Establishing Boundary KKT Convergence of Mirror Descent through Reparameterization

Establishing Boundary KKT Convergence of Mirror Descent through Reparameterization Kuangyu Ding∗

Kim-Chuan Toh†

August 10, 2026

arXiv:2608.07248v1 [math.OC] 7 Aug 2026

Abstract Sequence convergence to a boundary Karush–Kuhn–Tucker (KKT) point has long remained unclear for nonconvex mirror descent with Legendre kernels. The difficulty is that the Legendre gradient blows up at the boundary. Recent work [16] shows that mirror descent can accumulate at non-KKT boundary points despite decreasing objective values, precluding a convergence guarantee in general. Despite this negative result, mirror descent remains effective in many real applications. Motivated by this contrast, we address the boundary difficulty directly and establish KKT convergence of mirror descent for a broad class of structured nonconvex problems. We analyze mirror descent in reparameterized variables, where the Hessian metric is flattened and remains nondegenerate as the boundary is approached. Under extension and definability conditions jointly coupling the objective, the Legendre kernel, and the feasible geometry, the reparameterized sequence has finite length and converges, thereby recovering convergence to a KKT point of the original iterates. Our general framework applies to some concrete instances: Shannon entropy, Fermi–Dirac entropy, and power kernels on polyhedron.

Keywords. Mirror descent, mirror flow, Kurdyka–Lojasiewicz inequality, circuit decomposition, reparameterization, KKT stationarity.

1

Introduction

We consider the following smooth nonconvex problem with interval and affine constraints: min f (x), x∈X

L := {x ∈ Rn : Ax = b},

X := C ∩ L,

(1)

Q where C := ni=1 Ii and each Ii = (ℓi , ui ) is an open interval, possibly unbounded. Here A ∈ Rm×n has full row rank and b ∈ Rm . We assume that X ◦ := C ∩ L = ̸ ∅, so X ◦ = ri X. The objective f is C 1 on a neighborhood of X. By the sum rule of normal cones, NX (x) = NC (x) + NL (x) for every x ∈ X. Accordingly, x⋆ ∈ X is a KKT point of (1) if and only if there exists λ⋆ ∈ Rm such that 0 ∈ ∇f (x⋆ ) + AT λ⋆ + NC (x⋆ ).

(2)

The vector ∇f (x) + AT λ is the Lagrangian gradient associated with λ. Consider mirror descent generated by a kernel from the following standard class. ∗

School of Industrial Engineering, Purdue University, West Lafayette, IN 47907, USA. Email: kuangyud@u.nus.edu. Department of Mathematics and Institute of Operations Research and Analytics, National University of Singapore, Singapore 119076. Email: mattohkc@nus.edu.sg. †

1

Definition 1.1 (Legendre kernel [31, Section 26]). A proper closed convex function ϕ : Rn → (−∞, +∞] is essentially smooth if int(dom ϕ) ̸= ∅, it is differentiable throughout int(dom ϕ), and ∥∇ϕ(xν )∥ → ∞ for every sequence xν ∈ int(dom ϕ) converging to a point of the boundary of dom ϕ. It is essentially strictly convex if it is strictly convex on every convex subset of dom ∂ϕ, where ∂ϕ denotes the subdifferential of ϕ. We call ϕ a Legendre kernel on C if it has both properties and int(dom ϕ) = C and cl(dom ϕ) = C. P In this work, we focus on Legendre kernels on C of the separable form ϕ(x) := ni=1 hi (xi ). Here each hi is a one-dimensional Legendre kernel on Ii . Definition 1.2 (Bregman divergence). For x ∈ C and y ∈ dom ϕ, the Bregman divergence generated by ϕ is Dϕ (y, x) := ϕ(y) − ϕ(x) − ⟨∇ϕ(x), y − x⟩ . Starting from the interior point x0 ∈ X ◦ , mirror descent generates   1 xk+1 = Tαk (xk ) := arg min ⟨∇f (xk ), y − xk ⟩ + Dϕ (y, xk ) , y∈L αk

αk > 0,

(3)

where the constraint C̄ is enforced by the Legendre kernel ϕ. The essential smoothness of ϕ ensures that every well-defined mirror update lies in X ◦ , while accumulation points may lie in the relative boundary. The first-order condition for the mirror subproblem yields, for some λk+1 ∈ Rm , ∇ϕ(xk+1 ) − ∇ϕ(xk ) + ∇f (xk ) + AT λk+1 = 0, αk

Axk+1 = b.

(4)

Formally replacing the dual increment in (4) by a time derivative suggests the mirror flow d ∇ϕ(x(t)) + ∇f (x(t)) + AT λ(t) = 0, dt

Ax(t) = b.

(5)

When ϕ is C 2 on C, (5) becomes the Hessian–Riemannian gradient flow on the affine space L [1,2,7,9]: ∇2 ϕ(x(t))ẋ(t) + ∇f (x(t)) + AT λ(t) = 0, Aẋ(t) = 0, which can be interpreted as the gradient flow on L endowed with the Riemannian metric induced by ∇2 ϕ on C. The Bregman geometry is especially natural for optimization over probability distributions and couplings. On the simplex, the Shannon entropy kernel yields normalized multiplicative updates. On a transportation polytope, the entropic mirror step is a Kullback–Leibler projection that can be computed by Sinkhorn scaling [12, 26]. These projections underlie algorithms in computational optimal transport and in problems based on the Gromov–Wasserstein distance, including graph matching and barycenter computation [20, 23, 27, 29]. The feasible sets contain lower-dimensional faces representing sparse distributions and couplings, and solutions of interest need not be strictly positive. Boundary limits are therefore natural in these problems, not pathological corner cases. However, convergence of mirror descent for nonconvex problems to a boundary KKT point has long remained unclear. As the iterates approach the boundary, the Legendre gradient blows up, so the mirror-step optimality condition cannot be passed directly to the boundary limit. Existing work often circumvents this boundary difficulty by assuming the iterates or solutions of interest to remain in the interior, modifying the original problem (e.g., by adding an entropic regularizer or imposing explicit constraint strictly away from the boundary), or exploiting special problem structure to establish that the solutions of interest lie in the interior. Recent work [16] constructs a counterexample in which a mirror descent sequence has a non-KKT boundary accumulation point, as detailed in Section 1.1. 2

1.1

Challenges in boundary KKT recovery

A fundamental convergence guarantee for optimizationalgorithms is that every accumulation point is KKT stationary. For Euclidean methods (e.g. gradient descent, the proximal point method, and the proximal gradient method), this requirement is typically certified through a generic update xk+1 = xk − αk Gαk (xk ),

(6)

where Gα is a stationarity residual. Under the standard assumptions for each method, Gαk (xk ) → 0, and the associated optimality relation can be passed along any convergent subsequence to obtain KKT stationarity. By contrast, the Bregman stationarity residuals used for mirror descent depend on ∇ϕ and are defined only on the interior X ◦ . Even if these residuals tend to zero along interior iterates, a boundary accumulation point need not be KKT stationary. Stationarity measures in mirror descent. Existing nonconvex analysis of Bregman methods typically formulates stationarity through an interior residual. A basic choice is the Bregman displacement Rα (x) := Dϕ (Tα (x), x) of one mirror step, used in composite optimization and variational-inequality analysis [13, 14]. Another choice uses a Bregman proximal point of the full objective [34]. For α > 0, let pα (x) ∈ arg miny∈X {f (y) + α−1 Dϕ (y, x)} and define ∆α (x) :=

Dϕ (x, pα (x)) + Dϕ (pα (x), x) . α2

If ∇ϕ is M -Lipschitz on X, proximal optimality at x b := pα (x) gives dist2 (0, ∇f (b x) + NX (b x)) ≤ M ∆α (x). Both residuals are defined only in the interior, and this estimate requires a uniform Lipschitz bound on ∇ϕ. Hence this framework does not directly certify KKT stationarity at a boundary limit. Spurious stationary points. Even when Rα is extended to the boundary, its vanishing need not imply KKT stationarity. Take C = Rn+ and, for x̄ ∈ X, set P B(x̄) := {i : x̄i = 0}, I(x̄) := {i : x̄i > 0}, and M (x̄) := {y ∈ Rn : yi = x̄i for i ∈ B(x̄)}. For ϕ(x) = i h(xi ), the boundary extension in [10] freezes the zero coordinates:     1 X T α (x̄) ∈ arg min ⟨∇f (x̄), y − x̄⟩ + Dh (yi , x̄i ) . (7)  α y∈X∩M (x̄)  i∈I(x̄)

Its residual is Rα (x̄) := indexed by I(x̄):

P

i∈I(x̄) Dh ([T α (x̄)]i , x̄i ). Its vanishing certifies only the first-order equalities

Rα (x̄) = 0

⇐⇒

∃λ ∈ Rm : [∇f (x̄) + AT λ]i = 0

i ∈ I(x̄) .

KKT stationarity additionally requires the boundary inequalities indexed by B(x̄):

∃λ ∈ Rm : [∇f (x̄) + AT λ]i = 0 i ∈ I(x̄) , [∇f (x̄) + AT λ]j ≥ 0 j ∈ B(x̄) .

(8)

(9)

A point satisfying (8) but not (9) is a spurious stationary point. As shown in [15], the spurious stationary points are precisely the non-KKT equilibria of the mirror flow (5). The finite-horizon trapping and finite-exit results in [10, 15] leave the asymptotic question open: they neither prove convergence to a KKT point nor provide a counterexample. For further discussion of spurious stationary points and their relation to mirror-flow equilibria, see [15, 16]. 3

Circumventing the boundary difficulty. Prior convergence analysis has generally avoided this difficulty through interiority conditions, modifications of the original problem, or additional structural assumptions. For example, the analysis in [24] requires the entire cluster set of the iterates to remain in the interior. For stochastic mirror descent with random reshuffling, Qiu et al. [28] establish last-iterate convergence under boundedness of {∇ϕ(xk )}, a condition that excludes boundary accumulation. Other approaches modify the problem itself, either by adding an entropic regularizer or a barrier term to force the relevant solution into the interior, or by explicitly imposing additional constraints that keep the relevant variables away from the boundary of dom ϕ [24,33]. The boundary issue can also be absent when problem-specific structure guarantees that local minimizers are interior [18]. None of these results establishes convergence of the iterates to a KKT point when the limit lies on the boundary. Benign behavior under sequence convergence. Under the standing constraint qualification, it follows from [15, Proposition 5.3] that every convergent sequence generated by (3), with positive nonsummable stepsizes, has a KKT limit. In short, xk → x⋆ ,

αk > 0,

∞ X

αk = ∞

=⇒

x⋆ satisfies the KKT condition.

k=0

Thus, boundary KKT convergence follows once sequence convergence is established. Several structured settings exhibit related benign behavior. For exponentiated gradient with an Armijo line search, the objective values converge to the optimal value for a convex differentiable objective provided that the sequence has a strictly positive limit point [21]. Convergence of the iterates is available for convex mirror descent under square-summable, nonsummable stepsizes [17]. Under variational coherence, the last iterate of stochastic mirror descent converges almost surely to a minimizer [35]. Early continuous-time results established trajectory convergence for Hessian– Riemannian flows under quasiconvexity and for related singular Riemannian barrier flows with convex objectives on polyhedra [1, 2]. More recent continuous-time results establish trajectory convergence to a KKT point when the boundary equilibria are isolated [15]. However, for a general nonconvex objective, sequence convergence is far from trivial: the iterates may wind indefinitely near the boundary, as shown by the counterexample in [16]. In nonconvex optimization, the Kurdyka–Lojasiewicz (KL) inequality is a standard tool for proving finite length and hence sequence convergence [3, 5]. However, for mirror descent, the inverse mirror metric may degenerate at the boundary, so the same argument need not yield finite length. To illustrate this obstruction, consider mirror flow in the unconstrained setting L = Rn . Then the mirror flow satisfies ∇2 ϕ(x(t))ẋ(t) + ∇f (x(t)) = 0, f (x(t)) ↓ f∞ . Suppose that, on a nonstationary tail, the KL inequality ϑ′ (f (x) − f∞ ) ∥∇f (x)∥ ≥ 1 holds for a concave desingularizing function ϑ with ϑ(0) = 0 and ϑ′ > 0. When ∇2 ϕ = I, mirror flow reduces to Euclidean gradient flow, and −

d ϑ f (x(t)) − f∞ = ϑ′ f (x(t)) − f∞ ∥∇f (x(t))∥2 ≥ ∥∇f (x(t))∥ = ∥ẋ(t)∥ , dt

so integration yields finite length. For mirror flow, the KL inequality gives −

∇f, [∇2 ϕ]−1 ∇f d ϑ f (x(t)) − f∞ = ϑ′ f − f∞ ∇f, [∇2 ϕ]−1 ∇f ≥ ∥ẋ∥ . dt ∥∇f ∥ ∥[∇2 ϕ]−1 ∇f ∥

If ϕ is strongly convex and ∇ϕ is Lipschitz, then the coefficient in the last inequality is bounded away from zero. However, for Legendre kernels, [∇2 ϕ]−1 degenerates near the boundary of dom ϕ, 4

so this coefficient need not be bounded away from zero and this direct KL argument does not imply finite length. Non-KKT accumulation. Recent work [16] shows that the boundary obstruction is genuine: mirror descent can accumulate at non-KKT boundary points. In this counterexample, the objective is C ∞ and entropy-relatively smooth, and the mirror descent sequence is bounded with nonincreasing objective values and diminishing nonsummable stepsizes. Both the mirror-flow trajectory and the mirror descent sequence are nonconvergent: the trajectory approaches the boundary while winding indefinitely around a smooth closed curve of equilibria, and the sequence lies on this trajectory and shares its accumulation set, which contains a nonempty relatively open subset of non-KKT points. Motivated by these challenges, we aim to address the boundary difficulty directly. The counterexample shows that this requires ruling out nonconvergent boundary motion, which can produce non-KKT accumulation, whereas sequence convergence forces a KKT limit. The standard KL argument, however, may lose the length control needed for sequence convergence as the inverse mirror metric degenerates, while the usual stationarity measures need not certify the active-constraint inequalities. This raises the question: Can we establish convergence of mirror descent to a KKT point at the boundary of dom ϕ under verifiable joint conditions on the objective, the Legendre kernel, and the feasible geometry? Contributions.

Our contributions are threefold.

1. KKT convergence with boundary limits. We develop a metric-flattening reparameterization framework to handle the blow-up of ∇2 ϕ(x) at the boundary. Under a definable boundary extension andPa curvature condition for the kernels, we prove that mirror descent with αk ∈ (0, 1/L) and ∞ k=0 αk = ∞ has finite length in the reparameterized sequences and converges to a KKT point in the original sequence. We also derive convergence rates for the reparameterized sequence from the KL exponent of the reparameterized objective. We further establish convergence of mirror-flow trajectories to a KKT point. 2. Uniform bound on the Lagrangian gradient. We introduce a general conformal-circuit argument that yields the uniform bound ∥∇f (x) + AT λ∥∞ ≤ Γ on the Lagrangian gradient, where λ is the affine-constraint multiplier of the mirror subproblem. The constant Γ is independent of the distance from x to the boundary of dom ϕ. This uniform bound is a key ingredient in the KL argument proving finite length of the reparameterized sequence. 3. Examples and coordinatewise rates. We provide several concrete examples satisfying the conditions of our framework: Shannon entropy, Fermi–Dirac entropy, and power kernels. For these examples, transferring the KL rates from the reparameterized variables shows that different coordinates of the original sequence may converge at different rates, depending on whether their limits lie in the interior or on the boundary. Independent related work. While finalizing this manuscript, we became aware of independent work by Chen and So, newly posted on arXiv [11]. The main overlap concerns Shannon-entropy mirror descent on X = {x ∈ Rn+ : Ax = b}. In this setting, both works are built around the same square-root reparameterization: their scaled Kurdyka–Lojasiewicz framework can be interpreted √ through z = 2 x, the reparameterization used in this work. The two works differ mainly in scope. This difference also reflects a different perspective on the boundary difficulty. The analysis in [11] formulates its convergence mechanism directly in the original variables through scaled sufficient-decrease and relative-error conditions together 5

with a scaled Kurdyka–Lojasiewicz property, whereas our starting point is the metric-flattening reparameterization. By flattening the Hessian metric, this reparameterization removes the boundary degeneracy of [∇2 ϕ]−1 and, in continuous time, converts mirror flow into the Euclidean subgradient flow of the reparameterized objective. This geometric viewpoint underlies the broader scope of our framework. Beyond this common Shannon-entropy case, our framework covers a more general class of objective functions and kernels through a metric-flattening reparameterization. It applies to separable Legendre kernels on interval–affine feasible sets, with Shannon entropy, Fermi–Dirac entropy, and power kernels as examples. For this general framework, we establish finite length of the reparameterized sequence and KKT convergence of the original iterates. A key ingredient enabling this generality is a conformal circuit decomposition that uniformly bounds the Lagrangian gradient ∇f (x) + AT λ as x approaches the boundary of dom ϕ. The approach in [11] instead exploits the multiplicative entropy update directly. We further derive convergence rates for the general reparameterized sequence across the full range θ ∈ [0, 1) of Kurdyka–Lojasiewicz exponents. For the three kernel examples above, we transfer these rates to the original variables and show that different coordinates may converge at different rates, depending on whether their limits lie in the interior or at active endpoints. Our results also allow positive nonsummable stepsizes that may tend to zero, whereas the result in [11] requires a positive lower bound. Finally, we additionally establish finite length and KKT convergence for the continuous mirror flow. Notation. For an integer r ≥ 1, write [r] := {1, . . . , r}, Rr+ := {u ∈ Rr : ui ≥ 0 for all i ∈ [r]}, and Rr++ := {u ∈ Rr : ui > 0 for all i ∈ [r]}. The vector 1r denotes the all-ones vector. For u, v ∈ Rr , their Hadamard product is denoted by u ⊙ v. For v ∈ Rr , Diag(v) denotes the diagonal matrix with diagonal v. For a differentiable map F : Rp → Rq , DF (x) ∈ Rq×p denotes its Jacobian at x. Scalar functions and powers applied to vectors are understood componentwise. For a set S, cl S and ri S denote its closure and relative interior. When S is closed and convex, NS (x) denotes its convex normal cone at x. For a vector u, supp(u) := {i : ui ̸= 0}. Unless indicated otherwise, ⟨·, ·⟩, ∥·∥, and dist are the Euclidean inner product, norm, and induced point-to-set distance. The remainder of the paper is organized as follows. Section 2 presents the variational notions used in the analysis. Section 3 develops the metric flattening and definable boundary extension. Section 4 establishes finite length in the reparameterized variables and KKT convergence. Section 5 applies the general framework to Shannon entropy, Fermi–Dirac entropy, and power kernels. Section 6 concludes the paper. Some technical proofs are deferred to the appendix.

2

Preliminaries

This section presents some preliminaries used in the convergence analysis. Definition 2.1 (Limiting subdifferential). Let E : Rd → (−∞, +∞] be proper and lower semicontinuous, and let z ∈ dom E. Its Fréchet and limiting subdifferentials [32] are ( ) E(w) − E(z) − ⟨v, w − z⟩ b ∂E(z) := v : lim inf ≥0 , w→z ∥w − z∥ w̸=z   there exist z ν → z and v ν → v such that ∂E(z) := v : . ν) b E(z ν ) → E(z) and v ν ∈ ∂E(z Definition 2.2 (Definability [6, Definitions 6–7]). An o-minimal structure on the real field is a sequence O = (Or )r≥1 , where each Or is a Boolean algebra of subsets of Rr , satisfying the following properties: 6

(i) if A ∈ Or , then A × R ∈ Or+1 and R × A ∈ Or+1 ; (ii) if A ∈ Or+1 , then its projection onto the first r coordinates belongs to Or ; (iii) every real algebraic subset of Rr belongs to Or ; (iv) the sets in O1 are precisely the finite unions of points and intervals. A set A ⊆ Rr is definable in O if A ∈ Or , and a function is definable if its graph is definable. An extended-real-valued function E is definable if dom E and the graph of its restriction to dom E are definable. Throughout, all definability conditions refer to one fixed o-minimal structure. We write Ran,exp for the o-minimal expansion of the real field by restricted real-analytic functions and the exponential function. Definable sets are closed under finite unions, complements, and coordinate projections. In particular, the image of a definable set under a definable map and the closure of a definable set are definable. Every proper lower-semicontinuous definable function satisfies the KL property [5, 6, 19], as below. Definition 2.3 (Kurdyka–Lojasiewicz property). The function E has the Kurdyka–Lojasiewicz (KL) property at a point z̄ with ∂E(z̄) ̸= ∅ if there exist a neighborhood U of z̄, a number η > 0, and a continuous concave function ϑ : [0, η) → R+ , with ϑ(0) = 0, ϑ ∈ C 1 (0, η), and ϑ′ > 0, such that ϑ′ (E(z) − E(z̄)) dist(0, ∂E(z)) ≥ 1 (10) whenever z ∈ U and E(z̄) < E(z) < E(z̄) + η. It has KL exponent θ ∈ [0, 1) at z̄ if ϑ can be chosen as ϑ(s) = c0 s1−θ for some c0 > 0. Equivalently, dist(0, ∂E(z)) ≥ c(E(z) − E(z̄))θ locally for some c > 0. Definition 2.4 (Relative smoothness [8]). For L ≥ 0, we say that f is L-smooth relative to ϕ on X ◦ if |f (y) − f (x) − ⟨∇f (x), y − x⟩| ≤ LDϕ (y, x) (x, y ∈ X ◦ ). (11) The Legendre kernel and relative smoothness yields the following well-posedness and standard one-step descent estimate. Proposition 2.5 (Well-definedness [4, Lemma 2]). If X is compact, then for every x ∈ X ◦ and α > 0, the mirror subproblem (3) has a unique minimizer Tα (x) ∈ X ◦ . Lemma 2.6. Let x ∈ X ◦ and x+ := Tα (x), and suppose that f is L-smooth relative to ϕ. Then   1 1 + − L Dϕ (x+ , x) + Dϕ (x, x+ ). (12) f (x) − f (x ) ≥ α α
Proof. By the optimality condition (4), ⟨∇f (x), x+ − x⟩ = −α−1 Dϕ (x+ , x) + Dϕ (x, x+ ) . Combining this equality with (11) yields (12).

3

Metric-Flattening Reparameterization

This section develops the general framework of a metric-flattening reparameterization with a definable boundary extension, used in Section 4 to establish convergence. The reparameterization removes the boundary degeneracy of [∇2 ϕ(x)]−1 .

7

To handle the degeneracy of [∇2 ϕ(x)]−1 at the boundary, metricP we use a coordinatewise Q flattening reparameterization for the separable kernel ϕ = i hi on C = i Ii . Throughout Sections 3 and 4, assume each hi is a C 3 Legendre kernel on Ii = (ℓi , ui ) with h′′i > 0. Consequently, h′i (t) → −∞ at a lower endpoint and h′i (t) → +∞ at a upper endpoint. We use the lowersemicontinuous extensions of hi and ϕ, with values in R ∪ {+∞}. Fix x̄i ∈ Ii and define the metric-flattening reparameterization Z xi q Si (xi ) := h′′i (u) du, S(x) := (S1 (x1 ), . . . , Sn (xn )). (13) x̄i

Each Si is a C 2 increasing diffeomorphism from Ii onto an open interval Ji . Thus S is a C 2 diffeomorphism from C onto S(C), with z = S(x) and x = S −1 (z). The Jacobian of S −1 at z ∈ S(C) is  n 1  D(S −1 )(z) = Diag  q (14) −1 ′′ hi (Si (zi )) i=1

Consequently, D(S −1 )(z)T ∇2 ϕ(S −1 (z))D(S −1 )(z) = In .

(15)

The reparameterized interior feasible set is N ◦ := S(X ◦ ) = {z ∈ S(C) : AS −1 (z) = b}. By (15), S transforms the Hessian metric into the identity on S(C). The transformed metric is defined on all of Rn and therefore remains nondegenerate as points of N ◦ approach its boundary. From a dynamical-systems perspective, the reparameterization converts mirror flow into a subgradient flow. Related connections between mirror flow and reparameterized gradient flow are also studied in [15, 22]. Suppose that S −1 extends continuously to N := cl N ◦ , and retain the same notation for this extension. Define the reparameterized objective E : Rn → (−∞, +∞] by ( f (S −1 (z)), z ∈ N , E(z) := (16) +∞, z∈ / N. For z ∈ N ◦ , the sets N and N ◦ agree locally. The full row rank of AD(S −1 )(z) and standard subdifferential calculus then give [32, Exercise 10.10] ∂E(z) = {D(S −1 )(z)T (∇f (S −1 (z)) + AT µ) : µ ∈ Rm }, dist(0, ∂E(z)) = minm D(S −1 )(z)T (∇f (S −1 (z)) + AT µ) . µ∈R

(17) (18)

Let x(t) ∈ X ◦ solve (5), set z(t) := S(x(t)), and let λ(t) be the corresponding multiplier. Since ẋ(t) = D(S −1 )(z(t))ż(t), multiplying (5) by D(S −1 )(z(t))T and using (15) give
0 = ż(t) + D(S −1 )(z(t))T ∇f (x(t)) + AT λ(t) . By (17), this identity yields ż(t) ∈ −∂E(z(t)).

(19)

Thus mirror flow becomes the Euclidean subgradient flow of E in the reparameterized variables. Under the KL property of E, this equivalence provides an intuition to proving finite length and convergence of the reparameterized trajectory and motivates the analogous analysis of mirror descent below. To recover limits in the original variables, S −1 must extend continuously to the boundary of N ◦ . The KL argument additionally requires compactness and definability. We formalize these conditions below. 8

Definition 3.1 (Definable boundary extension). Let S : C → S(C) be the metric-flattening reparameterization in (13). We say that S admits a definable boundary extension for X = cl C ∩ L if the following conditions hold in a common o-minimal structure: (i) N := cl N ◦ ⊂ Rn is compact and definable. (ii) The inverse restriction S −1 |N ◦ : N ◦ → X ◦ extends to a continuous definable map N → X. The compactness condition is not essential to the convergence argument and may be replaced by boundedness of the generated reparameterized sequence, which can often be obtained from descent and level-boundedness. For convienience of presentation, we retain the global compactness assumption in the main statements. The boundary-extension condition can be verified coordinatewise as follows. Lemma 3.2. For each i, let Ii := {xi : x ∈ X ◦ } and Ji := Si (Ii ). Suppose that, for every i, cl Ji is compact and Si−1 : Ji → Ii extends to a continuous definable map cl Ji → cl Ii . Then S admits a definable boundary extension in the sense of Definition 3.1. Moreover, ( ) n Y ◦ −1 N = cl S(X ) = z ∈ cl Ji : AS (z) = b . i=1

The extended inverse S −1 : N → X is continuous, definable, and onto. Proof. For each i, the map Si−1 : Ji → Ii is a strictly increasing bijection. Density and compactness show that its continuous extension is onto cl Ii , while strict monotonicity gives injectivity. It is therefore a homeomorphism from cl Ji to cl Ii , and its inverse continuously and definably extends Si . Taking products yields continuous definable extensions of S −1 and S, for which we retain the same notation. Since X ◦ = C ∩ L is semialgebraic and S is definable, S(X ◦ ) is definable. The closure of Q ◦ a definable set is definable, so N = cl S(X ) is a definable closed subset of the compact set i cl Ji . To verify Definition 3.1(ii), fix z ∈ N and choose xν ∈ X ◦ with S(xν ) → z. Continuity gives xν → S −1 (z), and hence S −1 (z) ∈ X because X is closed. Thus S −1 |N ◦ extends to a continuous definable map N → X, so S admits a continuous definable boundary extension. It remains to identify N . Set ( ) n Y −1 e := z ∈ N cl Ji : AS (z) = b . i=1

e follows from S −1 (N ) ⊆ X. Conversely, fix z ∈ N e and set x := S −1 (z). Then The inclusion N ⊆ N ◦ xi ∈ cl Ii ⊆ cl Ii and Ax = b, so x ∈ X. For any x b ∈ X , the points xt := (1 − t)x + tb x belong to X ◦ ◦ for t ∈ (0, 1], and continuity gives S(xt ) → S(x) = z. Hence z ∈ cl S(X ) = N . Applying the same argument to any x ∈ X gives S(x) ∈ N and S −1 (S(x)) = x, so the extended inverse is onto. The following proposition provides examples of definable boundary extensions. Proposition 3.3. The following settings P satisfy the definable boundary-extension condition. In each case, hi = h for every i, so ϕ(x) = ni=1 h(xi ). (i) Shannon entropy h(x) = x(log x − 1) on (0, ∞), with the compact feasible set X = {x ≥ 0 : Ax = b}. (ii) Fermi–Dirac entropy h(x) = x log x+(1−x) log(1−x) on (0, 1), with X = {x ∈ [0, 1]n : Ax = b}.

9

(iii) For any fixed p ∈ (1, 2), the power kernel h = hp , where hp (x) = −x2−p /[(p − 1)(2 − p)] on (0, ∞), with the compact feasible set X = {x ≥ 0 : Ax = b}. √ Proof. (i) Shannon entropy. We have h′′ (x) = 1/x, so S ′ (x) = 1/ x. Choosing the additive constant so that the continuous extension satisfies S(0) = 0 gives √ S(x) = 2 x,

S −1 (z) =

z2 . 4

Since X = cl X ◦ is compact, every cl Ii and cl Ji = Si (cl Ii ) is compact. Moreover, Si−1 (zi ) = zi2 /4 extends continuously and semialgebraically to cl Ji . Lemma 3.2 therefore yields a definable boundary extension, with N = {z ≥ 0 : A(z ⊙ z/4) = b}. p (ii) Fermi–Dirac entropy. Here h′′ (x) = 1/[x(1 − x)], so S ′ (x) = 1/ x(1 − x). The normalization S(0) = 0 gives √ S(x) = 2 arcsin x, S −1 (z) = sin2 (z/2). The set X = cl X ◦ is compact, and the continuous extension of Si satisfies cl Ji = Si (cl Ii ) ⊆ [0, π]. Thus cl Ji is compact. The inverse Si−1 (zi ) = sin2 (zi /2) extends continuously and definably to this interval. Lemma 3.2 therefore yields a definable boundary extension. With S −1 (z) = sin2 (z/2), we similarly obtain N = {z ∈ [0, π]n : A sin2 (z/2) = b}. (iii) Power kernels. Fix p ∈ (1, 2). Then h′′p (x) = x−p , and h′p (x) = −x1−p /(p − 1) → −∞ as x ↓ 0, so hp is a Legendre kernel on (0, ∞). Since Sp′ (x) = x−p/2 and 2 − p > 0, the normalization Sp (0) = 0 gives   2 − p 2/(2−p) 2 (2−p)/2 −1 x , Sp (z) = z . Sp (x) = 2−p 2 Since X = cl X ◦ is compact, every cl Ii is compact. The continuous extension of Sp satisfies cl Ji = Sp (cl Ii ), so cl Ji is compact. The inverse Sp−1 (z) = ((2 − p)z/2)2/(2−p) also extends continuously and definably to zero. Lemma 3.2 therefore yields a definable boundary extension. With Sp−1 (z) = ((2 − p)z/2)2/(2−p) , we similarly obtain ( N =

z≥0:A



2−p z 2

2/(2−p)

) =b .

All sets and maps above are definable in the common o-minimal expansion Ran,exp . Hence all conditions in Definition 3.1 hold in each case.

4

Reparameterized finite length and KKT convergence

This section establishes finite length in the reparameterized sequence and KKT convergence of mirror descent and mirror flow. We apply the KL inequality to the reparameterized objective to prove finite length and convergence of the reparameterized sequence or trajectory. Continuity of the extended inverse S −1 then recovers convergence in the original sequence. We first present the main results. We then derive the estimates required for the proof, using a conformal circuit decomposition to obtain the key boundary-uniform estimate for the Lagrangia gradient, and prove the theorem.

10

4.1

Main convergence results

Suppose that the metric-flattening reparameterization S in (13) admits a definable boundary extension in the sense of Definition 3.1, and retain the reparameterized objective E from (16). The analysis relies on the following curvature bound on (1/h′′i )′ :  ′ |h′′′ (t)| 1 κi := sup (t) = sup ′′i , κ := max κi . (20) ′′ 2 i hi t∈Ii t∈Ii (hi (t)) Now, we are ready to present the main result. Theorem 4.1 (Finite length of S(xk ) and KKT convergence). Let each hi be a C 3 Legendre kernel on Ii with h′′i > 0, and assume that κ < ∞ and that the associated metric-flattening reparameterization S admits a definable boundary extension. Let f be C 1 on a neighborhood of X and L-smooth relative to ϕ on X ◦ for some L ≥ 0, with f |X definable in the same o-minimal P structure. Starting from x0 ∈ X ◦ , let xk+1 := Tαk (xk ), where 0 < αk ≤ ᾱ < ∞, ᾱL < 1, and ∞ k=0 αk = ∞. Then ∞ X

∥S(xk+1 ) − S(xk )∥ < ∞.

(21)

k=0

Consequently, S(xk ) → z⋆ ∈ N and xk → x⋆ := S −1 (z⋆ ) ∈ X, with 0 ∈ ∂E(z⋆ ). Moreover, f (xk ) is nonincreasing and converges to f (x⋆ ). The limit x⋆ is a KKT point of (1). The proof is given in Section 4.2. It first derives the one-step estimates required for the reparameterized sequence and then applies the standard KL finite-length argument [3, 5]. Corollary 4.2 (Convergence rates). Under the assumptions of Theorem 4.1, suppose in addition that E has KL exponent Pk−1 θ ∈ [0, 1) at z⋆ . Then there exist an index k0 and a constant γ0 > 0 such that, with τk := j=k0 αj and rk := f (xk ) − f (x⋆ ), the following conclusions hold. (i) If 0 ≤ θ < 1/2, there exists k1 ≥ k0 such that xk = x⋆ for every k ≥ k1 . (ii) If θ = 1/2, there is M > 0 such that, for all sufficiently large k, rk ≤ M e−γ0 τk ,

∥S(xk ) − z⋆ ∥ ≤ M e−γ0 τk /2 .

(iii) If 1/2 < θ < 1, there is M > 0 such that, for all sufficiently large k, −1/(2θ−1)

rk ≤ M τk

,

−(1−θ)/(2θ−1)

∥S(xk ) − z⋆ ∥ ≤ M τk

.

The proof is given in Appendix A. For mirror flow, the same reparameterization yields finite length. Theorem 4.3. Suppose each hi is a C 3 Legendre kernel on Ii with h′′i > 0, and suppose that the associated metric-flattening reparameterization S admits a definable boundary extension. Let f be C 2 on a neighborhood of X, and suppose that f |X is definable in the same o-minimal structure. For every x(0) ∈ X ◦ , the mirror flow (5) has a unique global solution in X ◦ . Its reparameterized trajectory z(t) = S(x(t)) satisfies Z ∞

∥ż(t)∥ dt < ∞.

(22)

0

Consequently, z(t) → z⋆ ∈ N and x(t) → x⋆ = S −1 (z⋆ ) ∈ X, with 0 ∈ ∂E(z⋆ ). The limit x⋆ is a KKT point of (1). The proof is given in Appendix B. 11

4.2

Proof of Theorem 4.1

We first derive boundary-uniform one-step estimates and then use the KL property to prove finite length of the reparameterized sequence. 4.2.1

Conformal circuit decomposition

Direct bounds on the affine-constraint multipliers or the dual sequence {∇ϕ(xk )} are generally impossible near the boundary: the Legendre gradient blows up there, and the multipliers may diverge as well. We instead control the corresponding Lagrangian gradient by decomposing the mirror-step displacement into conformal circuits [25, 30], defined as follows. Definition 4.4 (A-circuit). Let a nonzero c ∈ ker A be an A-circuit if its support is inclusionminimal among supports of nonzero vectors in ker A (that is, no nonzero vector in ker A has support strictly contained in supp(c)). Two A-circuits are equivalent if they differ by a nonzero scalar factor, and the resulting finite set of equivalence classes is denoted by C(A). For u, v ∈ Rn , write u ⊑ v if ui vi ≥ 0,

ui = 0 whenever vi = 0.

(23)

We use the following conformal decomposition. Lemma 4.5 (Conformal circuit decomposition [25, 30]). Every nonzero w ∈ ker A can be written as w = c1 + · · · + cq ,

q ≤ min{dim ker A, |supp(w)|},

where each cℓ is an A-circuit and cℓ ⊑ w. To control the Lagrangian gradient uniformly, we introduce the following circuit seminorm, which is invariant under shifts in range(AT ). Definition 4.6 (Circuit seminorm). If ker A ̸= {0}, define ∥g∥cir(A) := max

|⟨c, g⟩|

[c]∈C(A) minj∈supp(c) |cj |

.

(24)

If ker A = {0}, set ∥g∥cir(A) := 0. Proposition 4.7. For g ∈ Rn , the quantity ∥g∥cir(A) is well defined. The map g 7→ ∥g∥cir(A) is a seminorm, and there exists χA < ∞, depending only on A, such that ∥g∥cir(A) ≤ χA ∥g∥∞ . Moreover, g + AT µ cir(A) = ∥g∥cir(A) for every µ ∈ Rm . Proof. The claim is immediate when ker A = {0}. Suppose that ker A = ̸ {0}. Each circuit class is determined by its support. Indeed, let u and v be circuits with the same support J, and fix j ∈ J. If u and v are not scalar multiples, then vj u − uj v ̸= 0 and A(vj u − uj v) = 0,

[vj u − uj v]j = 0.

Thus vj u − uj v is a nonzero vector in ker A with supp(vj u − uj v) ⊆ J \ {j}, contradicting the support minimality of u. Hence two circuits with the same support are scalar multiples and belong to the same circuit class. Since [n] has only finitely many subsets, C(A) is finite. The quotient in (24) and the ratio below are invariant under rescaling c 7→ τ c, where τ = ̸ 0. Set ∥c∥1

χA := max

[c]∈C(A) minj∈supp(c) |cj |

12

which is finite because C(A) is finite. Hölder’s inequality and (24) give the bound. Nonnegativity, absolute homogeneity, and the triangle inequality follow from (24) directly, so this quantity is a seminorm. Finally, Ac = 0 gives c, g + AT µ = ⟨c, g⟩ for every circuit c, which proves the shift invariance. The following lemma provides a uniform bound on the Lagrangian gradient, even as the sequence approaches the boundary. P Lemma 4.8. For ϕ(x) = ni=1 hi (xi ) with h′′i > 0 on Ii , let x, x+ ∈ X ◦ , g ∈ Rn , and α > 0 satisfy ∇ϕ(x+ ) − ∇ϕ(x) + α(g + AT λ) = 0 for some λ ∈ Rm . Then g + AT λ

∞

≤ ∥g∥cir(A) .

(25)

Proof. For every i ∈ [n],
α gi + [AT λ]i = h′i (xi ) − h′i (x+ i ). T Since h′i is strictly increasing, xi − x+ i and gi + [A λ]i have the same sign and vanish simultaneously. + T If ker A = {0}, then x = x and hence g + A λ = 0. Hence (25) follows from Definition 4.6. ̸ 0, so j ∈ supp(x − x+ ). Lemma 4.5 applied to Suppose ker A ̸= {0}. Fix j with gj + [AT λ]j = + + x − x yields a circuit c ⊑ x − x with j ∈ supp(c). Thus ci (gi + [AT λ]i ) = |ci | gi + [AT λ]i for every i. Since Ac = 0, D E X T T |ci | gi + [A λ]i = c, g + A λ = ⟨c, g⟩ . i

Therefore gj + [AT λ]j ≤

|⟨c, g⟩| |⟨c, g⟩| ≤ ≤ ∥g∥cir(A) . |cj | minℓ∈supp(c) |cℓ |

Taking the maximum over j proves (25). 4.2.2

Sufficient decrease and relative-error bounds

Based on the preceding conformal circuit decomposition, we now establish the one-step estimates required by the finite-length argument under KL property. We first give a lower bound on the symmetric Bregman divergence in terms of the reparameterized displacement. Proposition 4.9. For every x, y ∈ C, Dϕ (y, x) + Dϕ (x, y) ≥ ∥S(y) − S(x)∥2 . Proof. For one coordinate, let Dh (v, u) := h(v) − h(u) − h′ (u)(v − u). Then Z v Dh (v, u) + Dh (u, v) =

′′

 Z v

h (t) dt u

 1 dt

u

Z v 2 p ′′ h (t) dt = (S(v) − S(u))2 . ≥ u

Summation over the coordinates proves (26). The following lemma transfers the circuit estimate to the reparameterized variables.

13

(26)

Lemma 4.10. Fix i ∈ [n] with κi < ∞. Suppose xi , x+ i ∈ Ii and si ∈ R satisfy

Then

′ h′i (x+ i ) = hi (xi ) − si .

(27)

|si | |s | κi |si |/2 p ′′i e−κi |si |/2 p ′′ ≤ Si (x+ . i ) − Si (xi ) ≤ e hi (xi ) hi (xi )

(28)

Proof. For u between xi and x+ i , (20) gives
′ |h′′′ (u)| |h′′′ (u)| ≤ κi h′′i (u). = h′′i (u) ′′i log h′′i (u) = i′′ hi (u) (hi (u))2 Integrating from xi to u yields log Hence

h′′i (u) ≤ κi h′i (u) − h′i (xi ) ≤ κi |si | . h′′i (xi ) e−κi |si |/2 1 eκi |si |/2 p ′′ ≤ p ′′ ≤ p ′′ . hi (xi ) hi (u) hi (xi )

Since h′′i (u) > 0, multiplying the preceding bound by h′′i (u) gives h′′ (u) e−κi |si |/2 ′′ p ′′ hi (u) ≤ p i ′′ = hi (xi ) hi (u)

q eκi |si |/2 ′′ h′′i (u) ≤ p ′′ hi (u). hi (xi ) p ′′ ′ Integrating this inequality in u from xi to x+ hi and (h′i )′ = h′′i , gives i , and using Si = eκi |si |/2 e−κi |si |/2 ′ + ′ p ′′ p ≤ ) − S (x ) hi (xi ) − h′i (xi ) ≤ Si (x+ h′i (x+ i i i i ) − hi (xi ) . hi (xi ) h′′i (xi ) Using (27) proves (28). Proposition 4.11. Suppose that the metric-flattening reparameterization S in (13) admits a definable boundary extension. Let f ∈ C 1 on a neighborhood of X, define E by (16), and set Γ := maxx∈X ∥∇f (x)∥cir(A) < ∞. Assume κ < ∞, fix ᾱ < ∞, and let x+ := Tα (x) for x ∈ X ◦ and 0 < α ≤ ᾱ. With z := S(x) and z + := S(x+ ), dist(0, ∂E(z)) ≤

exp(κᾱΓ/2) + z −z . α

(29)

The constant is universal and independent of the distance to the boundary. Proof. Note that z ∈ N ◦ and x = S −1 (z). Let λ be the mirror-step multiplier and write r := ′ ∇f (x) + AT λ. Lemma 4.8 and (4) give |ri | ≤ Γ and h′i (x+ i ) = hi (xi ) − αri , respectively. Applying Lemma 4.10 with si = αri gives |r | 1 exp(κᾱΓ/2) + p ′′i ≤ eκi α|ri |/2 zi+ − zi ≤ zi − z i . α α hi (xi ) Taking µ = λ in (18) and using (14) yields dist(0, ∂E(z)) ≤ D(S −1 )(z)r ≤ This proves (29). 14

exp(κᾱΓ/2) + z −z . α

4.2.3

Finite length of the reparameterized sequence

The following lemma follows the standard KL finite-length argument, with the difference that the stepsizes may vary and need not be bounded away from zero. Lemma 4.12. Let K ⊂ Rd be compact and E : Rd → R ∪ {+∞} be proper, P∞lower-semicontinuous, continuous on K = dom E, and KL. Suppose zk ∈ K, 0 < αk ≤ ᾱ < ∞, k=0 αk = ∞, and there are c1 , c2 > 0 such that c1 ∥zk+1 − zk ∥2 , αk c2 dist(0, ∂E(zk )) ≤ ∥zk+1 − zk ∥ . αk

E(zk ) − E(zk+1 ) ≥

Then

(30) (31)

P∞

k=0 ∥zk+1 − zk ∥ < ∞. Consequently, zk converges to a point z⋆ ∈ K and 0 ∈ ∂E(z⋆ ).

Proof. P Since E is continuous on the compact set K, it is Pbounded below. (30) gives E(zk ) ↓ E∞ ∈ R and k ∥zk+1 − zk ∥2 /αk < ∞. Since αk ≤ ᾱ, also k ∥zk+1 − zk ∥2 < ∞ and ∥zk+1 − zk ∥ → 0. Combining the two estimates gives E(zk ) − E(zk+1 ) ≥

c1 αk dist(0, ∂E(zk ))2 . c22

(32)

P P Thus k αk dist(0, ∂E(zk ))2 < ∞. Since k αk = ∞, this implies lim inf k dist(0, ∂E(zk )) = 0. Thus there is a subsequence zkj → z⋆ ∈ K such that dist(0, ∂E(zkj )) → 0. (31) ensures that ∂E(zk ) ̸= ∅ for every k. Since each limiting subdifferential is closed in finite dimensions, choose vkj ∈ ∂E(zkj ) with vkj = dist(0, ∂E(zkj )). Thus zkj → z⋆ , vkj → 0, and continuity on K gives E(zkj ) → E(z⋆ ) = E∞ . Sequential closedness of the limiting subdifferential yields 0 ∈ ∂E(z⋆ ). If E(zk ) = E∞ for some k, monotonicity gives E(zℓ ) = E∞ for every ℓ ≥ k. Equation (30) then gives zℓ+1 = zℓ , and (31) gives 0 ∈ ∂E(zk ). Thus the conclusion of the lemma already holds. Now assume E(zk ) > E∞ for every k. Apply the KL inequality at z⋆ , with neighborhood U , width η, and desingularizing function ϑ. Whenever zk ∈ U and 0 < E(zk ) − E∞ < η, 1 ≤ ϑ′ (E(zk ) − E∞ ) dist(0, ∂E(zk )) ≤

c2 ′ ϑ (E(zk ) − E∞ ) ∥zk+1 − zk ∥ . αk

(33)

Concavity of ϑ, (30), and (33) give ϑ(E(zk ) − E∞ ) − ϑ(E(zk+1 ) − E∞ ) ≥ ϑ′ (E(zk ) − E∞ )(E(zk ) − E(zk+1 )) c1 ′ c1 ϑ (E(zk ) − E∞ ) ∥zk+1 − zk ∥2 ≥ ∥zk+1 − zk ∥ . ≥ αk c2

(34)

Choose ρ > 0 such that B(z⋆ , 2ρ) ∩ K ⊂ U . For sufficiently large j, zkj − z⋆ < ρ,

0 < E(zkj ) − E∞ < η,

c2 ϑ(E(zkj ) − E∞ ) < ρ. c1

Suppose that N > kj is the first index for which zN ∈ / B(z⋆ , 2ρ). Monotonicity gives E(zk ) − E∞ < η P −1 for k = kj , . . . , N − 1, so (34) is valid for k = kj , . . . , N − 1. Therefore N k=kj ∥zk+1 − zk ∥ ≤

15

P −1 (c2 /c1 )ϑ(E(zkj ) − E∞ ) < ρ, which implies ∥zN − z⋆ ∥ ≤ zkj − z⋆ + N k=kj ∥zk+1 − zk ∥ < 2ρ, a contradiction. Thus the tail remains in the KL neighborhood. For every k ≥ kj , summing (34) gives ∞ X

∥zℓ+1 − zℓ ∥ ≤

ℓ=k

Thus z⋆ .

c2 ϑ(E(zk ) − E∞ ). c1

P∞

k=0 ∥zk+1 − zk ∥ < ∞, so zk is Cauchy. Since z⋆ is a cluster point, the sequence converges to

We are now ready to prove Theorem 4.1. Proof of Theorem 4.1. The boundary-extension condition makes X = S −1 (N ) compact. Proposition 2.5 therefore ensures that every mirror step is well defined and belongs to X ◦ . For zk := S(xk ), zk ∈ N = dom E and E(zk ) = f (xk ). By definition, E is proper and lower semicontinuous, continuous on N , and KL by definability. For every k, Lemma 2.6 and Proposition 4.9 give E(zk ) − E(zk+1 ) ≥

1 − αk L 1 − ᾱL ∥zk+1 − zk ∥2 ≥ ∥zk+1 − zk ∥2 . αk αk

Thus (30) holds with c1 = 1 − ᾱL, while Proposition 4.11 gives (31) with c2 = exp(κᾱΓ/2). Lemma 4.12 then gives (21), zk → z⋆ , and 0 ∈ ∂E(z⋆ ). Continuity of S −1 and E|N gives xk → x⋆ and f (xk ) → f (x⋆ ). Moreover, E(zk ) − E(zk+1 ) ≥ 0 for ᾱL < 1, so f (xk+1 ) ≤ f (xk ) for every k. Since {xk } is convergent, [15, Proposition 5.3] implies that x⋆ is a KKT point of (1).

5

Applications

In this section, we apply Theorem 4.1 to mirror descent with Shannon entropy, Fermi–Dirac entropy, and power kernels, and use Corollary 4.2 to derive rates for the original sequence {xk }.

5.1

Shannon entropy

Consider C := Rn++ and X := {x ∈ Rn+ : Ax = b}, with P X ◦ = X ∩ Rn++ ̸= ∅, and assume that X is compact. The Shannon entropy kernel is ϕ(x) := ni=1 xi (log xi − 1) for x ∈ Rn++ . The √ metric-flattening reparameterization and its inverse are S(x) = 2 x and S −1 (z) = z ⊙ z/4, where the square root is applied coordinatewise. Corollary 5.1. Let f be C 1 on a neighborhood of X, suppose that f |X is definable, and suppose ◦ that f is L-smooth relative to ϕ on x0 ∈ X ◦ , let xk+1 := Tαk (xk ), where P∞X . Starting from P √ √ ∞ 0 < αk ≤ ᾱ < ∞, ᾱL < 1, and k=0 αk = ∞. Then k=0 2 xk+1 − 2 xk < ∞, and xk converges to a KKT point x⋆ of f on X. Proof. Proposition 3.3 gives the semialgebraic boundary extension N = cl S(X ◦ ) = {z ∈ Rn+ : A(z ⊙ z/4) = b}. For h(x) = x(log x − 1), |h′′′ (x)| /(h′′ (x))2 = 1, so κ = 1. Compactness of X makes the circuit quantity Γ finite. Theorem 4.1 yields finite length and KKT convergence.

16

Corollary 5.2. Under the assumptions of Corollary 5.1, suppose that E has KL exponent θ ∈ [0, 1) √ at the limit z⋆ = 2 x⋆ . Let I0 := {i : x⋆,i = 0} and I+ := [n] \ I0 . If 0 ≤ θ < 1/2, then xk = x⋆ for P all sufficiently large k. If 1/2 ≤ θ < 1, take k0 and γ0 as in Corollary 4.2, let τk := k−1 j=k0 αj , and define ( exp(−γ0 τk /2), θ = 1/2, ρk := −(1−θ)/(2θ−1) τk , 1/2 < θ < 1, Then (xk − x⋆ )I+ = O(ρk ),

∥(xk )I0 ∥ = O(ρ2k ),

∥xk − x⋆ ∥ = O(ρk ).

Proof. Corollary 4.2 gives ∥zk − z⋆ ∥ = O(ρk ). Note that xk,i −x⋆,i = z⋆,i (zk,i −z⋆,i )/2+(zk,i −z⋆,i )2 /4. If i ∈ I+ , then z⋆,i > 0 and |xk,i − x⋆,i | = O(ρk ). If i ∈ I0 , then z⋆,i = 0 and xk,i = (zk,i − z⋆,i )2 /4 = O(ρ2k ). Since both coordinate sets are finite, these estimates give the claimed bounds. The coordinatewise rates for the original iterates can differ according to whether a coordinate converges to the boundary or remains in the interior. In this case, a coordinate converging to the boundary has the faster O(ρ2k ) rate, whereas a coordinate converging to a positive limit has rate O(ρk ). When αk ≡ α > 0 and 1/2 < θ < 1, these rates become O(k −2(1−θ)/(2θ−1) ) at the boundary and O(k −(1−θ)/(2θ−1) ) in the interior.

5.2

Fermi–Dirac entropy

Consider C := (0, 1)n and P X := {x ∈ [0, 1]n : Ax = b}, with X ◦ = X ∩ (0, 1)n ̸= ∅. The Fermi–Dirac entropy kernel is ϕ(x) := ni=1 [xi log xi + (1 − xi ) log(1 − xi )] for x ∈ (0, 1)n . Its metric-flattening √ reparameterization and inverse are S(x) = 2 arcsin x and S −1 (z) = sin2 (z/2), respectively, with all scalar operations applied coordinatewise. Corollary 5.3. Let f be C 1 on a neighborhood of X, suppose that f |X is definable, and suppose ◦ that f is L-smooth relative to ϕ on x0 ∈ X ◦ , let xk+1 := Tαk (xk ), where P∞X . Starting from P 0 < αk ≤ ᾱ < ∞, ᾱL < 1, and k=0 αk = ∞. Then ∞ k=0 ∥S(xk+1 ) − S(xk )∥ < ∞, and xk converges to a KKT point x⋆ of f on X. Proof. Proposition 3.3 gives the definable boundary extension N = cl S(X ◦ ) = {z ∈ [0, π]n : A sin2 (z/2) = b}. For the Fermi–Dirac kernel, |h′′′ (x)| /(h′′ (x))2 = |1 − 2x| ≤ 1. Compactness of X makes the circuit quantity Γ finite. The conclusion follows from Theorem 4.1. As in the Shannon-entropy case, the coordinatewise rates for the original iterates differ according to whether the limiting coordinate lies on the boundary or in the interior. Corollary 5.4. Under the assumptions of Corollary 5.3, suppose that E has KL exponent θ ∈ [0, 1) at the limit z⋆ = S(x⋆ ). Let I0 := {i : x⋆,i = 0}, I1 := {i : x⋆,i = 1}, and Iint := [n] \ (I0 ∪ I1 ). If 0 ≤ θ < 1/2, then xP k = x⋆ for all sufficiently large k. If 1/2 ≤ θ < 1, take k0 and γ0 as in Corollary 4.2, let τk := k−1 j=k0 αj , and define ( exp(−γ0 τk /2), θ = 1/2, ρk := −(1−θ)/(2θ−1) τk , 1/2 < θ < 1, Then ∥(xk − x⋆ )Iint ∥ = O(ρk ),

∥(xk )I0 ∥ = O(ρ2k ),

and ∥xk − x⋆ ∥ = O(ρk ). 17

∥(1n − xk )I1 ∥ = O(ρ2k ),

Proof. Corollary 4.2 gives ∥S(xk ) − z⋆ ∥ = O(ρk ). For each i ∈ [n],     zk,i + z⋆,i zk,i − z⋆,i xk,i − x⋆,i = sin sin . 2 2 If i ∈ Iint , this identity gives |xk,i − x⋆,i | = O(ρk ). If i ∈ I0 , then z⋆,i = 0 and xk,i = sin2 ((zk,i − z⋆,i )/2) = O(ρ2k ). If i ∈ I1 , then z⋆,i = π and 1 − xk,i = sin2 ((zk,i − z⋆,i )/2) = O(ρ2k ). The coordinatewise estimates give the final bound.

5.3

Power kernels

Consider C := Rn++ and X := {x ∈ Rn+ : Ax = b}, with X ◦ = X ∩ Rn++ = ̸ ∅, and assume that X is compact. For 1 ≤ p < 2, define hp , up to an affine term, by h′′p (x) = x−p on (0, ∞), with the closed extension normalized by hp (0) = 0 and hp (x) = +∞ for x < 0. Since h′′p > 0 and h′p (x) → −∞ as x ↓ 0, hp is a Legendre kernel on (0, ∞). The metric-flattening reparameterization and its inverse are 2 ′′ p−1 . Sp (x) = x1−p/2 /(1 − p/2) and Sp−1 (z) = ((1 − p/2)z)1/(1−p/2) . Moreover, h′′′ p (x) /(hp (x)) = px Because p < 2, both maps extend continuously to zero P and are definable in Ran,exp . For each i ∈ [n], fix pi ∈ [1, 2), and set ϕ(x) := ni=1 hpi (xi ) and S(x) := (Spi (xi ))ni=1 . Corollary 5.5. Let f be C 1 on a neighborhood of X, suppose that f |X is definable in Ran,exp , ◦ ◦ and suppose that f is L-smooth relative P∞to ϕ on X . Starting P∞from x0 ∈ X , let xk+1 := Tαk (xk ), where 0 < αk ≤ ᾱ < ∞, ᾱL < 1, and k=0 αk = ∞. Then k=0 ∥S(xk+1 ) − S(xk )∥ < ∞, and xk converges to a KKT point of f on X. Proof. The coordinate inverses above and Lemma 3.2 give the definable boundary extension N = cl S(X ◦ ) = {z ≥ 0 : AS −1 (z) = b}. ′′ 2 pi −1 , so compactness of X gives κ < ∞. Compactness of X also For hpi , h′′′ pi (x) /(hpi (x)) = pi x makes the circuit quantity Γ finite. Theorem 4.1 yields finite length and KKT convergence.

Corollary 5.6. Under the assumptions of Corollary 5.5, suppose that E has KL exponent θ ∈ [0, 1) at the limit z⋆ = S(x⋆ ). Let I0 := {i : x⋆,i = 0}, I+ := [n] \ I0 , and qi := 2/(2 − pi ) ≥ 2. If 0 ≤ θ < 1/2, then xk P = x⋆ for all sufficiently large k. If 1/2 ≤ θ < 1, take k0 and γ0 as in Corollary 4.2, let τk := k−1 j=k0 αj , and define ρk :=

( exp(−γ0 τk /2), −(1−θ)/(2θ−1) τk ,

θ = 1/2, 1/2 < θ < 1,

Then (xk − x⋆ )I+ = O(ρk ),

xk,i = O(ρqki )

(i ∈ I0 ),

∥xk − x⋆ ∥ = O(ρk ).

Proof. Corollary 4.2 gives ∥S(xk ) − z⋆ ∥ = O(ρk ). For each i ∈ [n],
q
q xk,i − x⋆,i = (1 − pi /2)Spi (xk,i ) i − (1 − pi /2)z⋆,i i . If i ∈ I+ , then z⋆,i > 0, and the mean-value theorem gives |xk,i − x⋆,i | = O(ρk ). If i ∈ I0 , then z⋆,i = 0 and xk,i = ((1 − pi /2)(Spi (xk,i ) − z⋆,i ))qi = O(ρqki ). This prove the bounds.

18

6

Conclusion

We prove that mirror descent converges to a KKT point for the nonconvex problem without excluding boundary limits. The result holds under verifiable conditions that jointly couple the objective, the Legendre kernel, and the feasible geometry. The key ingredient to establish the convergence is a metric-flattening reparameterization S that admits a definable boundary extension. Applying the KL argument to the reparameterized objective yields convergence of S(xk ). Continuity of S −1 then recovers convergence to the KKT point of the original sequence. We further apply our general framework to some concrete examples: Shannon entropy, Fermi–Dirac entropy, and power kernels. Future work may consider more general constraint geometries and genuinely nonseparable kernels, and extend mirror descent to broader Bregman-type methods, e.g. Bregman proximal point algorithms and Bregman ADMM, and their inexact variants. Acknowledgments. The authors used OpenAI’s ChatGPT for brainstorming and exploration during the early stages of this work and for language editing.

A

Proof of Corollary 4.2

Along the mirror descent sequence, rk = E(zk ) − E(z⋆ ). The KL-exponent condition and (32) give an index k0 and a constant γ0 > 0 such that rk − rk+1 ≥ γ0 αk rk2θ

(k ≥ k0 ).

(35)

If rk = 0 at some index, (30) gives zℓ = zk = z⋆ and rℓ = 0 for every ℓ ≥ k. Now assume rk > 0 on the tail under consideration. If 0 ≤ θ < 1/2, then Z rk 1−2θ 1−2θ rk − rk+1 = (1 − 2θ) t−2θ dt ≥ (1 − 2θ)rk−2θ (rk − rk+1 ) ≥ (1 − 2θ)γ0 αk . rk+1

P If rk remained positive, summation would contradict k αk = ∞. Hence rk1 = 0 for some k1 ≥ k0 . Since E(zk ) ↓ E(z⋆ ), (30) gives zk = zk1 for every k ≥ k1 . Passing to the limit yields zk1 = z⋆ , and therefore xk = x⋆ for every k ≥ k1 . If θ = 1/2, then rk+1 ≤ (1 − γ0 αk )rk . Since rk+1 > 0, this inequality gives 0 < 1 − γ0 αk < 1. Hence rk ≤ rk0

k−1 Y

(1 − γ0 αj ) ≤ rk0 exp(−γ0 τk ).

j=k0

If 1/2 < θ < 1, then (35) gives −(2θ−1)

rk+1

−(2θ−1)

− rk

Z rk = (2θ − 1) rk+1

t−2θ dt ≥ (2θ − 1)rk−2θ (rk − rk+1 ) ≥ (2θ − 1)γ0 αk .

Summation yields rk ≤



−1/(2θ−1) −(2θ−1) . rk0 + (2θ − 1)γ0 τk

The tail estimate in the proof of Lemma 4.12 gives, for all sufficiently large k, X ∥zk − z⋆ ∥ ≤ ∥zj+1 − zj ∥ ≤ M ϑ(rk ) = M rk1−θ , j≥k

and substituting the preceding energy bounds yields the corresponding rates for ∥S(xk ) − z⋆ ∥ in Corollary 4.2. 19

B

Proof of Theorem 4.3

Fix x(0) ∈ X ◦ . By [15, Proposition 3.1], (5) admits a global solution that remains in X ◦ . Since the induced vector field is locally C 1 , this solution is unique. For z(t) = S(x(t)), (19) gives −ż(t) ∈ ∂E(z(t)). To show that this is the minimum-norm element, let λ(t) be the multiplier in (5). Since ẋ(t) = D(S −1 )(z(t))ż(t), the Hessian form of (5) and (15) give
0 = D(S −1 )(z(t))T ∇2 ϕ(x(t))ẋ(t) + ∇f (x(t)) + AT λ(t)
= D(S −1 )(z(t))T ∇2 ϕ(S −1 (z(t)))D(S −1 )(z(t))ż(t) + D(S −1 )(z(t))T ∇f (x(t)) + AT λ(t)
= ż(t) + D(S −1 )(z(t))T ∇f (x(t)) + AT λ(t) . Therefore,
−ż(t) = D(S −1 )(z(t))T ∇f (x(t)) + AT λ(t) . Now fix v ∈ ∂E(z(t)). By (17), there exists η ∈ Rm such that
v = D(S −1 )(z(t))T ∇f (x(t)) + AT η . Subtracting the two equalities and setting µ = η − λ(t) yield v + ż(t) = D(S −1 )(z(t))T AT µ,

v = −ż(t) + D(S −1 )(z(t))T AT µ.

Differentiating AS −1 (z(t)) = b gives ⟨ż(t), v + ż(t)⟩ = AD(S −1 )(z(t))ż(t), µ = 0. Consequently, ∥v∥2 = ∥ż(t)∥2 + ∥v + ż(t)∥2 Thus −ż(t) is the minimum-norm element of ∂E(z(t)). By (17), ∇(f ◦ S −1 )(z(t)) ∈ ∂E(z(t)). Hence dist(0, ∂E(z(t))) = ∥ż(t)∥ , (36) d E(z(t)) = ∇(f ◦ S −1 )(z(t)), ż(t) = − ∥ż(t)∥2 . (37) dt R∞ Thus E(z(t)) ↓ E∞ ∈ R and 0 ∥ż(t)∥2 dt < ∞. Hence there exist times tj → ∞ such that ż(tj ) → 0. By compactness of N , after passing to a subsequence, z(tj ) → z⋆ ∈ N . Continuity of E on N and closedness of ∂E yield E(z⋆ ) = E∞ and 0 ∈ ∂E(z⋆ ). If E(z(t0 )) = E∞ for some t0 < ∞, monotonicity gives E(z(t)) = E∞ for every t ≥ t0 , and (36) implies ż(t) = 0 on [t0 , ∞). Now consider the case E(z(t)) > E∞ for every t ≥ 0. Let U , η, and ϑ be the neighborhood, width, and desingularizing function in the KL inequality at z⋆ . Whenever z(t) ∈ U and 0 < E(z(t)) − E∞ < η, (10) and (36) give −

d ϑ(E(z(t)) − E∞ ) = ϑ′ (E(z(t)) − E∞ ) ∥ż(t)∥2 ≥ ∥ż(t)∥ . dt

(38)

Choose ρ > 0 such that {z : ∥z − z⋆ ∥ ≤ 2ρ} ⊂ U . Since z(tj ) → z⋆ , E(z(tj )) → E∞ , and ϑ(s) → 0 as s ↓ 0, fix j such that ∥z(tj ) − z⋆ ∥ < ρ,

0 < E(z(tj )) − E∞ < η,

20

ϑ(E(z(tj )) − E∞ ) < ρ.

Define T := inf{t ≥ tj : ∥z(t) − z⋆ ∥ ≥ 2ρ}, with T = ∞ if the set is empty. For every s ∈ [tj , T ), monotonicity and (38) give Z s ∥ż(t)∥ dt ≤ ϑ(E(z(tj )) − E∞ ) − ϑ(E(z(s)) − E∞ ) < ρ. tj

Consequently,

Z s ∥z(s) − z⋆ ∥ ≤ ∥z(tj ) − z⋆ ∥ +

∥ż(t)∥ dt < 2ρ. tj

If T < ∞, letting s ↑ T contradicts the definition of T . Therefore T = ∞. Letting s → ∞ in the integral estimate gives Z ∞

∥ż(t)∥ dt ≤ ϑ(E(z(tj )) − E∞ ) < ∞, tj

which proves (22). Hence z(t) converges. Since z(tj ) → z⋆ , its limit is z⋆ . Continuity of S −1 then gives x(t) → x⋆ . It remains to prove that x⋆ is a KKT point. Integrating (5) and dividing by t > 0 give ∇ϕ(x(t)) − ∇ϕ(x(0)) 1 + t t

Z t

∇f (x(s)) ds ∈ range(AT ).

(39)

0

Since x(t) → x⋆ , the integral average converges to ∇f (x⋆ ). The constraint qualification C ∩ L ̸= ∅ implies NC (x⋆ ) ∩ range(AT ) = {0}. (40) Indeed, choose x b ∈ C ∩ L. If v = AT µ ∈ NC (x⋆ ), then ⟨v, x b − x⋆ ⟩ = 0. For all sufficiently small ε > 0, normality applied to x b + εv ∈ C gives 0 ≥ ⟨v, x b + εv − x⋆ ⟩ = ε ∥v∥2 , proving (40). If x⋆ ∈ C, the first term in (39) tends to zero, so ∇f (x⋆ ) ∈ range(AT ) and the KKT condition follows. Suppose that x⋆ lies on the boundary. Essential smoothness gives ∥∇ϕ(x(t))∥ → ∞. For all sufficiently large t, let ν(t) :=

∇ϕ(x(t)) − ∇ϕ(x(0)) , ∥∇ϕ(x(t))∥

γ(t) :=

∥∇ϕ(x(t))∥ . t

Every cluster point of ν(t) is a unit vector in NC (x⋆ ) [1, Lemma 4.1]. The function γ is bounded. Otherwise, along some tj → ∞, one would have γ(tj ) → ∞ and ν(tj ) → v. Dividing (39) by γ(tj ) would give v ∈ range(AT ), contradicting (40). Choose tj → ∞. After passing to a subsequence, let γ(tj ) → γ⋆ ≥ 0 and ν(tj ) → v. Then γ⋆ v ∈ NC (x⋆ ). Passing to the limit in (39) gives γ⋆ v + ∇f (x⋆ ) ∈ range(AT ). Hence there exists λ⋆ ∈ Rm such that 0 ∈ ∇f (x⋆ ) + AT λ⋆ + NC (x⋆ ), which is (2).

References [1] Felipe Alvarez, Jérôme Bolte, and Olivier Brahic. Hessian Riemannian gradient flows in convex programming. SIAM Journal on Control and Optimization, 43(2):477–501, 2004. [2] Hedy Attouch, Jérôme Bolte, Patrick Redont, and Marc Teboulle. Singular Riemannian barrier methods and gradient-projection dynamical systems for constrained optimization. Optimization, 53(5–6):435–454, 2004. [3] Hedy Attouch, Jérôme Bolte, and Benar Fux Svaiter. Convergence of descent methods for semi-algebraic and tame problems: Proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods. Mathematical Programming, 137:91–129, 2013.

21

[4] Heinz H. Bauschke, Jérôme Bolte, and Marc Teboulle. A descent lemma beyond Lipschitz gradient continuity: First-order methods revisited and applications. Mathematics of Operations Research, 42(2):330–348, 2017. [5] Jérôme Bolte, Aris Daniilidis, and Adrian S. Lewis. The Lojasiewicz inequality for nonsmooth subanalytic functions with applications to subgradient dynamical systems. SIAM Journal on Optimization, 17(4):1205–1223, 2007. [6] Jérôme Bolte, Aris Daniilidis, Adrian S. Lewis, and Masahiro Shiota. Clarke subgradients of stratifiable functions. SIAM Journal on Optimization, 18(2):556–572, 2007. [7] Jérôme Bolte and Edouard Pauwels. Curiosities and counterexamples in smooth convex optimization. Mathematical Programming, 195(1–2):553–603, 2022. [8] Jérôme Bolte, Shoham Sabach, Marc Teboulle, and Yakov Vaisbourd. First order methods beyond convexity and Lipschitz gradient continuity with applications to quadratic inverse problems. SIAM Journal on Optimization, 28(3):2131–2151, 2018. [9] Immanuel M. Bomze, Panayotis Mertikopoulos, Werner Schachinger, and Mathias Staudigl. Hessian barrier algorithms for linearly constrained optimization problems. SIAM Journal on Optimization, 29(3):2100–2127, 2019. [10] He Chen, Jiajin Li, and Anthony Man-Cho So. Spurious stationarity and hardness results for Bregman proximaltype algorithms. arXiv preprint arXiv:2404.08073, 2026. [11] He Chen and Anthony Man-Cho So. On the iterate convergence of Bregman projected gradient method. arXiv preprint arXiv:2608.05035, 2026. [12] Marco Cuturi. Sinkhorn distances: Lightspeed computation of optimal transport. In Advances in Neural Information Processing Systems 26, 2013. [13] Cong D. Dang and Guanghui Lan. On the convergence properties of non-Euclidean extragradient methods for variational inequalities with generalized monotone operators. Computational Optimization and Applications, 60(2):277–310, 2015. [14] Kuangyu Ding, Jingyang Li, and Kim-Chuan Toh. Nonconvex stochastic Bregman proximal gradient method with application to deep learning. Journal of Machine Learning Research, 26(39):1–44, 2025. [15] Kuangyu Ding and Kim-Chuan Toh. On exploration of an interior mirror descent flow for stochastic nonconvex constrained problem. arXiv preprint arXiv:2507.15264v3, 2025. [16] Kuangyu Ding and Kim-Chuan Toh. Non-KKT accumulation in entropic mirror descent, 2026. arXiv:2608.01658. [17] Thinh T. Doan, Subhonmesh Bose, D. Hoa Nguyen, and Carolyn L. Beck. Convergence of the iterates in mirror descent methods. IEEE Control Systems Letters, 3(1):114–119, 2019. [18] Yuehaw Khoo, Tianyun Tang, and Kim-Chuan Toh. A bregman ADMM for Bethe variational problem. arXiv preprint arXiv:2502.04613, 2025. [19] Krzysztof Kurdyka. On gradients of functions definable in O-minimal structures. Annales de l’Institut Fourier, 48(3):769–783, 1998. [20] Jiajin Li, Jianheng Tang, Lemin Kong, Huikang Liu, Jia Li, Anthony Man-Cho So, and Jose Blanchet. A convergent single-loop algorithm for relaxation of Gromov–Wasserstein in graph data. In International Conference on Learning Representations, 2023. [21] Yen-Huan Li and Volkan Cevher. Convergence of the exponentiated gradient method with Armijo line search. Journal of Optimization Theory and Applications, 181(2):588–607, 2019. [22] Zhiyuan Li, Tianhao Wang, Jason D. Lee, and Sanjeev Arora. Implicit bias of gradient descent on reparametrized models: On equivalence to mirror descent. In Advances in Neural Information Processing Systems 35, pages 34626–34640, 2022. [23] Facundo Mémoli. Gromov–Wasserstein distances and the metric approach to object matching. Foundations of Computational Mathematics, 11(4):417–487, 2011. [24] Mahesh Chandra Mukkamala, Jalal Fadili, and Peter Ochs. Global convergence of model function based Bregman proximal minimization algorithms. Journal of Global Optimization, 83(4):753–781, 2022. [25] Stefan Müller and Georg Regensburger. Elementary vectors and conformal sums in polyhedral geometry and their relevance for metabolic pathway analysis. Frontiers in Genetics, 7:90, 2016. [26] Gabriel Peyré and Marco Cuturi. Computational optimal transport. Foundations and Trends in Machine Learning, 11(5–6):355–607, 2019.

22

[27] Gabriel Peyré, Marco Cuturi, and Justin Solomon. Gromov–Wasserstein averaging of kernel and distance matrices. In Proceedings of the 33rd International Conference on Machine Learning, volume 48 of Proceedings of Machine Learning Research, pages 2664–2672, 2016. [28] Junwen Qiu, Leilei Mei, and Junyu Zhang. Shuffling the stochastic mirror descent via dual lipschitz continuity and kernel conditioning. arXiv preprint arXiv:2603.16042, 2026. [29] Gabriel Rioux, Ziv Goldfeld, and Kengo Kato. Entropic Gromov–Wasserstein distances: Stability and algorithms. Journal of Machine Learning Research, 25(363):1–52, 2024. [30] R. Tyrrell Rockafellar. The elementary vectors of a subspace of RN . In R. C. Bose and T. A. Dowling, editors, Combinatorial Mathematics and Its Applications, pages 104–127. University of North Carolina Press, Chapel Hill, NC, 1969. [31] R. Tyrrell Rockafellar. Convex Analysis, volume 28 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1970. [32] R. Tyrrell Rockafellar and Roger J.-B. Wets. Variational Analysis. Springer, Berlin, 1998. [33] Meyer Scetbon, Gabriel Peyré, and Marco Cuturi. Linear-time Gromov–Wasserstein distances using low-rank couplings and costs. In Proceedings of the 39th International Conference on Machine Learning, volume 162 of Proceedings of Machine Learning Research, pages 19347–19365, 2022. [34] Siqi Zhang and Niao He. On the convergence rate of stochastic mirror descent for nonsmooth nonconvex optimization. arXiv:1806.04781, 2018. [35] Zhengyuan Zhou, Panayotis Mertikopoulos, Nicholas Bambos, Stephen P. Boyd, and Peter W. Glynn. On the convergence of mirror descent beyond stochastic convex programming. SIAM Journal on Optimization, 30(1):687–716, 2020.

23

Related documents

• Establishing Boundary KKT Convergence of Mirror Descent through Reparameterization#785211
• Establishing Boundary KKT Convergence of Mirror Descent through Reparameterization#783620
• Runge--Kutta generalized Convolution Quadrature for sectorial problems#1069661
• Estimating differential pistons for the Extremely Large Telescope using focal plane imaging and a residual network#635219
• Studying the mirror acceleration via kinetic simulations of relativistic plasma turbulence#670345