Vector-Field-Driven Mirror Update

• The paper introduces a vector-field-driven generalization of mirror descent that replaces the gradient with an arbitrary vector field while preserving Bregman geometry.
• It details both data-driven learned mirror maps via input-convex neural networks and derivative-free finite-difference methods, emphasizing robustness and improved convergence.
• The study provides theoretical guarantees using relative smoothness and certificate-driven backtracking, offering practical insights into certified optimization algorithms.

A vector-field-driven mirror update is a generalization of the classical mirror descent algorithm in convex optimization, in which the usual gradient term is replaced by a potentially arbitrary vector field. This approach accommodates derivative-free optimization, learned vector fields, and domain-informed updates, while preserving the geometric structure induced by a mirror map and its associated Bregman divergence. The vector-field-driven mirror update framework subsumes both traditional mirror descent with explicit gradients and modern data-driven or zeroth-order variants, and provides interfaces for rigorous analysis, algorithmic certification, and practical implementation (Tan et al., 2022, Hayashi, 31 Jan 2026).

1. Mirror Descent and Its Vector-Field Generalization

In classical mirror descent, the iteration is defined for a convex set $\mathcal{X}\subseteq\mathbb{R}^n$ and a strictly convex, differentiable distance-generating function (mirror potential) $\psi:\mathcal{X}\to\mathbb{R}$, with the associated mirror map $\nabla\psi:\mathcal{X}\to(\mathbb{R}^n)^*$. The update for minimizing a convex differentiable objective $f:\mathcal{X}\to\mathbb{R}$ is

• (Dual update) $y_k = \nabla\psi(x_k) - t_k\nabla f(x_k)$,
• (Primal update) $x_{k+1} = (\nabla\psi)^{-1}(y_k)$.

The framework replaces the gradient $\nabla f(x)$ by a general vector field $\Omega(x)$, yielding the generalized update: $$\theta_k = \nabla\psi(x_k),\quad \theta_{k+1} = \theta_k - t_k\Omega(x_k),\quad x_{k+1} = (\nabla\psi)^{-1}(\theta_{k+1})$$ or equivalently,

$$x_{k+1} = \nabla\psi^*\big( \nabla\psi(x_k) - t_k\,\Omega(x_k) \big)$$

where $\psi:\mathcal{X}\to\mathbb{R}$0 denotes the Fenchel-conjugate of $\psi:\mathcal{X}\to\mathbb{R}$1. The Bregman divergence induced by $\psi:\mathcal{X}\to\mathbb{R}$2 is

$\psi:\mathcal{X}\to\mathbb{R}$3

This formulation allows the design of updates driven by functional data, finite-difference oracles, or learned vector fields (Tan et al., 2022, Hayashi, 31 Jan 2026).

2. Instantiations: Data-Driven and Derivative-Free Vector Fields

Two prominent instantiations of the vector-field-driven framework have been established.

a) Data-driven learned mirror maps: The mirror potential $\psi:\mathcal{X}\to\mathbb{R}$4 is parameterized by an input-convex neural network (ICNN), providing a convex and differentiable function with learnable parameters. The vector field $\psi:\mathcal{X}\to\mathbb{R}$5 replaces the Euclidean identity in gradient descent, adapting the geometry to data. Since the inverse mirror map $\psi:\mathcal{X}\to\mathbb{R}$6 is tractable only for simple $\psi:\mathcal{X}\to\mathbb{R}$7, a separate neural network $\psi:\mathcal{X}\to\mathbb{R}$8 is trained to approximate this inverse (Tan et al., 2022).

b) Deterministic zeroth-order oracles: The vector field $\psi:\mathcal{X}\to\mathbb{R}$9 is constructed using deterministic central finite differences, e.g.

$\nabla\psi:\mathcal{X}\to(\mathbb{R}^n)^*$0

for $\nabla\psi:\mathcal{X}\to(\mathbb{R}^n)^*$1, with $\nabla\psi:\mathcal{X}\to(\mathbb{R}^n)^*$2 function value evaluations per iteration. Uniform dominance over cones around the direction of the true gradient is ensured via robust conic scaling, yielding a certified vector field appropriate for Bregman geometry (Hayashi, 31 Jan 2026).

3. Relative Smoothness and Trajectory-Wise Certification

Global and a posteriori guarantees for vector-field-driven mirror updates rely on relative smoothness-type inequalities with respect to the pair $\nabla\psi:\mathcal{X}\to(\mathbb{R}^n)^*$3. Define the mixed Bregman-style discrepancy: $\nabla\psi:\mathcal{X}\to(\mathbb{R}^n)^*$4 The global relative smoothness condition requires

$\nabla\psi:\mathcal{X}\to(\mathbb{R}^n)^*$5

ensuring monotonic decrease of $\nabla\psi:\mathcal{X}\to(\mathbb{R}^n)^*$6. In practice, the weaker trajectory-wise (a posteriori) property is certified along the realized iterates: $\nabla\psi:\mathcal{X}\to(\mathbb{R}^n)^*$7 If additionally, the objective $\nabla\psi:\mathcal{X}\to(\mathbb{R}^n)^*$8 is star-convex outside a punctured neighborhood $\nabla\psi:\mathcal{X}\to(\mathbb{R}^n)^*$9 of a minimizer $f:\mathcal{X}\to\mathbb{R}$0 (i.e., $f:\mathcal{X}\to\mathbb{R}$1 for $f:\mathcal{X}\to\mathbb{R}$2), the following last-iterate guarantee holds: $f:\mathcal{X}\to\mathbb{R}$3 Thus, the framework provides explicit last-iterate certificates even with nonstandard vector fields (Hayashi, 31 Jan 2026).

4. Learning Vector Fields and Mirror Potentials

Parameterizing the mirror potential via an ICNN, as in

$f:\mathcal{X}\to\mathbb{R}$4

with convex non-decreasing activations and elementwise nonnegativity, results in convex and $f:\mathcal{X}\to\mathbb{R}$5-strongly convex (with added $f:\mathcal{X}\to\mathbb{R}$6) $f:\mathcal{X}\to\mathbb{R}$7. Learning proceeds by minimizing a finite-horizon objective aggregating the empirical task loss and a forward-backward consistency penalty for the approximate inverse map $f:\mathcal{X}\to\mathbb{R}$8. The effective update is: $f:\mathcal{X}\to\mathbb{R}$9 Empirical performance matches or exceeds classical gradient-based algorithms across SVMs, multi-class classifiers, and image inverse problems, with accelerated convergence when consistency is enforced and step-sizes are learned (Tan et al., 2022).

5. Robust Conic Dominance and Derivative-Free Mirror Descent

The finite-difference instantiation constructs a vector field $y_k = \nabla\psi(x_k) - t_k\nabla f(x_k)$0, with $y_k = \nabla\psi(x_k) - t_k\nabla f(x_k)$1 from central differences and explicit scaling $y_k = \nabla\psi(x_k) - t_k\nabla f(x_k)$2 to achieve "robust conic dominance." For any cone $y_k = \nabla\psi(x_k) - t_k\nabla f(x_k)$3 about the gradient direction,

$y_k = \nabla\psi(x_k) - t_k\nabla f(x_k)$4

is enforced by solving a dominance problem over bounded-uncertainty sets $y_k = \nabla\psi(x_k) - t_k\nabla f(x_k)$5 (with $y_k = \nabla\psi(x_k) - t_k\nabla f(x_k)$6 from second differences). The α-scaling formula is provided in closed form. The resulting method ensures a generalized star-convexity interface except for a resolution-dependent exceptional set, outside which full certification is possible. Within a small neighborhood of the optimum, an $y_k = \nabla\psi(x_k) - t_k\nabla f(x_k)$7 error floor is explicitly characterized (Hayashi, 31 Jan 2026).

6. Theoretical Guarantees and Algorithmic Template

For the learned-mirror and finite-difference variants, the regret and last-iterate bounds are explicitly controlled by the geometry and the quality of vector field approximation:

• In the learned-mirror case, with backward map error $y_k = \nabla\psi(x_k) - t_k\nabla f(x_k)$8 bounded, the regret is

$y_k = \nabla\psi(x_k) - t_k\nabla f(x_k)$9

with sublinear or linear convergence up to an additive error (Tan et al., 2022).

• For certified finite-difference mirror descent, if star-convexity and trajectory certificates hold,

$x_{k+1} = (\nabla\psi)^{-1}(y_k)$0

Practical implementation employs certificate-driven backtracking for step sizes and explicit construction of the scaled vector field according to the dominance criterion (Hayashi, 31 Jan 2026).

7. Connections and Empirical Evaluations

Vector-field-driven mirror descent includes as special cases: classical mirror descent, Blahut-Arimoto-type information-geometric algorithms, data-driven learned optimizers, and deterministic zeroth-order (finite-difference) methods. In extensive empirical evaluation, the learned mirror descent framework exhibits superior convergence compared to gradient descent and Adam across support vector machine classification, multi-class linear classification, and total variation-based image restoration tasks, often requiring significantly fewer iterations to reach comparable accuracy and loss metrics (Tan et al., 2022). The finite-difference variant exposes a hidden geometric structure linking Bregman telescoping identities, a posteriori certificate-driven analysis, and robust conic geometry (Hayashi, 31 Jan 2026).

---

References:

• Data-Driven Mirror Descent with Input-Convex Neural Networks (Tan et al., 2022)
• Deterministic Zeroth-Order Mirror Descent via Vector Fields with A Posteriori Certification (Hayashi, 31 Jan 2026)