Reweighted ALPS (Re-ALPS) MCMC

• Re-ALPS is a Markov chain Monte Carlo (MCMC) method that efficiently samples multimodal distributions using warm-start points near distinct modes.
• It replaces Gaussian approximations with unbiased Monte Carlo estimation of local partition functions to rebalance mixture weights.
• The method offers non-asymptotic, polynomial-time mixing guarantees and superior performance in challenging non-Gaussian, heavy-tailed settings.

Reweighted Annealed Leap-Point Sampler (Re-ALPS) is a Markov chain Monte Carlo (MCMC) methodology designed for sampling efficiently from multimodal target distributions when a set of “warm start” points—each near a distinct mode—is available. Re-ALPS refines the Annealed Leap-Point Sampler (ALPS) by discarding its Gaussian approximation at each mode, instead employing explicit Monte Carlo estimation of local partition functions. This enables robust mixing across target distributions with non-Gaussian and heavy-tailed modes using only oracle access to the target density, and it is accompanied by non-asymptotic, polynomial-time mixing guarantees under natural warm-start and local mixing assumptions (Lee et al., 19 Dec 2025).

1. Underlying Algorithmic Scheme

Re-ALPS targets sampling from distributions of the form

$\pi(x) \propto e^{-V(x)},\quad x \in \Omega,$

given $M$ warm-start points $\{x_1,\dots,x_M\}$, each positioned near a different mode of $\pi$. The algorithm uses a simulated tempering ladder of inverse-temperatures $\beta_1 > \beta_2 > \cdots > \beta_L = 0$, interpolating between highly concentrated “cold” distributions and the true target. At the coldest level, a teleportation (mode-leap) kernel enables rapid transitions between warm starts. Intermediate levels employ local updates (e.g., Langevin or random-walk Metropolis) to ensure adequate within-component exploration.

Distinctively, Re-ALPS dispenses with ALPS’s Hessian-based Gaussian approximations, instead reweighting mixture components dynamically by periodically estimating partition functions via importance-sampling. These dynamic weights continually rebalance mode probabilities, preventing bottlenecks that can hinder exploration across modes.

2. Tilted Distributions, Weights, and Tempering

For each $\beta_i$, Re-ALPS defines a tempered, multi-component mixture

$$\widetilde\pi_{\beta_i}(x) = \pi(x) \sum_{k=1}^M w_{i,k} \, q_{\beta_i}(x-x_k),$$

where $q_{\beta_i}$ is an unnormalized tilting kernel (typically $q_\beta(x) = \exp(-\frac{\beta}{2}\|x\|^2)$ on $\mathbb{R}^d$), and $\{w_{i,k}\}$ are weights ensuring the mass of each mixture component remains well balanced. For normalization,

$$Z_{i,k} = \int_\Omega \pi(x) q_{\beta_i}(x-x_k) dx, \quad Z_i = \sum_{k} w_{i,k} Z_{i,k},$$

thus

$$\pi_{\beta_i}(x) = \frac{\widetilde\pi_{\beta_i}(x)}{Z_i}.$$

The ladder parameterizes a transition from distributions sharply concentrated around each warm start ($\beta_1$ large) to the target ($\beta_L=0$), supporting mixture-wide exploration at all levels. Weights $\{w_{i,k}\}$ are estimated so that $w_{i,k} Z_{i,k} \approx 1$, enforcing “component balance”.

3. Teleportation and Simulated Tempering Dynamics

Re-ALPS employs two classes of moves:

• Temperature swaps: At any level $i$, the chain $(X, i)$ proposes $i \mapsto i \pm 1$ with rate $\lambda$, accepted according to

$$\min\left\{1, \frac{r_{i'}}{r_i} \frac{\pi_{\beta_{i'}}(X)}{\pi_{\beta_i}(X)}\right\}.$$

• Leap-point (teleportation) jumps: At $i=1$ only, with rate $\gamma$, uniformly pick $j, j' \in \{1,\dots,M\}$ and propose $X_{\text{new}} = X - x_j + x_{j'}$, then accept with probability

$$\min\left\{1, \frac{q_{\beta_1}(X - x_{j'})}{q_{\beta_1}(X - x_j)}\right\}.$$

This structure ensures rapid cross-modal mixing at the coldest ladder rung while relying on conventional local MCMC at intermediate levels. The invariant distribution of the full chain is preserved via Metropolis-Hastings acceptance for all transitions, and detailed pseudocode summarizes the mixture of leap, tempering, and local steps.

4. Monte Carlo Estimation of Mixture Weights

To achieve mode balance at each tempering level, Re-ALPS requires estimates of the component partition functions $Z_{i,k}$. Unlike ALPS, which uses local Gaussian Laplace/Hessian approximations, Re-ALPS updates these estimates dynamically:

• Draw $N$ samples $\{(X_j, i_j)\}$ from the chain at the current level.
• Estimate:

$$\widehat Z_{i,k} = \frac{1}{N}\sum_{j=1}^N \widetilde\pi_{\beta_i,k}(X_j)\mathbf{1}\{i_j=i\}$$

where $$\widetilde\pi_{\beta_i,k}(x)=\pi(x)q_{\beta_i}(x-x_k)$$.

• Update component weights:

$$\widehat w_{i,k} = \bigl[\widehat Z_{i,k}\bigr]^{-1}\left/\sum_{k'=1}^M \bigl[\widehat Z_{i,k'}\bigr]^{-1}\right.$$

• Update level weights similarly for temperature-ladder transitions.

By Chebyshev- or chi-squared-type arguments, $O(1/\delta)$ samples suffice to estimate all ratios within $[1-\epsilon,1+\epsilon]$ with high probability, ensuring robust weight adaptation.

5. Non-Asymptotic Mixing and Theoretical Guarantees

Re-ALPS provides the first general polynomial-time mixing guarantee for warm-start MCMC in multi-modal settings, given the following assumptions:

Assumption A (Warm-start + Local Mixing):

1. $\pi = \sum_{k=1}^M \alpha_k \pi_k$ for unknown component distributions.
2. Each warm start $x_k$ satisfies

$$\int \pi_k(x) q_\beta(x-x_k) dx \ge c_{\text{tilt}} \int \pi(x) q_\beta(x-x_k) dx \quad \forall\beta\ge0,$$

ensuring a significant fraction of tilted mass aligns to the “correct” component.

1. Each $q_\beta$-tilted component satisfies a Poincaré (or log-Sobolev) inequality with constant $C_{\text{local}}$.
2. Local MCMC kernels (such as Langevin) decompose over components.

Assumption B (Ladder Closeness):

Adjacent tempered distributions and component marginals remain within constant chi-squared divergence, and initialization is $U$-warm with respect to the coldest mixture.

Main Theorem:

For

$$T = \Omega\left(\mathrm{poly}(U, C_{\text{local}}, M, L, c_{\text{tilt}}, \gamma, \lambda, 1/\epsilon, 1/\delta)\right)$$

and

$$N = \Omega\left(\mathrm{poly}(L, M, U, c_{\text{tilt}}, 1/\delta)\right)$$

samples per level for reweighting, the output marginal $\hat\pi$ satisfies $\mathrm{TV}(\hat\pi, \pi) \le \epsilon$ with probability at least $1 - \delta$.

The analysis proceeds via:

1. Establishing rapid chi-squared concentration to tilt-aligned “good” components.
2. Decomposing mixing time into local (continuous) and projected (discrete chain) mixing, exploiting the Poincaré inequalities.
3. Removing narrow bottlenecks in the projected Markov chain via balanced weighting.
4. Demonstrating the efficacy of the Monte Carlo estimation in maintaining statistical accuracy of weights.

6. Empirical Performance and Practical Comparisons

Comparative evaluation of Re-ALPS and original ALPS was performed on a three-mode five-dimensional target:

$$\pi(x)\propto 0.1\,\mathrm{Cauchy}(x;0,1) + 0.8\,\exp(-\|x\|^4) + 0.1\,N(x;15\cdot\mathbf{1}, I_5).$$

The quartic mode at $\|x\|^4$ is significantly sharper than local Hessian estimates. ALPS, using its Gaussian-Laplace leap proposals, achieved a global leap-acceptance of 0.1%, but accepted zero moves into the quartic mode, resulting in the chain being trapped in the Gaussian/Cauchy wells.

Conversely, Re-ALPS (using dynamic partition-function estimation and the same number of tempering levels) achieved leap-acceptance of 43.5% and empirical modal occupancy $[0.095,\,0.808,\,0.097]$, in tight agreement with the true $[0.1,0.8,0.1]$ weights. Visualizations showed Re-ALPS’s proposals distributed correctly over the difficult quartic mode, while ALPS's proposals almost entirely missed the true support.

7. Distinctive Methodological Features and Impact

Re-ALPS eliminates the reliance on Hessian-based (Gaussian) local mode approximations and leverages unbiased Monte Carlo for partition function updates. This extension addresses fundamental mixing hardness found in standard and tempered MCMC approaches, particularly in regimes where geometric complexity or non-Gaussianity preclude tractable analytic proposals.

This approach establishes the first polynomial mixing guarantee for generic warm-start multimodal ensembles, requires only black-box access to the target density, and allows application to a broader class of distributions exhibiting sharp, intricate, or heavy-tailed local geometries (Lee et al., 19 Dec 2025). It empirically overcomes the critical failure modes of ALPS in the presence of sharp or heavy-tailed modes. All implementation details and further experimental results are provided in the accompanying repository.