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From Tsallis to KL: Convergence and Error Estimates for Tsallis-Regularized Optimal Transport
Takeshi Suguro, Toshiaki Yachimura
TL;DR
For general f-divergences, the information projection objective does not generally coincide with the corresponding regularized optimal transport objective. For nonnegative bounded costs, the paper establishes qualitative and quantitative convergence results for the Tsallis-to-KL limit. For nonnegative costs, ε > 0, and {qk}k∈N ⊂ (1, 2] with qk ↓ 1, {Fk}k∈N Γ-converges to a limit.
Problem
For general f-divergences, the information projection objective does not generally coincide with the corresponding regularized optimal transport objective.
Source: §1 · p. 2Method
For nonnegative bounded costs, the paper establishes qualitative and quantitative convergence results for the Tsallis-to-KL limit.
Source: §1 · pp. 2, 5Results
For nonnegative costs, ε > 0, and {qk}k∈N ⊂ (1, 2] with qk ↓ 1, {Fk}k∈N Γ-converges to a limit.
Source: §1 · p. 6Takeaways & Limitations
The results characterize the Tsallis-to-KL limit for the stated cost setting and fixed regularization parameter.
Source: §1 · pp. 5–6The stated convergence result assumes Polish spaces and a nonnegative bounded continuous cost, with ε > 0 fixed.
Source: §1 · pp. 2, 6
Abstract
from arXiv · showhide
We study the Tsallis-to-Kullback--Leibler (KL) limit for entropy-regularized optimal transport with nonnegative bounded continuous costs. Fixing the regularization parameter $\varepsilon > 0$, we first derive an exact variational reformulation of Tsallis-regularized optimal transport in terms of the Tsallis information projection onto the set of couplings. The formula isolates an explicit correction term and thereby explains why, unlike in the KL case, the regularized transport problem and the corresponding information projection problem do not coincide exactly. We also establish existence and uniqueness for the Tsallis information projection. We then prove, with respect to the narrow topology, the $\Gamma$-convergence of the Tsallis-regularized functionals to the KL-regularized functional as $q\downarrow1$, together with narrow convergence of their unique minimizers. Finally, we obtain explicit error estimates of order $O(q-1)$ for both the regularized optimal transport values and the associated information projection values. These results quantify the passage from Tsallis regularization to the classical KL setting and clarify the relation between entropic regularization and information projection for $1 < q \leq 2$.
1 Introduction
The introduction frames Tsallis regularization as a way to study the gap between regularized optimal transport and information projection, which coincide in the KL case. It states qualitative convergence to the KL problem and quantitative error estimates as q approaches 1.
1 Introduction
- Unlike the KL case, a general f-divergence projection objective does not in general coincide with its regularized optimal transport objective; Tsallis divergence provides a q-parameterized case approaching KL.The Tsallis divergence reduces to KL as q↓1, making it a test case for the gap between the two formulations.Source: §1 · pp. 2–3
- For 1 < q ≤2, the Tsallis projection problem has a finite infimum in (1.6), which is attained, and its minimizer is unique.Source: §1 · pp. 4–5
- The stated results concern nonnegative bounded continuous costs on Polish probability spaces, with ε > 0 fixed and q decreasing to 1 within (1, 2].These are the conditions stated for the qualitative convergence theorem.Source: §1 · pp. 2, 6
- As q↓1, the Tsallis-regularized functionals Γ-converge to the KL-regularized functional, and their unique minimizers converge narrowly.The stated setting uses Polish spaces, nonnegative bounded continuous costs, and fixed ε > 0.Source: §1 · p. 6
- The regularized optimal transport values and corresponding information projection values have error estimates of order O(q −1) as q↓1.The introduction treats these as two separate estimates.Source: §1 · p. 6
2 Preliminaries
The preliminaries connect Tsallis-specific logarithm identities to the relation between Tsallis projection and regularized transport, then collect dual-potential properties used in later comparisons.
2.1 Basic properties of the q-logarithm and q-exponential functions
- For x, y > 0, the section records non-additivity identities for logq and log2−q, and expresses Tsallis relative entropy using the q-logarithm when µ ≪ ν and q > 1.Source: §2.1 · p. 7
- The non-additivity formula yields a relation between the Tsallis projection problem and Tsallis-regularized optimal transport.Source: §2.1 · p. 7
2.2 Schr¨odinger potentials for Tsallis-regularized optimal transport
- For bounded costs and ε > 0, the Tsallis-regularized transport dual admits a maximizer, with associated potentials bounded in magnitude by 2∥c∥∞.Source: §2.2 · p. 10
- Both Tsallis- and KL-regularized optimal couplings satisfy the same upper L∞ bound, while the KL density also has a positive lower bound.Source: §2.2 · p. 11
3 Γ-convergence from Tsallis to KL regularization
The section establishes the Tsallis-to-KL Γ-convergence argument by proving liminf and recovery-sequence bounds. It also uses equicoercivity and uniqueness to conclude narrow convergence of minimizers.
- The Γ-convergence proof combines a liminf inequality with recovery sequences built from bounded-density approximations and a diagonal choice as qk ↓ 1.The recovery sequence is chosen to converge narrowly while controlling the difference between Tsallis and KL divergences.Source: §3 · pp. 11, 12, 13, 15
- The functionals are lower semicontinuous and equicoercive because their sublevel sets lie in the narrowly compact coupling set.Source: §3 · p. 16
- Strict convexity of the entropy terms on the coupling set gives unique minimizers for the Tsallis and KL functionals.Source: §3 · p. 16
- Every cluster point of the Tsallis minimizers is the unique KL minimizer, so the minimizers converge narrowly.Source: §3 · p. 16
4 Error estimates for the Tsallis-to-KL limit
The section establishes O(q-1) error estimates for both Tsallis-regularized optimal transport values and the corresponding information projection values. It treats these two value comparisons in separate subsections.
4.1 Error estimate for the regularized optimal transport values
- The Tsallis-regularized and KL-regularized optimal transport values differ by O(q-1).The estimate is derived by comparing the entropy integrands.Source: §4 · p. 16, §4.1 · pp. 17–18
4.2 Error estimate for the information projection values
- The corresponding Tsallis and KL information projection values also have an O(q-1) error bound.The proof establishes an upper bound with a constant independent of q and a lower bound of -||c||∞(q-1).Source: §4 · p. 16, §4.2 · pp. 20–21
4.1 Error estimate for the regularized optimal transport values
- For the transport-value estimate, the proof sets a=q-1 and bounds the pointwise difference between the two entropy integrands.For 0<x<1, it uses u=-a log x to bound e^(-u)-1+u.Source: §4.1 · pp. 17–18