Geometric Annealing Path

• Geometric annealing path is a log-linear interpolation between distributions that minimizes a convex combination of KL divergences.
• It underpins algorithms like AIS, SMC, and quantum annealing by balancing variance control and sampling efficiency.
• Generalizations using q-paths and spline-based methods enhance convergence and reduce computational errors in complex state spaces.

A geometric annealing path is a canonical construction in Monte Carlo, optimization, and physics-based algorithms for navigating between probability distributions, energy landscapes, or quantum Hamiltonians by interpolating in a manner that directly reflects underlying geometric or information-theoretic structure. The geometric path notably arises as the log-linear or exponential interpolation between an initial (often tractable) distribution and a target (possibly intractable) distribution, and possesses deep variational, information-theoretic, and optimality properties across inference, statistical mechanics, dynamical systems, and quantum computation.

1. Formal Definition and Variational Characterization

The geometric annealing path between two unnormalized densities, $p_0(x)$ and $p_1(x)$, is defined pointwise as

$$p_t(x) \propto \exp\big[(1-t)\log p_0(x) + t\log p_1(x)\big] = p_0(x)^{1-t}\,p_1(x)^t, \qquad t\in[0,1].$$

This path is a special case of the quasi-arithmetic mean under the logarithm embedding. It possesses a variational characterization as the unique minimizer of a convex combination of Bregman divergences: for $t\in[0,1]$, $p_t$ is the unique minimizer of

$F(\mu) = (1-t) D_{KL}(p_0:\mu) + t D_{KL}(p_1:\mu)$

over the space of densities. Thus, the geometric path is the KL-barycenter of its endpoints. This geometric mean interpolation is the steepest-descent path under constrained variance for the infinitesimal reduction of KL divergence in algorithms such as Annealed Importance Sampling (Brekelmans et al., 2022, Goshtasbpour et al., 2023).

2. Geometric Path in Information Geometry and Divergences

The geometric annealing path is naturally situated within the broader context of Bregman divergences and information geometry. The general variational framework extends to arbitrary monotonic embeddings $\varphi$, yielding:

$$\mu_t(x) = \varphi^{-1}\big( (1-t)\varphi(p_0(x)) + t\varphi(p_1(x)) \big).$$

For $\varphi(u) = \log u$, this reduces to the geometric mean interpolation. The minimization above arises for general $\rho$–$p_1(x)$0 representational Bregman divergences, connecting divergence minimization and the construction of annealing/sequencing paths. Generalizations such as $p_1(x)$1-paths (using power means and deformed logarithms), which interpolate between the geometric and arithmetic means, minimize convex combinations of $p_1(x)$2-divergences (Brekelmans et al., 2022, Masrani et al., 2021).

Path Type	Interpolation Formula	Minimizes (Divergence)
Geometric	$p_1(x)$3	Weighted KL
Arithmetic	$p_1(x)$4	Quadratic/Bregman (Euclidean)
$p_1(x)$5-Path	$p_1(x)$6	Weighted $p_1(x)$7-divergence

The geometric path maintains closure under exponential families and avoids heavy-tailed degeneracies encountered in arithmetic mixtures, ensuring smooth interpolation in statistical and physical parameterizations (Chehab et al., 2023).

3. Algorithmic Schedules and Optimality

In Monte Carlo algorithms such as AIS, SMC, and PT, and in population annealing, intermediate distributions along a geometric path are used to gradually shift from a proposal/initial to a target/final distribution, with each step chosen to control the statistical error or communication barrier. The geometric mean balances variance and overlap optimally in many asymptotic regimes: it transforms the exponential scaling of mean squared error with endpoint separation—typical in naive (arithmetic or direct jump) schemes—into polynomial scaling in the parameter distance, as shown by Fisher-Rao or information-geometric metrics (Chehab et al., 2023, Goshtasbpour et al., 2023). Optimal path schedules in multidimensional parameter spaces minimize the thermodynamic length or dissipation by following geodesics in a Riemannian metric built from fluctuations and relaxation times, providing minimum-entropy-production or minimum-irreversible-work protocols (Barzegar et al., 2024).

4. Generalizations: Power Means and Optimized Paths

Alternative interpolation schemes generalize the geometric path. $p_1(x)$8-paths parameterize a continuum between arithmetic $p_1(x)$9 and geometric $$p_t(x) \propto \exp\big[(1-t)\log p_0(x) + t\log p_1(x)\big] = p_0(x)^{1-t}\,p_1(x)^t, \qquad t\in[0,1].$$0 means, admitting tunable mass-covering-concentrating behavior via the deformed logarithm and $$p_t(x) \propto \exp\big[(1-t)\log p_0(x) + t\log p_1(x)\big] = p_0(x)^{1-t}\,p_1(x)^t, \qquad t\in[0,1].$$1-exponential families. For sampling multimodal or badly overlapping endpoints, small deviations from $$p_t(x) \propto \exp\big[(1-t)\log p_0(x) + t\log p_1(x)\big] = p_0(x)^{1-t}\,p_1(x)^t, \qquad t\in[0,1].$$2 can significantly improve estimation, as shown in SMC and AIS empirical results (Masrani et al., 2021).

In Parallel Tempering, while geometric interpolation is standard, it can be highly suboptimal when the endpoint measures are nearly mutually singular—optimized nonlinear or spline-based paths can dramatically improve round-trip rates and mixing, surpassing the fundamental performance limitations of linear-in-log annealing (Syed et al., 2021).

5. Geometric Annealing in Physics and Optimization

In simulated annealing and convex optimization, the geometric path structure appears as the bridge between random-walk sampling and deterministic optimization: the Gibbs family $$p_t(x) \propto \exp\big[(1-t)\log p_0(x) + t\log p_1(x)\big] = p_0(x)^{1-t}\,p_1(x)^t, \qquad t\in[0,1].$$3 interpolates between maximum-entropy and minimum-energy configurations as $$p_t(x) \propto \exp\big[(1-t)\log p_0(x) + t\log p_1(x)\big] = p_0(x)^{1-t}\,p_1(x)^t, \qquad t\in[0,1].$$4. There is a mathematically precise equivalence between the geometric annealing path (in simulated annealing) and the central path of interior-point algorithms with entropic/universal barriers, connecting temperature schedules, Dikin ellipsoids, and Newton steps (Abernethy et al., 2015).

In path-space measure transport for stochastic optimal control, trust-region-constrained steps on KL interpolations correspond to geometric annealing in path space, providing robust and adaptive update schedules that maintain control over sampling error in high dimensions (Blessing et al., 17 Aug 2025). For population annealing, the geometric path arises as the analytic solution to discrete-time Schrödinger bridges, with each step performing the thermodynamically optimal reweighting and mutation procedure and the entire path being an entropic geodesic in probability law space (Ohzeki, 17 Mar 2026).

6. Geometric Annealing in Dynamical Systems and Quantum Computation

Annealing paths are central to the theoretical operation of dynamical analog solvers (e.g., Coherent Ising Machines) and quantum annealers. In the CIM, geometric paths in the control-parameter space (temperature/gain) track the critical geometric phase transitions of the energy landscape (from convex to rough to rigid), providing schedule guidelines that yield rapid convergence to near-ground states for SK spin glasses (Yamamura et al., 2023, Zhou et al., 14 Mar 2026). Geometric schedules outperform linear or gain-only paths in tracking the barrier-dominated regime to the global minimum in $$p_t(x) \propto \exp\big[(1-t)\log p_0(x) + t\log p_1(x)\big] = p_0(x)^{1-t}\,p_1(x)^t, \qquad t\in[0,1].$$5 time asymptotically in $$p_t(x) \propto \exp\big[(1-t)\log p_0(x) + t\log p_1(x)\big] = p_0(x)^{1-t}\,p_1(x)^t, \qquad t\in[0,1].$$6.

In quantum annealing, geometric phases manifest as non-stoquastic interactions in effective Hamiltonians. Non-adiabatic Aharonov-Anandan phases induce geometric terms in the Hamiltonian evolution that depend solely on the path in parameter space, not its temporal parameterization. This enables the engineering of non-stoquastic couplings by geometric path design—implications include quantum speedups and the circumvention of classical simulation barriers (Vinci et al., 2017).

7. Theoretical Significance and Implications

Geometric annealing paths are canonical due to their emergence as variational solutions to optimal transport (OT) and entropy minimization problems, their information-geometric interpretation (entropic geodesics, KL-barycenters), and their algorithmic and physical optimality in annealing processes across diverse domains. They unify inference, optimization, stochastic control, and dynamical system analysis under a single geometric-variational umbrella, with extensive implications for algorithmic efficiency and the physical limits of computation (Brekelmans et al., 2022, Abernethy et al., 2015, Chehab et al., 2023, Ohzeki, 17 Mar 2026).

A plausible implication is that further generalizations (e.g., multi-parameter geodesics in thermodynamic metrics, or geometric path planning in quantum Hamiltonian control) will continue to provide principled strategies for bridging complex state spaces in high-dimensional and nonconvex settings, setting the ground for new developments in computational physics, machine learning, and quantum information processing.