Inexact Dual Averaging Methods

• Inexact dual averaging is an optimization approach that recovers primal iterates using perturbed dual information, incorporating errors from compression or approximate solves.
• The method integrates error-feedback, consensus mechanisms, and adaptive subproblem strategies to maintain convergence bounds similar to exact dual averaging.
• It finds applications in distributed optimization, composite convex problems, and structured learning, achieving convergence rates like O(1/√T) under controlled perturbations.

Inexact dual averaging denotes a class of dual-averaging methods in which the dual information used to recover primal iterates is not exact. The inexactness may arise from message compression with error feedback, approximate solution of the dual-averaging subproblem, consensus-induced oracle noise in distributed optimization, or approximate inner minimization in Lagrangian dual methods. Across these settings, the central objective is to preserve the averaging structure of dual averaging while quantifying how perturbations enter convergence bounds, feasibility guarantees, and structural properties of the iterates (Gao et al., 3 Oct 2025, Huang et al., 2024, Liu et al., 2019, Necoara et al., 2013).

1. Problem formulations and exact baseline

A canonical composite formulation is

$\min_{x\in\mathcal X}\;F(x)=f(x)+\psi(x),$

where $f:\mathbb R^d\to\mathbb R$ is convex and $L$-smooth, and $\psi:\mathbb R^d\to\mathbb R\cup\{+\infty\}$ is a closed convex regularizer or an indicator of a constraint set $\mathcal X$ (Gao et al., 3 Oct 2025). In the structured-learning setting, the same template appears as

$\min_{x\in\mathbb R^n}\;F(x):=f(x)+r(x),$

with $r$ encoding sparsity, low rank, or related structure (Huang et al., 2024).

Exact dual averaging maintains an aggregate gradient and then solves a regularized primal recovery subproblem. One formulation uses

$$G_k=\sum_{i=1}^k s_i\,\nabla f_{\xi_i}(x_{i-1}),\qquad \alpha_k=\sum_{i=1}^k s_i,$$

followed by

$$x_k=\arg\min_x\left\{\langle G_k,x\rangle+\tfrac12\|x-x_0\|_{H_k}^2+\alpha_k r(x)\right\}.$$

When $H_k=\eta^{-1}I$, this reduces to the classical proximal-DA subproblem with closed-form prox-updates for many $f:\mathbb R^d\to\mathbb R$0 (Huang et al., 2024). In the composite error-feedback setting, the exact dual-averaging reference point is the recursion $f:\mathbb R^d\to\mathbb R$1, where the primal iterate is recovered from a subproblem involving $f:\mathbb R^d\to\mathbb R$2, the accumulated regularizer, and a prox-function $f:\mathbb R^d\to\mathbb R$3 through its Bregman divergence (Gao et al., 3 Oct 2025).

The same averaging principle also appears in distributed and constrained settings. In distributed dual averaging, each agent solves a primal step of the form

$f:\mathbb R^d\to\mathbb R$4

where $f:\mathbb R^d\to\mathbb R$5 is the prox-function and $f:\mathbb R^d\to\mathbb R$6 is a local dual estimate (Liu et al., 2019). In dual methods for smooth convex programs with complicating constraints, averaging is applied to approximate primal and dual sequences generated from inexact dual gradients (Necoara et al., 2013).

2. Sources of inexactness

The literature treats several distinct perturbation mechanisms. Some modify the gradient accumulation itself; some perturb the primal recovery subproblem; some replace exact global information with local surrogates; and some solve the inner Lagrangian problem only approximately.

Setting	Inexactness mechanism	Control condition
Composite optimization with compression	$f:\mathbb R^d\to\mathbb R$7, $f:\mathbb R^d\to\mathbb R$8, $f:\mathbb R^d\to\mathbb R$9	$L$0, $L$1
Adaptive dual averaging for structured learning	$L$2 is an $L$3-approximate subproblem solution	$L$4, $L$5, $L$6
Distributed dual averaging	$L$7 from incomplete consensus	$L$8, $L$9
Inexact dual first-order methods	$\psi:\mathbb R^d\to\mathbb R\cup\{+\infty\}$0 approximately minimizes the inner Lagrangian	$\psi:\mathbb R^d\to\mathbb R\cup\{+\infty\}$1

These models share a common feature: the error is not ignored but embedded into the analysis. In the error-feedback construction, the identity

$\psi:\mathbb R^d\to\mathbb R\cup\{+\infty\}$2

ensures that compression errors are carried forward explicitly (Gao et al., 3 Oct 2025). In RAMDA, inexactness is made implementable through a subgradient-residual test and an objective-nondecrease exclusion criterion (Huang et al., 2024). In distributed dual averaging, consensus mismatch is reinterpreted as an inexact-gradient oracle (Liu et al., 2019). In inexact dual first-order methods for constrained convex optimization, approximate inner minimization induces an approximate dual function $\psi:\mathbb R^d\to\mathbb R\cup\{+\infty\}$3 and approximate dual gradient $\psi:\mathbb R^d\to\mathbb R\cup\{+\infty\}$4, together with the inequality

$\psi:\mathbb R^d\to\mathbb R\cup\{+\infty\}$5

for any $\psi:\mathbb R^d\to\mathbb R\cup\{+\infty\}$6 (Necoara et al., 2013).

3. Algorithmic realizations

In the composite error-feedback scheme, the state consists of a dual vector $\psi:\mathbb R^d\to\mathbb R\cup\{+\infty\}$7, an error-memory vector $\psi:\mathbb R^d\to\mathbb R\cup\{+\infty\}$8, and a primal iterate $\psi:\mathbb R^d\to\mathbb R\cup\{+\infty\}$9. With a prox-function $\mathcal X$0 that is $\mathcal X$1-strongly convex with respect to some norm $\mathcal X$2, and Bregman divergence

$\mathcal X$3

the iteration is

$\mathcal X$4

$\mathcal X$5

$\mathcal X$6

$\mathcal X$7

An equivalent weighted form replaces $\mathcal X$8 by $\mathcal X$9 and $\min_{x\in\mathbb R^n}\;F(x):=f(x)+r(x),$0 by $\min_{x\in\mathbb R^n}\;F(x):=f(x)+r(x),$1 (Gao et al., 3 Oct 2025).

RAMDA retains the dual-averaging subproblem but introduces both adaptive preconditioning and inexact subproblem solves. Writing

$\min_{x\in\mathbb R^n}\;F(x):=f(x)+r(x),$2

the method computes only an approximate minimizer $\min_{x\in\mathbb R^n}\;F(x):=f(x)+r(x),$3. To obtain $\min_{x\in\mathbb R^n}\;F(x):=f(x)+r(x),$4, a proximal-gradient loop is applied to $\min_{x\in\mathbb R^n}\;F(x):=f(x)+r(x),$5, where

$\min_{x\in\mathbb R^n}\;F(x):=f(x)+r(x),$6

using

$\min_{x\in\mathbb R^n}\;F(x):=f(x)+r(x),$7

The iteration stops when the prescribed subgradient-residual and objective checks are satisfied (Huang et al., 2024).

In the smooth distributed setting, the local dual estimate is updated through a second-order consensus recursion: $\min_{x\in\mathbb R^n}\;F(x):=f(x)+r(x),$8

$\min_{x\in\mathbb R^n}\;F(x):=f(x)+r(x),$9

This is then coupled with the dual-averaging primal step

$r$0

(Liu et al., 2019).

In the constrained smooth convex case, inexact dual first-order methods operate on the dual problem

$r$1

where $r$2. The outer loop uses approximate gradients $r$3, generated by approximately solving the inner problem. The IDG scheme updates

$r$4

and forms averaged primal and dual iterates

$r$5

The IDFG variant adds a fast-gradient momentum step and a different averaging rule for $r$6 (Necoara et al., 2013).

4. Convergence theory

The central technical question is how the perturbation enters the dual-averaging telescoping argument. In the composite error-feedback setting, the main result for fixed weights $r$7 and constant $r$8 is stated for

$r$9

Under convexity and $$G_k=\sum_{i=1}^k s_i\,\nabla f_{\xi_i}(x_{i-1}),\qquad \alpha_k=\sum_{i=1}^k s_i,$$0-smoothness of $$G_k=\sum_{i=1}^k s_i\,\nabla f_{\xi_i}(x_{i-1}),\qquad \alpha_k=\sum_{i=1}^k s_i,$$1, closed convexity of $$G_k=\sum_{i=1}^k s_i\,\nabla f_{\xi_i}(x_{i-1}),\qquad \alpha_k=\sum_{i=1}^k s_i,$$2, $$G_k=\sum_{i=1}^k s_i\,\nabla f_{\xi_i}(x_{i-1}),\qquad \alpha_k=\sum_{i=1}^k s_i,$$3-strong convexity of $$G_k=\sum_{i=1}^k s_i\,\nabla f_{\xi_i}(x_{i-1}),\qquad \alpha_k=\sum_{i=1}^k s_i,$$4, and an unbiased compression operator satisfying

$$G_k=\sum_{i=1}^k s_i\,\nabla f_{\xi_i}(x_{i-1}),\qquad \alpha_k=\sum_{i=1}^k s_i,$$5

one has

$$G_k=\sum_{i=1}^k s_i\,\nabla f_{\xi_i}(x_{i-1}),\qquad \alpha_k=\sum_{i=1}^k s_i,$$6

With $$G_k=\sum_{i=1}^k s_i\,\nabla f_{\xi_i}(x_{i-1}),\qquad \alpha_k=\sum_{i=1}^k s_i,$$7 and the standard error-feedback estimate

$$G_k=\sum_{i=1}^k s_i\,\nabla f_{\xi_i}(x_{i-1}),\qquad \alpha_k=\sum_{i=1}^k s_i,$$8

this yields

$$G_k=\sum_{i=1}^k s_i\,\nabla f_{\xi_i}(x_{i-1}),\qquad \alpha_k=\sum_{i=1}^k s_i,$$9

and, ignoring constants,

$$x_k=\arg\min_x\left\{\langle G_k,x\rangle+\tfrac12\|x-x_0\|_{H_k}^2+\alpha_k r(x)\right\}.$$0

which matches the classical exact composite dual averaging rate. The paper further remarks that one can refine the result to show an $$x_k=\arg\min_x\left\{\langle G_k,x\rangle+\tfrac12\|x-x_0\|_{H_k}^2+\alpha_k r(x)\right\}.$$1 rate when $$x_k=\arg\min_x\left\{\langle G_k,x\rangle+\tfrac12\|x-x_0\|_{H_k}^2+\alpha_k r(x)\right\}.$$2 is also strongly convex (Gao et al., 3 Oct 2025).

The proof is based on a perturbed dual-averaging template. Defining

$$x_k=\arg\min_x\left\{\langle G_k,x\rangle+\tfrac12\|x-x_0\|_{H_k}^2+\alpha_k r(x)\right\}.$$3

one obtains

$$x_k=\arg\min_x\left\{\langle G_k,x\rangle+\tfrac12\|x-x_0\|_{H_k}^2+\alpha_k r(x)\right\}.$$4

and then bounds the error term using Young’s inequality and strong convexity of $$x_k=\arg\min_x\left\{\langle G_k,x\rangle+\tfrac12\|x-x_0\|_{H_k}^2+\alpha_k r(x)\right\}.$$5 (Gao et al., 3 Oct 2025).

In RAMDA, the convergence statement is asymptotic rather than purely ergodic. Under standard smoothness and bounded-variance assumptions on $$x_k=\arg\min_x\left\{\langle G_k,x\rangle+\tfrac12\|x-x_0\|_{H_k}^2+\alpha_k r(x)\right\}.$$6, step-sizes $$x_k=\arg\min_x\left\{\langle G_k,x\rangle+\tfrac12\|x-x_0\|_{H_k}^2+\alpha_k r(x)\right\}.$$7 satisfying

$$x_k=\arg\min_x\left\{\langle G_k,x\rangle+\tfrac12\|x-x_0\|_{H_k}^2+\alpha_k r(x)\right\}.$$8

momentum-averaging weights $$x_k=\arg\min_x\left\{\langle G_k,x\rangle+\tfrac12\|x-x_0\|_{H_k}^2+\alpha_k r(x)\right\}.$$9 with $H_k=\eta^{-1}I$0, and $H_k=\eta^{-1}I$1, the iterates almost surely converge to a stationary point $H_k=\eta^{-1}I$2 of $H_k=\eta^{-1}I$3. The method also attains the familiar $H_k=\eta^{-1}I$4 rate, or $H_k=\eta^{-1}I$5 in convex cases, in terms of the squared norm of a limiting-subgradient measure (Huang et al., 2024).

In smooth distributed dual averaging, the decisive condition is that the second-order consensus matrix

$H_k=\eta^{-1}I$6

satisfies $H_k=\eta^{-1}I$7, together with

$H_k=\eta^{-1}I$8

Then the ergodic average $H_k=\eta^{-1}I$9 satisfies

$f:\mathbb R^d\to\mathbb R$00

and each local average $f:\mathbb R^d\to\mathbb R$01 also converges to $f:\mathbb R^d\to\mathbb R$02 at rate $f:\mathbb R^d\to\mathbb R$03 (Liu et al., 2019).

For inexact dual first-order methods with averaging, the outer rates depend on the specific scheme. For IDG,

$f:\mathbb R^d\to\mathbb R$04

while IDFG yields

$f:\mathbb R^d\to\mathbb R$05

The corresponding analyses also provide estimates on primal feasibility violation and primal suboptimality. The error propagation is mild in the sense summarized in the paper: IDG does not accumulate inner errors, whereas IDFG accumulates them linearly in $f:\mathbb R^d\to\mathbb R$06 but is faster in terms of $f:\mathbb R^d\to\mathbb R$07 (Necoara et al., 2013).

5. Regularizers, constraints, and structural effects

The nonsmooth or constrained part is not a peripheral detail in inexact dual averaging. In the composite error-feedback formulation, the dual-averaging subproblem has the form

$f:\mathbb R^d\to\mathbb R$08

so the regularizer or indicator enters directly into the accumulated objective. The paper emphasizes that the summation structure naturally accommodates $f:\mathbb R^d\to\mathbb R$09, that the nonsmooth $f:\mathbb R^d\to\mathbb R$10 accumulates linearly in $f:\mathbb R^d\to\mathbb R$11, and that no virtual iterate is needed (Gao et al., 3 Oct 2025). This point is methodologically significant because the paper also argues that the basic error-feedback mechanism and its standard analysis no longer stand when a composite part is involved (Gao et al., 3 Oct 2025).

In structured neural network training, the regularizer is used to induce sparsity, low rank, or related architecture-level structure. RAMDA connects inexact dual averaging to manifold identification through partial smoothness. If $f:\mathbb R^d\to\mathbb R$12 is partly smooth at a solution $f:\mathbb R^d\to\mathbb R$13 relative to an active manifold $f:\mathbb R^d\to\mathbb R$14, and prox-regularity together with the nondegeneracy condition

$f:\mathbb R^d\to\mathbb R$15

hold, then the inexact DA iterates $f:\mathbb R^d\to\mathbb R$16 almost surely lie in $f:\mathbb R^d\to\mathbb R$17 for all sufficiently large $f:\mathbb R^d\to\mathbb R$18 (Huang et al., 2024). The paper interprets this as eventual attainment of the ideal structure induced by the regularizer at the stationary point of asymptotic convergence (Huang et al., 2024).

Constraint handling is equally central in the distributed and Lagrangian settings. Distributed dual averaging is described as attractive for handling constraints and dynamic networks (Liu et al., 2019). In distributed model predictive control, complicating constraints are moved into the cost through Lagrange multipliers, and primal feasibility can be enforced through constraint tightening: $f:\mathbb R^d\to\mathbb R$19 Under Slater, the choice

$f:\mathbb R^d\to\mathbb R$20

preserves strict feasibility of the tightened problem, and if $f:\mathbb R^d\to\mathbb R$21 and $f:\mathbb R^d\to\mathbb R$22 are $f:\mathbb R^d\to\mathbb R$23, the returned $f:\mathbb R^d\to\mathbb R$24 is feasible for the original constraints and satisfies $f:\mathbb R^d\to\mathbb R$25 (Necoara et al., 2013).

6. Relations to proximal methods, limitations, and application domains

A recurrent comparison is between dual averaging and per-step proximal or mirror-descent updates. The composite error-feedback work contrasts dual averaging with proximal methods of the form

$f:\mathbb R^d\to\mathbb R$26

noting that proximal methods rely on per-step composite contractions, whereas dual averaging carries past gradients in $f:\mathbb R^d\to\mathbb R$27 and solves a global subproblem at each step. This global subproblem is what enables the clean telescoping argument used in the inexact analysis (Gao et al., 3 Oct 2025).

The same section also identifies a limitation of prior error-feedback practice: standard EF methods are effective for smooth unconstrained optimization with compression, but fail in the broader setting of composite optimization. The reason is presented as a fundamental limitation in both the method and its analysis technique. The proposed combination of Dual Averaging with EControl is described as the first provable integration of error-feedback into composite dual averaging, with a strong convergence analysis in the general composite setting (Gao et al., 3 Oct 2025).

RAMDA broadens the inexact-dual-averaging perspective in a different direction. Compared with exact dual averaging, it allows diagonal adaptive $f:\mathbb R^d\to\mathbb R$28, introduces momentum averaging via $f:\mathbb R^d\to\mathbb R$29, and tolerates inexact prox-solve. Compared with other inexact proximal methods, its subgradient-norm criterion is described as fully implementable, as not requiring local smoothness of $f:\mathbb R^d\to\mathbb R$30, and as allowing nonconvex $f:\mathbb R^d\to\mathbb R$31 (Huang et al., 2024). The associated proximal-gradient subsolver has complexity $f:\mathbb R^d\to\mathbb R$32 per PG iteration for the gradient plus the cost of the prox, and the iteration count needed to achieve $f:\mathbb R^d\to\mathbb R$33 is $f:\mathbb R^d\to\mathbb R$34 when $f:\mathbb R^d\to\mathbb R$35 is convex and $f:\mathbb R^d\to\mathbb R$36 when $f:\mathbb R^d\to\mathbb R$37 is nonconvex (Huang et al., 2024).

The application range in the cited works is broad but technically coherent. Potential applications beyond composite convex optimization include constrained distributed optimization, safe reinforcement learning, stochastic approximation under resource constraints, and settings with inexact or incremental oracles that can be viewed as a form of compression error (Gao et al., 3 Oct 2025). Empirical studies for RAMDA report large-scale modern computer vision, language modeling, and speech tasks, including structured ResNet on ImageNet, sparse Transformer-XL on WikiText, and sparse Tacotron2 on LJSpeech; the PG-based inexact subsolver cuts the per-epoch cost of adaptive structured training by $f:\mathbb R^d\to\mathbb R$38–$f:\mathbb R^d\to\mathbb R$39 over near-exact solvers, with negligible loss in final accuracy (Huang et al., 2024). In distributed smooth optimization, the proposed method is examined on a large-scale LASSO problem, where primal residual and objective error decay like $f:\mathbb R^d\to\mathbb R$40 (Liu et al., 2019). In distributed MPC, inexact dual averaging is integrated with a parallel coordinate-descent inner solver and a tightening procedure that preserves feasibility guarantees (Necoara et al., 2013).