Research map / Optimization / Nonconvex and smooth optimization
A connection between Tempering and Entropic Mirror Descent
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Shows that tempering SMC is entropic mirror descent on the reverse KL, gives convergence rates for tempering iterates, and derives adaptive tempering rules (constant KL/chi-square between consecutive temperatures) that justify ESS-based practice.
Problem
Choosing tempering (annealing) schedules for SMC/AIS sampling and understanding the optimization interpretation and convergence of tempering iterates, which previously lacked theory beyond specific target families.
Approach
Tempering iterates are exactly entropic mirror descent on KL(.|pi) with step sizes gamma_n=(lambda_n-lambda_{n-1})/(1-lambda_{n-1}); this yields a KL convergence rate and a principled ODE-based adaptive schedule where consecutive KL divergences are kept constant, with SMC importance weights implementing the mirror descent update.
Key results
- Tempering iterates satisfy KL(mu_n|pi) <= (1-lambda_n)/lambda_1 * KL(pi|mu_0), giving an explicit convergence rate for the tempering sequence.
- Adaptive tempering with constant consecutive KL yields a number of tempering steps scaling as O(d^{1/2}) for i.i.d. Gaussian targets.
- On a narrow Gaussian target (N(1,0.1^2 I)), SMC needs 5 tempering steps while the compared adaptive scheme needs 6257 steps with worse approximations.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use the adaptive SMC tempering rule instead of the alternative constant-rate KL strategy | Tempering for the narrow Gaussian target · Narrow Gaussian: π = N(1_d, 0.1²I), μ₀ = N(0_d, I) | tempering steps: 6257 -> 5 | Helped |
Surprising findings
- The number of tempering steps scales as O(sqrt(d)), contrary to the O(d) scaling suggested by prior SMC stability analysis.
Limitations
- Theoretical analysis for the SMC case assumes Markov kernels are mu_n-invariant.
- Explicit adaptive schedules are analytic only for simple Gaussian cases; general targets need particle-based numerical approximation.
- Experiments are limited to toy Gaussian targets in moderate dimension (d=25).
Experimental setup
Metrics: KL divergence convergence rate number of tempering steps particle approximation quality
Baselines: ESS-based adaptive tempering annealed importance sampling particle mirror descent SRAIS parallel tempering
Builds on: Sequential Monte Carlo samplers mirror descent on measures annealed importance sampling Langevin Monte Carlo
Closest papers by meaning
- Mirror Langevin Monte Carlo: the Case Under Isoperimetry (NeurIPS 2021 · similarity 0.90)
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- Mirror Descent Beyond Euclidean Stability: An Exponential Separation in Initialization Sensitivity (NeurIPS 2026 · similarity 0.85)
- Constrained Sampling with Primal-Dual Langevin Monte Carlo (NeurIPS 2024 · similarity 0.85)
- Tuning Sequential Monte Carlo Samplers via Greedy Incremental Divergence Minimization (ICML 2025 · similarity 0.84)
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