Inverse Stochastic Gradient Algorithms

• Inverse stochastic gradient algorithms are methods that repurpose stochastic gradient information to address inverse problems, learn inverse operators, or recover latent objectives.
• They integrate techniques such as ML-SGD, stochastic variance reduction, and inverse Hessian learning to improve convergence rates and regularization in both Hilbert and Banach spaces.
• These methods find practical applications in imaging, reinforcement learning, and Bayesian inference, enabling effective handling of noisy and complex data dynamics.

Inverse stochastic gradient algorithms are a heterogeneous family of methods in which stochastic-gradient information is used in an inverse role rather than only as a direct descent direction. In the recent literature represented here, this role appears in three principal forms: stochastic-gradient methods for inverse problems and inverse learning; stochastic methods that explicitly learn and apply an inverse Hessian or inverse-curvature operator; and adaptive inverse-reinforcement-learning schemes that reconstruct an unknown reward from observed stochastic-gradient dynamics. A closely related but distinct line studies stochastic or inexact gradients inside accelerated first-order optimization without addressing inverse problems or inverse-gradient constructions (Fonseca et al., 2022, Carlon et al., 10 Jul 2025, Krishnamurthy et al., 2020, Choudhary et al., 23 Sep 2025).

1. Terminological scope

The literature does not use a single uniform meaning for the expression. The representative usages in the cited works are organized below.

Usage	Characteristic formulation	Representative papers
Inverse problems solved by stochastic gradients	Recover an unknown $f$ or $x^\dagger$ from a forward operator $A$ or $F$ using SGD, mini-batch SGD, or variance-reduced stochastic gradients	(Fonseca et al., 2022, Abhishake et al., 2024, Jin et al., 16 Oct 2025, Jin et al., 19 Jan 2026)
Inverse-curvature stochastic optimization	Learn $H_k \approx (\nabla^2F(x_k))^{-1}$ and update $x_{k+1}=x_k-\eta_k H_k\upsilon_k$	(Carlon et al., 10 Jul 2025)
Inverse learning of objectives from gradient dynamics	Observe a forward stochastic-gradient learner and reconstruct its reward $R(\theta)$ via passive Langevin dynamics	(Krishnamurthy et al., 2020, Krishnamurthy, 6 Jul 2025)

A useful boundary case is "Inexact and Stochastic Gradient Optimization Algorithms with Inertia and Hessian Driven Damping" (Choudhary et al., 23 Sep 2025). That work explicitly states that it is not about an “inverse stochastic gradient algorithm” in the usual sense of inverse problems, inverse operators, or an algorithm that inverts gradients; its contribution lies instead in accelerated first-order optimization with inexact and stochastic gradient information. This suggests that the phrase is best treated as an umbrella label for several adjacent constructions rather than as the name of a single canonical algorithmic template (Choudhary et al., 23 Sep 2025).

2. Statistical inverse problems and adjoint-based stochastic gradients

A direct inverse-problem interpretation appears in statistical inverse learning. In "Statistical Learning and Inverse Problems: A Stochastic Gradient Approach" (Fonseca et al., 2022), the unknown is a function $f^\circ\in H$ linked to the observations through the bounded linear forward map $A:H\to L^2(X)$ and the model

$Y = A[f^\circ](X) + \epsilon.$

The inverse problem is reformulated as population-risk minimization,

$x^\dagger$0

and the gradient has the adjoint form

$x^\dagger$1

Under a kernel representation $x^\dagger$2, the paper derives the unbiased stochastic gradient

$x^\dagger$3

This yields the Hilbert-space update

$x^\dagger$4

with the excess-risk bound

$x^\dagger$5

For $x^\dagger$6, the paper obtains an $x^\dagger$7 rate, and it also proposes ML-SGD, which smooths each gradient snapshot by fitting a base learner such as smoothing splines or regression trees before updating (Fonseca et al., 2022).

The same inverse-learning viewpoint persists in the nonlinear random-design setting. "Gradient-Based Non-Linear Inverse Learning" (Abhishake et al., 2024) studies

$x^\dagger$8

and analyzes both full GD and mini-batch SGD with constant step size on the empirical risk

$x^\dagger$9

Its key operator is the tangent-kernel covariance

$A$0

with tangent kernel

$A$1

Under a source condition $A$2 and the effective-dimension bound

$A$3

the paper proves high-probability bounds for GD and mini-batch SGD up to the stopping time

$A$4

and states that these stopping rules yield minimax-optimal rates within the classical RKHS framework (Abhishake et al., 2024).

3. Iterative regularization, variance reduction, and Banach-space formulations

For linear inverse problems, stochastic variance reduction has been reinterpreted as iterative regularization rather than only as finite-sum optimization. "An Analysis of Stochastic Variance Reduced Gradient for Linear Inverse Problems" (Jin et al., 2021) studies the least-squares inverse problem

$A$5

with

$A$6

Its SVRG update is

$A$7

and the paper derives the bias-variance form

$A$8

Balancing the two terms yields an order-optimal stopping rule in the regularization-theoretic sense, while the variance of the SVRG iterate error is smaller than that of SGD (Jin et al., 2021).

"On the convergence of stochastic variance reduced gradient for linear inverse problems" (Jin et al., 16 Oct 2025) extends this viewpoint to Hilbert spaces and introduces regularized SVRG (rSVRG), where the exact forward operator is replaced by a low-rank approximation $A$9, typically a truncated SVD. Standard SVRG is shown to be regularizing with a suitable a priori stopping rule, whereas rSVRG has a built-in regularization effect and can achieve optimal convergence rates in terms of the noise level without early stopping. Under the source condition

$F$0

and the truncation rule $F$1, the paper obtains the optimal noise-dependent scale

$F$2

for rSVRG, while standard SVRG remains order-optimal only under a stopping rule and mainly for rougher solutions (Jin et al., 16 Oct 2025).

The Banach-space generalization appears in "Stochastic Gradient Descent for Nonlinear Inverse Problems in Banach Spaces" (Jin et al., 19 Jan 2026). There the inverse equation is split into components $F$3, and the stochastic update uses duality maps: $F$4 The Lyapunov quantity is the Bregman distance

$F$5

and under $F$6-convexity, uniform smoothness, a tangential cone condition, and appropriate step sizes, the paper proves almost sure convergence of the exact-data iterates to the $F$7-minimum-distance solution, convergence in expectation, and a regularizing property for noisy data under an a priori stopping rule. Under conditional stability it derives rates such as

$F$8

Its numerical examples on Schlieren tomography and electrical impedance tomography also show that Banach-space choices near $F$9 can be more effective than Hilbert-space choices for sparse solutions and impulsive noise (Jin et al., 19 Jan 2026).

A practical spectral criterion for deciding whether stochastic gradients are advantageous in imaging inverse problems is given in "The Practicality of Stochastic Optimization in Imaging Inverse Problems" (Tang et al., 2019). For least-squares imaging problems with Hessian $H_k \approx (\nabla^2F(x_k))^{-1}$0, the paper defines the stochastic-acceleration factor

$H_k \approx (\nabla^2F(x_k))^{-1}$1

and states that, for an imaging inverse problem, if and only if its Hessian matrix has a fast-decaying eigenspectrum, then the stochastic gradient methods can be more advantageous than deterministic methods. It also shows that a good minibatch scheme typically has relatively low correlation within each of the minibatches (Tang et al., 2019).

4. Indirect stochastic gradients and stochastic-gradient samplers

A different sense of inversion arises when the gradient itself is unavailable and must be reconstructed through sampling. "Stochastic Approximate Gradient Descent via the Langevin Algorithm" (Qiu et al., 2020) considers objectives

$H_k \approx (\nabla^2F(x_k))^{-1}$2

when unbiased stochastic gradients cannot be trivially be obtained because $H_k \approx (\nabla^2F(x_k))^{-1}$3 is hard to sample from. The paper uses an underdamped Langevin chain

$H_k \approx (\nabla^2F(x_k))^{-1}$4

to construct the approximate gradient

$H_k \approx (\nabla^2F(x_k))^{-1}$5

and then performs projected descent

$H_k \approx (\nabla^2F(x_k))^{-1}$6

The gradient estimator is biased in finite time but asymptotically accurate, with bounds

$H_k \approx (\nabla^2F(x_k))^{-1}$7

The paper proves $H_k \approx (\nabla^2F(x_k))^{-1}$8 outer convergence in the convex case and a stationary-point condition in the nonconvex case (Qiu et al., 2020).

Stochastic-gradient MCMC enters inverse problems more directly in "Multi-variance replica exchange stochastic gradient MCMC for inverse and forward Bayesian physics-informed neural network" (Lin et al., 2021). That work proposes multi-variance replica exchange stochastic gradient Langevin dynamics, or m-reSGLD, for multimodal inverse problems and Bayesian PINNs. The two replicas use different temperatures and different energy estimators: $H_k \approx (\nabla^2F(x_k))^{-1}$9 which corresponds in inverse problems to a high-fidelity forward solver for the cold chain and a low-fidelity forward solver for the hot chain. The paper derives an unbiased estimator of the swap rate under heterogeneous noisy energies and reports inverse-problem experiments in which the method captures multiple modes more cost-effectively than same-fidelity replica exchange or single-chain SGLD (Lin et al., 2021).

5. Inverse curvature and inverse-Hessian stochastic methods

The most literal use of the phrase appears in stochastic quasi-Newton methods that learn an inverse Hessian. "Efficient Stochastic BFGS methods Inspired by Bayesian Principles" (Carlon et al., 10 Jul 2025) replaces the SGD step

$x_{k+1}=x_k-\eta_k H_k\upsilon_k$0

by the inverse-preconditioned update

$x_{k+1}=x_k-\eta_k H_k\upsilon_k$1

where $x_{k+1}=x_k-\eta_k H_k\upsilon_k$2 is intended to approximate $x_{k+1}=x_k-\eta_k H_k\upsilon_k$3. The paper’s central idea is Bayesian assimilation of noisy curvature information through a posterior over inverse Hessians,

$x_{k+1}=x_k-\eta_k H_k\upsilon_k$4

with a soft secant likelihood $x_{k+1}=x_k-\eta_k H_k\upsilon_k$5, pair precision $x_{k+1}=x_k-\eta_k H_k\upsilon_k$6, and regularization parameter $x_{k+1}=x_k-\eta_k H_k\upsilon_k$7. The resulting stochastic BFGS and stochastic L-BFGS updates are designed to learn an inverse Hessian approximation even with small batch sizes, while the curvature filter

$x_{k+1}=x_k-\eta_k H_k\upsilon_k$8

prevents large eigenvalues and noise amplification. The paper proves positive definiteness and bounded eigenvalues under curvature filters, states $x_{k+1}=x_k-\eta_k H_k\upsilon_k$9 cost for S-BFGS and $R(\theta)$0 cost for L-S-BFGS when the memory $R(\theta)$1 is treated as constant, and reports experiments up to dimension $R(\theta)$2 in which L-S-BFGS outperforms SdLBFGS and oLBFGS on all tested datasets (Carlon et al., 10 Jul 2025).

This inverse-curvature interpretation should be distinguished from Hessian-driven damping in first-order acceleration. "Inexact and Stochastic Gradient Optimization Algorithms with Inertia and Hessian Driven Damping" (Choudhary et al., 23 Sep 2025) studies the convex Hilbert-space problem

$R(\theta)$3

through a discretization of

$R(\theta)$4

Its discrete algorithms I-IGAHD and S-IGAHD remain first-order, do not compute Hessians explicitly, and establish rates such as

$R(\theta)$5

together with gradient summability under deterministic or stochastic perturbations. The paper explicitly states that it is not about inverse stochastic gradient algorithms in the sense of inverse problems or inverse-gradient constructions, but it remains relevant as background on noisy-gradient acceleration (Choudhary et al., 23 Sep 2025).

6. Inverting gradient dynamics: latent-gradient inference and adaptive IRL

An inverse interpretation of stochastic gradients also appears when the gradients themselves are treated as latent variables. "A Latent Variational Framework for Stochastic Optimization" (Casgrain, 2019) formulates stochastic optimization as a partially observed stochastic control problem in which the optimizer cannot directly observe $R(\theta)$6 and only sees noisy gradient observations $R(\theta)$7. The optimality system is a forward-backward stochastic differential equation whose drift depends on the filtered estimate

$R(\theta)$8

By choosing different priors and observation models for latent clean gradients, the framework recovers SGD, mirror descent, Kalman gradient descent, and momentum. This establishes a direct connection between stochastic optimization algorithms and a secondary Bayesian inference problem on gradients (Casgrain, 2019).

The adaptive inverse-reinforcement-learning formulation turns that perspective into an explicit inverse stochastic gradient algorithm. In "Langevin Dynamics for Adaptive Inverse Reinforcement Learning of Stochastic Gradient Algorithms" (Krishnamurthy et al., 2020) and the later monograph "Inverse Reinforcement Learning using Revealed Preferences and Passive Stochastic Optimization" (Krishnamurthy, 6 Jul 2025), the forward learner follows

$R(\theta)$9

while the inverse learner seeks to reconstruct $f^\circ\in H$0 from the observed stochastic-gradient dynamics. The main passive Langevin recursion has the form

$f^\circ\in H$1

where $f^\circ\in H$2 is a localization kernel, $f^\circ\in H$3 is the forward initialization density, and $f^\circ\in H$4 is a temperature parameter. The paper proves weak convergence to a generalized Langevin diffusion whose stationary density is

$f^\circ\in H$5

so that

$f^\circ\in H$6

for the empirical density $f^\circ\in H$7. Multi-kernel variants, active misspecified-gradient variants, non-reversible variants, and Markov-switching extensions are developed to improve high-dimensional performance and tracking in non-stationary environments (Krishnamurthy et al., 2020, Krishnamurthy, 6 Jul 2025).

Taken together, these works indicate that inverse stochastic gradient algorithms are best understood not as one method but as a research area organized around a common inversion principle: the stochastic gradient may be filtered through an adjoint operator, regularized as an inverse solver, reconstructed by sampling, premultiplied by a learned inverse curvature operator, or observed passively in order to recover the hidden objective that generated it (Fonseca et al., 2022, Carlon et al., 10 Jul 2025, Krishnamurthy et al., 2020).