary for continuous-time proximal gradient descent.

2. Dissipation, stationarity, and exponential convergence

A central structural property is monotonicity of the objective along trajectories. For the proximal gradient dynamics

$f$8

with $f$9 continuously differentiable, $g$0 convex, closed, and proper, and $g$1, the upper right Dini derivative satisfies

$g$2

so $g$3 is nonincreasing and trajectories converge to the set

$g$4

(Gokhale et al., 2024). The finite-valued assumption on $g$5 can be removed: when $g$6 may take the value $g$7, $g$8 is forward invariant, and whenever the cost is finite its Dini derivative is still nonpositive.

This continuous-time theory differs in a notable way from standard discrete-time step-size analysis. In the discrete proximal gradient method, convergence often requires $g$9 when $$x_{n+1}=\operatorname{prox}_{\gamma H}\bigl(x_n-\gamma \nabla F(x_n)\bigr)$$0 is $$x_{n+1}=\operatorname{prox}_{\gamma H}\bigl(x_n-\gamma \nabla F(x_n)\bigr)$$1-Lipschitz. In the continuous-time analysis above, the ODE converges for all $$x_{n+1}=\operatorname{prox}_{\gamma H}\bigl(x_n-\gamma \nabla F(x_n)\bigr)$$2; no upper bound like $$x_{n+1}=\operatorname{prox}_{\gamma H}\bigl(x_n-\gamma \nabla F(x_n)\bigr)$$3 is needed for monotonicity and convergence results (Gokhale et al., 2024). A common misconception is therefore to transfer the discrete step-size restriction directly to the continuous-time model.

Beyond monotonicity, two exponential convergence mechanisms are identified. The first is a condition directly on the vector field: $$x_{n+1}=\operatorname{prox}_{\gamma H}\bigl(x_n-\gamma \nabla F(x_n)\bigr)$$4 which yields

$$x_{n+1}=\operatorname{prox}_{\gamma H}\bigl(x_n-\gamma \nabla F(x_n)\bigr)$$5

The second is the proximal Polyak–Łojasiewicz condition

$$x_{n+1}=\operatorname{prox}_{\gamma H}\bigl(x_n-\gamma \nabla F(x_n)\bigr)$$6

which implies

$$x_{n+1}=\operatorname{prox}_{\gamma H}\bigl(x_n-\gamma \nabla F(x_n)\bigr)$$7

The paper further shows that the new vector-field condition implies a Kurdyka–Łojasiewicz condition with exponent $$x_{n+1}=\operatorname{prox}_{\gamma H}\bigl(x_n-\gamma \nabla F(x_n)\bigr)$$8, and therefore also implies the proximal PL condition (Gokhale et al., 2024).

The framework also extends to time-varying optimization problems

$$x_{n+1}=\operatorname{prox}_{\gamma H}\bigl(x_n-\gamma \nabla F(x_n)\bigr)$$9

If for each $$\dot x=-x+\operatorname{prox}_{\alpha g}\bigl(x-\alpha \nabla f(x)\bigr),$$0 the pair $$\dot x=-x+\operatorname{prox}_{\alpha g}\bigl(x-\alpha \nabla f(x)\bigr),$$1 satisfies the proximal PL condition with the same constant $$\dot x=-x+\operatorname{prox}_{\alpha g}\bigl(x-\alpha \nabla f(x)\bigr),$$2, then with

$$\dot x=-x+\operatorname{prox}_{\alpha g}\bigl(x-\alpha \nabla f(x)\bigr),$$3

the dynamics satisfy

$$\dot x=-x+\operatorname{prox}_{\alpha g}\bigl(x-\alpha \nabla f(x)\bigr),$$4

and hence

$$\dot x=-x+\operatorname{prox}_{\alpha g}\bigl(x-\alpha \nabla f(x)\bigr),$$5

This is a cost-tracking result rather than a direct state-tracking bound (Gokhale et al., 2024).

The Euclidean dynamics have been instantiated in several model classes. For the LASSO problem, the proximal operator is soft-thresholding and the cost decays exponentially because the problem is known to satisfy the proximal PL condition. For matrix recovery with nuclear norm regularization and for nonconvex matrix factorization with nuclear norm regularization, the cost is nonincreasing and trajectories converge to stationary points, while whether the cost decays exponentially is left as an open question. A numerical experiment on a feed-forward neural network with $$\dot x=-x+\operatorname{prox}_{\alpha g}\bigl(x-\alpha \nabla f(x)\bigr),$$6 regularization exhibits a trajectory moving along a path where the cost is monotonically decreasing, consistent with the general theorem (Gokhale et al., 2024).

3. Wasserstein-space realizations

A measure-theoretic realization replaces Euclidean space by the $$\dot x=-x+\operatorname{prox}_{\alpha g}\bigl(x-\alpha \nabla f(x)\bigr),$$7-Wasserstein space $$\dot x=-x+\operatorname{prox}_{\alpha g}\bigl(x-\alpha \nabla f(x)\bigr),$$8. The composite problem becomes

$$\dot x=-x+\operatorname{prox}_{\alpha g}\bigl(x-\alpha \nabla f(x)\bigr),$$9

where $0\in \nabla f(x)+\partial g(x)$0 is convex and smooth in the Euclidean sense, and $0\in \nabla f(x)+\partial g(x)$1 is proper, lower semicontinuous, and convex along generalized geodesics in Wasserstein space (Salim et al., 2020). In this setting, continuous-time gradient flow is defined through a Wasserstein EVI, and the JKO minimizing movement scheme

$0\in \nabla f(x)+\partial g(x)$2

plays the role of a proximal operator.

The forward–backward scheme for $0\in \nabla f(x)+\partial g(x)$3 performs a forward transport step

$0\in \nabla f(x)+\partial g(x)$4

followed by a backward JKO step

$0\in \nabla f(x)+\partial g(x)$5

Equivalently,

$0\in \nabla f(x)+\partial g(x)$6

This is the precise Wasserstein counterpart of Euclidean proximal gradient descent (Salim et al., 2020).

Under $0\in \nabla f(x)+\partial g(x)$7-smoothness and $0\in \nabla f(x)+\partial g(x)$8, the forward step map is the optimal transport map $0\in \nabla f(x)+\partial g(x)$9. Under the standing assumptions on $G$0, the JKO step satisfies the resolvent-like relation

$G$1

mirroring the Hilbert-space identity for the proximal operator. The analysis yields a discrete EVI

$G$2

from which standard rates follow: in the convex case,

$G$3

and in the strongly convex case,

$G$4

The authors emphasize that these are discrete-time analogues of the continuous-time rates for the Wasserstein gradient flow of $G$5, and that the rates match those of proximal gradient in Hilbert spaces (Salim et al., 2020).

The JKO term also supplies the Wasserstein proximal operator for nonsmooth functionals. Examples include internal energies

$G$6

for absolutely continuous measures $G$7, with negative entropy and power entropies as special cases. In a sampling example with

$G$8

the objective is, up to a constant, the KL divergence to the standard Gaussian, the continuous-time Wasserstein gradient flow is the Fokker–Planck equation for Langevin diffusion, and the forward–backward scheme becomes a deterministic drift step followed by a JKO entropy step. In the Gaussian case, the JKO step and the entire scheme admit closed form, permitting exact tracking of means, covariances, and $G$9, and empirically verifying linear convergence (Salim et al., 2020).

A recurrent point of clarification is that continuous-time and discrete-time behaviors need not coincide. The paper explicitly notes that standard Langevin Monte Carlo, interpreted as a forward-flow splitting of the KL gradient flow, is biased and does not inherit the continuous-time convergence rate, whereas the proposed forward–backward scheme is presented as an unbiased splitting whose rates match the continuous-time gradient flow rates (Salim et al., 2020).

4. Infinite-dimensional stochastic control and feedback policies

In continuous-time stochastic control, proximal gradient descent appears on an infinite-dimensional Hilbert space of controls. The controlled state solves

$G=F+H$0

with control-affine drift

$G=F+H$1

and the cost is

$G=F+H$2

The smooth and nonsmooth parts are separated as $G=F+H$3, with $G=F+H$4 convex, possibly nonsmooth, and allowed to take the value $G=F+H$5 to encode constraints or divergence regularizers (Reisinger et al., 2022).

At the open-loop level, the proximal operator is pointwise: $G=F+H$6 and the proximal gradient iteration is

$G=F+H$7

where the gradient of $G=F+H$8 is represented through the reduced Hamiltonian and an adjoint FBSDE. At the policy level, for feedback maps $G=F+H$9, the Proximal Policy Gradient Method updates

$G:\mathbb R^d\to(-\infty,+\infty]$0

The forward term is therefore a BSDE-derived gradient step, and the backward term is a pointwise proximal step (Reisinger et al., 2022).

The BSDE is not auxiliary bookkeeping; it is the gradient mechanism. The adjoint process $G:\mathbb R^d\to(-\infty,+\infty]$1 solves a backward stochastic differential equation driven by $G:\mathbb R^d\to(-\infty,+\infty]$2, and the key identity is that

$G:\mathbb R^d\to(-\infty,+\infty]$3

is proportional, up to the factor $G:\mathbb R^d\to(-\infty,+\infty]$4, to $G:\mathbb R^d\to(-\infty,+\infty]$5 evaluated at the induced control. Uniform bounds on $G:\mathbb R^d\to(-\infty,+\infty]$6, Lipschitz continuity of $G:\mathbb R^d\to(-\infty,+\infty]$7, stability with respect to $G:\mathbb R^d\to(-\infty,+\infty]$8, and bounds on $G:\mathbb R^d\to(-\infty,+\infty]$9 obtained via Malliavin calculus provide the regularity estimates needed for contraction arguments (Reisinger et al., 2022).

Under suitable structural conditions, the paper proves linear convergence to a stationary point. The conditions are organized as six mechanisms: small horizon, large discount, strong convexity in control, weak dependence on state in costs, weak control effect on dynamics, and strong dissipativity in state dynamics. Under these assumptions and for

$$\partial G(x)=\{y: G(x)+\langle y,z-x\rangle \le G(z),\ \forall z\},$$0

there exist $$\partial G(x)=\{y: G(x)+\langle y,z-x\rangle \le G(z),\ \forall z\},$$1 and $$\partial G(x)=\{y: G(x)+\langle y,z-x\rangle \le G(z),\ \forall z\},$$2 such that

$$\partial G(x)=\{y: G(x)+\langle y,z-x\rangle \le G(z),\ \forall z\},$$3

The induced control $$\partial G(x)=\{y: G(x)+\langle y,z-x\rangle \le G(z),\ \forall z\},$$4 is a stationary point of $$\partial G(x)=\{y: G(x)+\langle y,z-x\rangle \le G(z),\ \forall z\},$$5 in the sense of Fréchet subdifferentials in $$\partial G(x)=\{y: G(x)+\langle y,z-x\rangle \le G(z),\ \forall z\},$$6 (Reisinger et al., 2022).

The same framework accommodates control constraints, sparse/Lasso control, and entropy or $$\partial G(x)=\{y: G(x)+\langle y,z-x\rangle \le G(z),\ \forall z\},$$7-divergence regularization. The paper further states that adding entropy-type regularization can make $$\partial G(x)=\{y: G(x)+\langle y,z-x\rangle \le G(z),\ \forall z\},$$8 strongly convex in the control even when $$\partial G(x)=\{y: G(x)+\langle y,z-x\rangle \le G(z),\ \forall z\},$$9 itself is only semi-convex or concave in $f$00, and that increasing a fictitious discount factor improves contraction constants. These observations are used to justify reinforcement-learning heuristics according to which entropy regularization or fictitious discounting accelerates policy gradient convergence (Reisinger et al., 2022).

5. Time-varying and constrained extensions

Time variation introduces a tracking problem rather than a pure asymptotic minimization problem. In unconstrained composite dynamics, the time-varying proximal PL estimate

$f$01

shows that proximal gradient dynamics track the instantaneous optimal cost with an error controlled by the speed of variation $f$02 (Gokhale et al., 2024).

For linearly constrained time-varying convex optimization, an online proximal-ADMM scheme replaces exact subproblem minimization by proximal gradient steps and a perturbed dual ascent. At time $f$03, the updates are

$f$04

$f$05

and

$f$06

The paper itself focuses on discrete time and does not explicitly derive an ODE, but the algorithmic structure “lends itself naturally to a continuous-time interpretation,” yielding differential inclusions for the primal variables together with a damped dual equation

$f$07

in a natural small-step limit (Zhang et al., 2020).

The convergence statement is a tracking result to a perturbed KKT trajectory. With $f$08 and $f$09 the approximate KKT point, the paper proves

$f$10

and therefore

$f$11

In the static case, the additive term vanishes and the method converges linearly to the approximate KKT point. In the time-varying case, the paper states that the discrete analysis corresponds, in continuous-time language, to exponential stability for the static problem and input-to-state stability for the time-varying case (Zhang et al., 2020).

A further point of interpretation concerns the perturbation parameter $f$12. The distance between the approximate KKT primal variables and the exact KKT primal variables is $f$13, so the total error splits into a tracking error and a regularization error. This is the constrained analogue of the tradeoff between stabilization and exactness that often accompanies proximal and damped primal–dual dynamics (Zhang et al., 2020).

6. Saddle-point generalizations, acceleration, and related continuous-time models

Continuous-time proximal gradient ideas also extend beyond pure minimization to monotone inclusions and minimax problems. In equality-constrained convex optimization, the proximal augmented Lagrangian method

$f$14

is equivalent to the implicit GDA step

$f$15

or, at operator level,

$f$16

This is a proximal point iteration on the monotone operator $f$17, and in the unconstrained case it reduces to implicit gradient descent (Liu et al., 10 Jun 2026).

The first-order variable-step continuous-time limit is the ODE

$f$18

whose Lyapunov analysis uses the variational inequality measure $f$19 rather than the usual objective gap. The solution satisfies

$f$20

and the discrete scheme achieves the corresponding $f$21 last-iterate rates. A second-order ODE family parameterized by $f$22,

$f$23

yields Nesterov-type implicit GDA schemes with $f$24 last-iterate convergence for the primal–dual objective gap. Specializing to $f$25 produces an explicit GDA scheme with an $f$26 last-iterate rate under inverse strong monotonicity of $f$27 (Liu et al., 10 Jun 2026).

This saddle-point formulation shifts the perspective from pure convex optimization to minimax optimization. The paper explicitly states that the underlying potential function must be shifted from the conventional objective gap to a variational inequality measure. In that sense, it generalizes continuous-time proximal gradient descent from composite minimization to operator splitting on monotone saddle operators (Liu et al., 10 Jun 2026).

A related but distinct line of work studies continuous-time models of discrete gradient methods through backward error analysis. The principal flow

$f$28

is presented as a continuous flow that captures divergent and oscillatory behaviors of gradient descent, including escaping local minima and saddle points, and depends explicitly on the Hessian eigendecomposition (Rosca et al., 2023). Although this is not a proximal model, it is discussed as a blueprint for understanding and generalizing to proximal, implicit, or regularized flows. This suggests that continuous-time proximal-gradient models may likewise benefit from formulations that retain finite-step-size effects rather than working only in the $f$29 limit (Rosca et al., 2023).