One-Step Forward-Backward Methods

• One-Step FB methods are algorithmic strategies that combine a single forward (explicit) and backward (implicit) update to solve stochastic, optimization, and monotone inclusion problems.
• They offer high computational efficiency by using predictor–corrector schemes in FBSDE discretizations and prox/grad queries in optimization, achieving second-order convergence under smoothness conditions.
• Extensions including accelerated, interpolated, and backtracking variants enhance convergence rates and adaptivity, making these methods versatile for a range of numerical and theoretical applications.

The term "One-Step Forward-Backward (FB)" encompasses a spectrum of algorithmic strategies, theoretical envelopes, and numerical discretization schemes, each grounded in the unifying principle of a single composite step mixing a "forward" (explicit, typically gradient or nonexpansive) operator or update with a "backward" (implicit, typically monotone or proximal) component. One-step FB methods are central to stochastic analysis (notably FBSDEs), modern convex and nonconvex optimization (proximal splitting, envelope methods), and monotone inclusion problems, with extensions to explicit accelerated and interpolated variants. This entry surveys the principal one-step FB methodologies, with formal algorithmic characterizations, stability and convergence properties, and illustrative computational findings.

1. One-Step FB Schemes for FBSDEs: Explicit Predictor–Corrector Construction

For decoupled forward–backward stochastic differential equations

$$\begin{cases} X_t = X_0 + \int_0^t b(s,X_s)\,ds + \int_0^t \sigma(s,X_s)\,dW_s, \ Y_t = \Phi(X_T) + \int_t^T f(s,Y_s,Z_s)\,ds - \int_t^T Z_s\,dW_s, \end{cases}$$

the one-step explicit second-order scheme (Han et al., 2024) discretizes these dynamics via a two-phase backward recursion:

• Predictor: computes $(Y_{i+1-\alpha}, Z_{i+1-\alpha})$ at an off-grid internal location using one conditional expectation with parameter $\alpha\in (0,1]$.
• Corrector: integrates the predicted value, updating $(Y_i, Z_i)$ at the time grid point using explicit conditional expectations of increments.

Formally, for grid points $t_i$, denoting $\Delta W_i = W_{t_{i+1}} - W_{t_i}$, the update reads: $$\begin{aligned} Y_{i+1-\alpha} &= \mathbb{E}_{t_{i+1-\alpha}}[Y_{i+1}+\alpha h f_{i+1}], \ Z_{i+1-\alpha} &= \mathbb{E}_{t_{i+1-\alpha}}\big[\tfrac{1}{\alpha h}Y_{i+1}\Delta W_{i+1-\alpha, i+1}+f_{i+1}\Delta W_{i+1-\alpha, i+1}\big], \ \widetilde f_{i+1-\alpha} &= f(t_{i+1-\alpha}, Y_{i+1-\alpha}, Z_{i+1-\alpha}), \ Y_i &= \mathbb{E}_{t_i}\big[Y_{i+1}+\tfrac{h}{2\alpha}\widetilde f_{i+1-\alpha}+h(1-\tfrac{1}{2\alpha})f_{i+1}\big], \ Z_i &= \mathbb{E}_{t_i}\big[\tfrac{2}{h}Y_{i+1}\Delta W_i+\tfrac{1}{\alpha}\widetilde f_{i+1-\alpha}\Delta W_{i,i+1-\alpha}+\tfrac{2\alpha-1}{\alpha}f_{i+1}\Delta W_i-Z_{i+1}\big]. \end{aligned}$$ The parameter $\alpha$ interpolates between explicit schemes ($\alpha\to 0$) and Crank–Nicolson ($\alpha=1$), yielding a fully explicit, second-order globally convergent family.

2. Stability and Error Analysis in One-Step FB Schemes

Stability is characterized via "permutation stability": small terminal and driver perturbations yield proportionally small backward errors in all solution components. Under $f\in C^3_b$, the backward error satisfies

$$\mathbb{E}\left[|\varepsilon_{y,i}|^2 + h\sum_{\ell=i}^{N-1}|\varepsilon_{z,\ell}|^2\right] \le C\Bigl(\mathbb{E}[|\varepsilon_{y,N}|^2+h|\varepsilon_{z,N}|^2]+\sum_{\ell=i}^{N-1}\mathbb{E}[|R^\varepsilon y_\ell|^2+(h+h^2)|\widetilde R^\varepsilon y_\ell|^2 + h|R^\varepsilon z_\ell|^2]\Bigr).$$

The Itô–Taylor expansion up to $O(h^3)$ demonstrates local truncation errors of $O(h^3)$, resulting in global $L^2$-strong error $O(h^2)$ for both $Y$ and $Z$ under standard smoothness assumptions. Numerical experiments on logistic-type and FitzHugh-Nagumo drivers confirm order-two decay uniformly in $\alpha$ (Han et al., 2024).

3. One-Step FB in Optimization: Forward–Backward Envelope (FBE) and Its Evaluation

For optimization problems of the form $\min_x \{\varphi(x) = f(x) + g(x)\}$, with $f$ smooth (possibly nonconvex) and $g$ proper, closed, prox-bounded, the one-step FB envelope is the majorization–minimization model

$$\varphi_\lambda(x) = \inf_z \left[ f(x) + \langle\nabla f(x), z - x\rangle + \frac{1}{2\lambda}\|z - x\|^2 + g(z) \right],$$

where the infimum is attained at the proximal-gradient step $$z = \operatorname{prox}_{\lambda g}(x - \lambda \nabla f(x))$$. In closed form,

$$\varphi_\lambda(x) = f(x) + g(z_\lambda(x)) + \frac{1}{2\lambda}\|z_\lambda(x) - (x - \lambda \nabla f(x))\|^2,$$

and the explicit gradient is $$\nabla\varphi_\lambda(x) = Q_\lambda(x) R_\lambda(x)$$ with $R_\lambda(x) = \lambda^{-1}(x - z_\lambda(x))$ and $Q_\lambda(x) = I - \lambda \nabla^2 f(x)$ (Themelis et al., 2016).

The central property: a single evaluation of $f$, $\nabla f$, and $\operatorname{prox}_{\lambda g}$ suffices to compute both $\varphi_\lambda(x)$ and its gradient, a "one-step" principle underpinning higher-order and quasi-Newton optimization schemes without explicit second derivatives.

4. One-Step Forward-Backward in Splitting and Projective Methods

In the context of monotone inclusion $0 \in A x + B x$, with $A$ maximally monotone and $B$ $\beta$-cocoercive, the one-step forward–backward splitting reads: $$x^{k+1} = J_{\rho A}\big(x^k - \rho B x^k\big), \quad J_{\rho A} = (I + \rho A)^{-1}, \quad \rho \le 2\beta.$$ Recent advances have introduced projective splitting variants processing each cocoercive operator with only a single forward step per iteration. The single-forward-step projective splitting generalizes to block/multi-operator settings and recovers FB as the (unattainable) boundary case $\alpha \downarrow 0$ (Johnstone et al., 2019).

A notable algorithmic distinction is the ability to perform practical backtracking linesearch to adjust step sizes adaptively in the absence of known cocoercivity constants, enabling step sizes of up to $2\beta$.

5. Interpolated and Accelerated One-Step FB Methods

Extensions of the classical one-step FB include schemes which incorporate inertial (momentum) terms, interpolation via history, or deviations (auxiliary vectors): $$\begin{aligned} y_n &= x_n + \alpha_n(y_{n-1} - x_n) + u_n, \ z_n &= x_n + \alpha_n(p_{n-1} - x_n) + \cdots + v_n, \ p_n &= (M + \gamma_n A)^{-1}[M z_n - \gamma_n C y_n], \ x_{n+1} &= x_n + \lambda_n (p_n - z_n) + \Theta_n(z_{n-1} - p_{n-1}), \end{aligned}$$ with deviations $(u_n, v_n)$ controlled via a norm-safeguard (Sadeghi et al., 2022). By tuning a scalar parameter $e\in[0,1]$ governing power-law scaling, the method interpolates between classical FB ($e=0$) and optimal accelerated $O(1/n^2)$ rates ($e=1$), with best practical convergence often achieved at intermediate $e$ values.

6. Comparative Performance and Computational Impact

Large-scale computational tests affirm that one-step FB envelope-based quasi-Newton schemes, such as ZeroFPR (Themelis et al., 2016), attain significant reductions in prox/grad calls and wall-clock time relative to plain or accelerated FB, especially near minimizers and when high precision is required. In the FBSDE discretization context, the explicit one-step predictor–corrector framework yields stable, $O(h^2)$-convergent solutions for stiff drivers without requiring implicit nonlinear subproblems (Han et al., 2024). For composite monotone inclusions, single-forward-step splitting with backtracking exploits the $2\beta$ stepsize bound and delivers empirically faster convergence for strongly cocoercive blocks and non-uniform curvature regimes (Johnstone et al., 2019). Accelerated one-step FB schemes with interpolated deviation parameters outperform both classical and fully accelerated endpoints when tuned on target problems (Sadeghi et al., 2022).

7. Summary Table: One-Step FB Method Variants

Domain	Formal Scheme/Operator	Notable Property/Result
Stochastic FBSDE	Predictor–corrector explicit	Global $L^2$-order 2 accuracy, full explicitness, stability (Han et al., 2024)
Optimization (FBE)	$\varphi_\lambda$ envelope	One prox/grad query for value/gradient, enables higher-order schemes (Themelis et al., 2016)
Operator Splitting	FB splitting, projective	$2\beta$ stepsize for cocoercive blocks, backtracking capable (Johnstone et al., 2019)
Accelerated FB	Deviations/interpolation	Rate interpolation $e$ achieves $O(1/n^2)$, practical gain for $e\approx 0.5$ (Sadeghi et al., 2022)

Each variant retains the defining "one-step" architecture: a single, composite iteration that combines explicit (forward) and implicit (backward) mappings or updates, yielding efficiency, modularity, and extensibility across a wide variety of mathematical domains.