A small gradient is reassuring only if it means that the objective is nearly minimized. The Polyak–Łojasiewicz condition supplies precisely this guarantee [6]. It can yield exponential convergence even without convexity, as in Fast gradient flows without convexity.
Now replace a point by a probability distribution, Euclidean distance by quadratic Wasserstein distance, and the objective by relative entropy. The same condition becomes the logarithmic Sobolev inequality. This is more than a dictionary: one short argument shows that the entire distribution converges, not just its entropy. It also explains the implication from logarithmic Sobolev to Talagrand’s transport inequality [1].
Entropy has a gradient: Fisher information
Let \(\pi(x)=Z^{-1}e^{-V(x)}\) be a smooth positive probability density on \(\mathbb R^d\). We use natural logarithms and write
\[ \mathcal F(\rho)=\operatorname{KL}(\rho\mid\pi) =\int\rho\log\frac\rho\pi\,dx,\qquad \mathcal I(\rho\mid\pi)=\int\left|\nabla\log\frac\rho\pi\right|^2\rho\,dx. \]For the calculations, assume finite second moments and enough smoothness and decay to integrate by parts. If a density moves with velocity \(v\), the continuity equation is \(\partial_t\rho+\nabla\cdot(\rho v)=0\). Differentiating gives
\[ \frac d{dt}\mathcal F(\rho_t) =\int\nabla\log(\rho_t/\pi)\cdot v_t\,\rho_t\,dx. \]The tangent norm is \(\|v\|_\rho^2=\int|v|^2\rho\). Thus the Wasserstein gradient is represented by \(\nabla\log(\rho/\pi)\), and its squared norm is Fisher information. Its negative gradient flow is
\[ v_t=-\nabla\log(\rho_t/\pi),\qquad \partial_t\rho_t=\Delta\rho_t+\nabla\cdot(\rho_t\nabla V). \]This is the Fokker–Planck equation. Along a regular solution,
\[ -\frac d{dt}\mathcal F(\rho_t) =\mathcal I(\rho_t\mid\pi) =|\dot\rho_t|_{W_2}^2. \]The equality with metric speed uses the gradient velocity, not an arbitrary velocity solving the same continuity equation. Wasserstein gradient-flow theory makes this identity precise beyond smooth densities [2].
The P–L condition in disguise
Proposition 1 (the dictionary). Logarithmic Sobolev with constant \(\lambda>0\) is the inequality
\[ \mathcal I(\rho\mid\pi)\ge 2\lambda\operatorname{KL}(\rho\mid\pi). \]Equivalently,
\[ \tfrac12\|\operatorname{grad}_{W_2}\mathcal F(\rho)\|_\rho^2 \ge\lambda(\mathcal F(\rho)-\min\mathcal F). \]This is exactly Wasserstein P–L. Consequently,
\[ \mathcal F(\rho_t)\le e^{-2\lambda t}\mathcal F(\rho_0). \]Proof: substitute the gradient norm
The minimum of relative entropy is zero, attained only at \(\pi\). Substitute the gradient–Fisher identity above. Energy dissipation then gives \(\mathcal F'\le-2\lambda\mathcal F\), and Grönwall gives the decay estimate. \(\square\)
The constant convention matters: for the standard Gaussian, \(\lambda=1\). In the classical function formulation, putting \(\rho=h^2\pi/\int h^2d\pi\) turns this into \(\operatorname{Ent}_\pi(h^2)\le(2/\lambda)\int|\nabla h|^2d\pi\). This is the normalization of Gross’s Gaussian inequality [3].
Why the measures themselves converge
Exponential decay of an objective does not, by itself, imply convergence of its argument. A valley can contain many minimizers; a curve might keep travelling along it. Here the dissipation identity gives the missing control: the curve has finite remaining length.
Theorem 1 (finite length under Wasserstein P–L). Let \(\mathcal G:\mathcal P_2(\mathbb R^d)\to(-\infty,+\infty]\) be lower semicontinuous, with finite infimum \(\mathcal G_\star\). Suppose a global locally absolutely continuous flow, of finite initial energy, has locally absolutely continuous energy \(E\) and satisfies for almost every time
\[ E(t)=\mathcal G(\rho_t)-\mathcal G_\star,\qquad -E'(t)=|\dot\rho_t|_{W_2}^2=|\partial\mathcal G|^2(\rho_t), \qquad |\partial\mathcal G|^2(\rho_t)\ge2\lambda E(t). \]Here \(|\partial\mathcal G|\) is the descending metric slope. Then the flow converges in \(W_2\) to a minimizer \(\rho_\infty\), and
\[ W_2(\rho_t,\rho_\infty) \le\sqrt{\frac{2E(t)}\lambda} \le\sqrt{\frac{2E(0)}\lambda}\,e^{-\lambda t}. \]Proof: differentiate the square root of the energy
Write \(s(t)=|\dot\rho_t|_{W_2}\). As long as \(E(t)>0\), dissipation and P–L give
\[ -\frac d{dt}\sqrt{E(t)} =\frac{s(t)^2}{2\sqrt{E(t)}} \ge\sqrt{\lambda/2}\,s(t). \]Integrate from \(a\) to \(b\), and use that distance is bounded by the length of a curve:
\[ W_2(\rho_a,\rho_b) \le\int_a^b s(t)\,dt \le\sqrt{2/\lambda}\bigl(\sqrt{E(a)}-\sqrt{E(b)}\bigr). \]Also \(E'\le-2\lambda E\), so \(E(t)\to0\). The last bound makes the curve Cauchy as time tends to infinity. Completeness of \(\mathcal P_2\) gives a limit, and lower semicontinuity gives \(\mathcal G(\rho_\infty)\le\liminf\mathcal G(\rho_t)=\mathcal G_\star\). Let \(b\to\infty\) in the length bound. If energy reaches zero at a finite time, the dissipation identity makes the curve constant thereafter, with the same conclusion. \(\square\)
For entropy, the minimizer is uniquely \(\pi\). Applying the theorem at time zero gives
\[ W_2(\rho,\pi)^2\le\frac2\lambda\operatorname{KL}(\rho\mid\pi). \]This is Talagrand’s quadratic transportation inequality, with the same constant. The calculation proves it for initial data admitting the stated dissipative flow; standard approximation extends it to all finite-entropy data under the usual hypotheses. The full logarithmic-Sobolev-to-transport implication is due to Otto and Villani [1].
A Gaussian example attaining both rates
Take \(\pi=\mathcal N(0,I)\) and \(\rho_0=\mathcal N(m_0,I)\). The exact flow is \(\rho_t=\mathcal N(e^{-t}m_0,I)\). Hence
\[ \mathcal F(\rho_t)=\tfrac12e^{-2t}|m_0|^2,\qquad W_2(\rho_t,\pi)=e^{-t}|m_0|. \]The objective decays at rate two, the distance at rate one. Both constants in Theorem 1 are sharp.
Why the Gaussian logarithmic Sobolev constant is one
Here is a short semigroup proof for smooth positive densities \(f=d\rho/d\pi\) bounded above and away from zero, with sufficient derivative integrability. Let
\[ P_tf(x)=\mathbb E\,f(e^{-t}x+\sqrt{1-e^{-2t}}Z),\qquad Z\sim\pi. \]Gaussian invariance gives \(\int P_tg\,d\pi=\int g\,d\pi\). Differentiation gives \(\nabla P_tf=e^{-t}P_t\nabla f\). Weighted Cauchy–Schwarz therefore yields
\[ \frac{|\nabla P_tf|^2}{P_tf} \le e^{-2t}P_t\!\left(\frac{|\nabla f|^2}{f}\right). \]After integration, \(\mathcal I(P_tf\,\pi\mid\pi)\le e^{-2t}\mathcal I(f\pi\mid\pi)\). Entropy dissipation and \(P_tf\to1\) give
\[ \operatorname{KL}(f\pi\mid\pi) =\int_0^\infty\mathcal I(P_tf\,\pi\mid\pi)\,dt \le\tfrac12\mathcal I(f\pi\mid\pi). \]Boundedness justifies the entropy limit by dominated convergence. Truncation, smoothing, and lower semicontinuity extend the inequality to its natural finite-information domain. \(\square\)
More than entropy, more than convexity
Uniformly convex confinement. If \(\nabla^2V\succeq\lambda I\), the Bakry–Émery criterion gives logarithmic Sobolev with constant \(\lambda\) [4]. This is a sufficient condition, not a characterization: a bounded perturbation of a uniformly convex potential can preserve a logarithmic Sobolev inequality even when pointwise convexity is lost.
A genuinely nonconvex lifted loss. If \(|\nabla h|^2\ge2\lambda(h-h_\star)\), integration proves Wasserstein P–L for \(\mathcal G(\rho)=\int h\,d\rho\) whenever its slope has the expected gradient representation. For instance,
\[ h(x,y)=\tfrac12(y-x^2)^2,\qquad |\nabla h|^2=(1+4x^2)(y-x^2)^2\ge2h. \]Its minimizers are distributions supported on a parabola. A particle stays in its initial sublevel, which bounds \(|y-x^2|\); the equations \(x'=2x(y-x^2)\), \(y'=-(y-x^2)\) then rule out finite-time escape. Theorem 1 applies to distributional flows for which its finite-moment and full-dissipation assumptions hold. Different initializations can select different minimizing measures.
Confined internal energy. For \(m>1\), the functional
\[ \mathcal G(\rho)=\frac1{m-1}\int\rho^m\,dx+ \frac\lambda2\int|x|^2\rho\,dx \]is \(\lambda\)-displacement convex and satisfies Wasserstein P–L with constant \(\lambda\). Indeed, its geodesic subgradient inequality bounds the energy gap by \(|\partial\mathcal G|r-\lambda r^2/2\le|\partial\mathcal G|^2/(2\lambda)\), where \(r\) is the distance to a minimizer. Convexity supplies one route to P–L; the finite-length argument needs only P–L itself. See Optimal Transport for Machine Learners [5].
References
- Felix Otto and Cédric Villani. Generalization of an Inequality by Talagrand and Links with the Logarithmic Sobolev Inequality. Journal of Functional Analysis 173(2), 361–400, 2000.
- Luigi Ambrosio, Nicola Gigli, and Giuseppe Savaré. Gradient Flows in Metric Spaces and in the Space of Probability Measures. Second edition, Birkhäuser, 2008.
- Leonard Gross. Logarithmic Sobolev Inequalities. American Journal of Mathematics 97(4), 1061–1083, 1975.
- Dominique Bakry and Michel Émery. Diffusions hypercontractives. Séminaire de Probabilités XIX, 177–206, 1985.
- Gabriel Peyré. Optimal Transport for Machine Learners. Book manuscript, 2025–2026.
- Hamed Karimi, Julie Nutini, and Mark Schmidt. Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak–Łojasiewicz Condition. ECML PKDD, 2016.