Variational Lower Bounds

• Variational lower bounds are mathematically rigorous estimates derived via optimization techniques such as convex duality and Lagrangian relaxation, providing lower limits for complex functionals.
• They transform intractable variational problems into tractable convex formulations using semidefinite and sum-of-squares relaxations, enabling systematic approximation improvements.
• Applications range from quantum physics and statistical inference to machine learning and numerical PDEs, where these bounds certify optimality and enhance model performance.

A variational lower bound is a mathematically rigorous, typically optimization-derived lower estimate for a functional, objective, or quantity of interest. Variational lower bounds are fundamental tools in mathematical analysis, convex optimization, statistical inference, quantum physics, and machine learning. They serve to certify nontrivial minima for otherwise intractable variational problems—ranging from integrals over function spaces and expectation functionals in probabilistic models to quantum ground-state energies and information-theoretic metrics.

1. Fundamental Principles: Duality and Relaxation

The construction of variational lower bounds often utilizes convex duality, Lagrangian relaxation, or variational inequalities. For generic integral minimization over a function space,

$$F^* = \inf_{u\in\mathcal{A}} \int_\Omega f(x,u(x),\nabla u(x))\,dx,$$

the key strategy is to derive a dual or relaxed maximization problem whose optimal value provides a provable lower bound on the infimum. This typically involves

• introducing auxiliary fields (e.g., dual variables, Lagrange multipliers, test functions),
• enforcing constraints weakly (e.g., via integration by parts and suitable test spaces),
• reducing infinite-dimensional problems to tractable max–min or sup–inf forms.

For polynomial integrands and constraints, pointwise dual relaxation (PDR) converts the original nonconvex variational problem into a convex maximization over multipliers and auxiliary polynomials, subject to pointwise nonnegativity or sum-of-squares (SOS) constraints (Chernyavsky et al., 2021). The original primal and dual problems sandwich the true minimum, with the dual value sharp in important cases (e.g., quadratic energies, principal eigenvalues).

2. Semidefinite and Sum-of-Squares Relaxations

When the variational minimization is intractable because of nonconvexity or high-dimensionality, finite-dimensional SDP/SOS relaxations yield computable lower bounds. For example, the relaxation of the positivity constraint on a quantum many-body density matrix $\rho$ to positivity of moment matrices (i.e., expectation matrices of finite sets of operators) leads to

$$\min_{\rho} \mathrm{Tr}[H\rho]\quad\text{subject to}\;\mathrm{Tr}\,\rho=1,\;M(\rho)\succeq 0,$$

where $$M_{k\ell}(\rho) = \mathrm{Tr}[O_k^\dagger O_\ell \rho]$$ and $M\succeq 0$ is an SDP constraint (Baumgratz et al., 2011, Eisert, 2023). Increasing the operator set hierarchy gives systematically tighter bounds. In the purely polynomial case, SOS relaxations of PDR enforce nonnegativity over the feasible set by membership in quadratic modules of sums-of-squares polynomials. Feasibility at each relaxation degree yields a lower bound, and under suitable compactness or structure, convergence to the sharp value is provable (Chernyavsky et al., 2021).

For complex models—e.g., condensed-matter Hamiltonians or PDEs with functional constraints—block structure, symmetry (translational invariance), and moment matrix reductions enable bounding problems that would otherwise be computationally prohibitive (Baumgratz et al., 2011, Eisert, 2023).

3. Statistical and Information-Theoretic Bounds

Variational lower bounds support rigorous estimation in statistics and information theory. For entropy- and mutual-information–like quantities,

• The variational entropy inequality states $$-\int f(x)\log f(x)\,dx \le -\int f(x)\log g(x)\,dx$$, with equality iff $f=g$ almost everywhere, leading directly to tight bounds on output entropy under distortion constraints (Koch et al., 2015).
• In statistical learning, the classical Evidence Lower Bound (ELBO) for latent-variable models,

$$\log p(x) \geq \mathbb{E}_{q(z)}[\log p(x,z) - \log q(z)],$$

results from Jensen’s inequality and underpins variational inference.

• Enhanced lower bounds for, e.g., total variation distance between Bernoulli sums and Poisson laws, arise from variational optimization over Stein test functions, delivering concrete, closed-form improvements over classical results (Sason, 2013).

Recent work generalizes these approaches:

• Variational lower bounds for mutual information, such as the Barber–Agakov, Donsker–Varadhan, and InfoNCE (contrastive) objectives, enable tractable high-dimensional MI estimation with explicit bias–variance tradeoffs by interpolating families of bounds (Poole et al., 2019).
• Locally enhanced variational bounds for hierarchical latent variable models apply tightening (e.g., importance weighting) independently per local group, yielding both improved posterior fit and scalable stochastic optimization (Geffner et al., 2022).

4. Applications in Quantum and Statistical Physics

Variational lower bounds are extensively employed as certificates of optimality and performance guarantees in quantum many-body physics and quantum information:

• Moment-matrix SDP relaxations and hierarchies of local or global constraints (e.g., $p$-positivity in moment matrices) yield lower bounds on ground-state energies, with the possibility of systematic improvement (Baumgratz et al., 2011, Eisert, 2023).
• The Anderson bound and its refinements (via block structure and hierarchical SDP relaxations) provide $O(1)$-accurate lower bounds on energy densities for translationally invariant lattice Hamiltonians and act as benchmarks against which the quality of variational ansatz (e.g., tensor network states) is measured (Eisert, 2023).
• In variational quantum simulation, fidelity lower bounds are constructed via cost-function fidelities over basis and superposition states, with tight worst-case certification achieved using semidefinite relaxation over constraints encoded by measured overlaps (Park et al., 8 Sep 2025).

5. Variational Bounds in Machine Learning

Variational lower bounds underpin the training and evaluation of probabilistic generative models:

• Analytic lower bounds on the ELBO for binary VAEs, derived using Taylor expansion of the Bernoulli likelihood and optimization over encoder parameters, yield closed-form surrogate objectives facilitating model selection, initialization, and robust convergence properties (Sicks et al., 2020).
• In diffusion models, reweighted loss objectives are shown to correspond to weighted sums of cascaded variational lower bounds on the log-likelihood, resulting in provably tighter bounds than the standard ELBO while preserving tractable generation (Shi et al., 24 Nov 2025).
• Corrections via perturbation theory and cumulant expansions, properly modified to guarantee lower bounds (e.g., using concave surrogates of the log), tighten the approximation to the true log-evidence in variational inference and afford reduced bias in posterior variance estimation (Bamler et al., 2019).

6. Lower Bounds for Numerical PDE and Homogenization Problems

In the theory and computation of PDEs, homogenized coefficients, and functional inequalities, dual variational formulations directly yield lower bounds:

• A dual (infimum-over-divergence-free fields) variational problem for homogenized coefficients yields tight lower bounds on macroscopic effective parameters. The corresponding finite element discretizations, using divergence-free FE spaces, enable matching upper and lower bounds with optimal convergence rates—the projection-based lower bounds incur minimal additional computational cost beyond the primal solve (Gaynutdinova et al., 2022).
• Convex duality, in the PDR or SOS approaches, permits systematic, reliable lower bounding of global minima for nonconvex integral functionals and eigenvalues, under verifiable sharpness conditions (Chernyavsky et al., 2021).

7. Nonlocal Variational Bounds and $\Gamma$-Convergence

For nonlocal variational approximations, such as those arising in functionals related to total variation or BV space characterizations, sharp lim inf lower bounds can be established:

• By careful decomposition (absolutely continuous, jump, Cantor), slicing, and one-dimensional analysis, lower bounds for the lim inf of a class of nonlocal functionals are shown to reproduce the three part decomposition of the total variation, each with optimal geometric constants (Lahti, 2023).
• These results aid in the $\Gamma$-convergence analysis and provide structural insight for further generalizations to vectorial and spatially inhomogeneous cases.

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References:

• (Baumgratz et al., 2011) Baumgratz & Plenio. Lower Bounds for Ground States of Condensed Matter Systems.
• (Sason, 2013) Sason. Improved Lower Bounds on the Total Variation Distance for the Poisson Approximation.
• (Koch et al., 2015) Kostina & Verdu. Converse Bounds for Entropy-Constrained Quantization.
• (Bamler et al., 2019) Masegosa et al. Tightening Bounds for Variational Inference by Revisiting Perturbation Theory.
• (Poole et al., 2019) Poole et al. On Variational Bounds of Mutual Information.
• (Sicks et al., 2020) Sicks et al. A lower bound for the ELBO of the Bernoulli VAE.
• (Geffner et al., 2022) Geffner & Domke. Variational Inference with Locally Enhanced Bounds for Hierarchical Models.
• (Gaynutdinova et al., 2022) Moshnyaga et al. Efficient numerical method for reliable upper and lower bounds on homogenized parameters.
• (Chernyavsky et al., 2021) Fantuzzi et al. Convex relaxations of integral variational problems: pointwise dual relaxation and sum-of-squares optimization.
• (Eisert, 2023) Eisert. A note on lower bounds to variational problems with guarantees.
• (Park et al., 8 Sep 2025) Lin et al. Subspace Variational Quantum Simulation: Fidelity Lower Bounds as Measures of Training Success.
• (Shi et al., 24 Nov 2025) Shi & Titsias. Demystifying Diffusion Objectives: Reweighted Losses are Better Variational Bounds.
• (Lahti, 2023) Lahti. A sharp lower bound for a class of non-local approximations of the total variation.