Online Rolling Controlled SMC

• ORCSMC is a method that extends controlled SMC for real-time filtering and smoothing in hidden Markov models using a rolling window mechanism.
• It employs dual particle systems and twisting functions to adaptively balance bias and variance while managing bounded computational resources.
• Empirical results reveal significant variance reduction and improved stability compared to standard bootstrap particle filtering in high-dimensional settings.

Online Rolling Controlled Sequential Monte Carlo (ORCSMC) is an advanced methodology for real-time inference in general-state-space hidden Markov models. By leveraging controlled sequential Monte Carlo techniques with a rolling window mechanism, ORCSMC delivers adaptive filtering and smoothing with bounded computational resources while significantly reducing variance compared to standard particle filtering approaches (Xue et al., 1 Aug 2025).

1. Formulation and Model Setting

ORCSMC operates within the standard hidden Markov (state-space) model framework on $\mathbb{R}^d$, with observations in $\mathbb{R}^{d'}$. The model comprises latent states $X_t$ evolving via Markov transitions

$$X_1 \sim \mu(x_1), \quad X_t \mid X_{t-1} = x_{t-1} \sim f_t(x_t \mid x_{t-1}),$$

and conditionally independent observations

$Y_t \mid X_t = x_t \sim g_t(y_t \mid x_t).$

The primary inferential objectives are: (i) online filtering ($p(x_t \mid y_{1:t})$), (ii) offline smoothing ($p(x_{1:T}\mid y_{1:T})$), and (iii) the marginal data likelihood ($p(y_{1:t})$).

2. Controlled SMC and Twisting Functions

The central mechanism underlying controlled SMC is the introduction of a time-indexed sequence of strictly positive twisting (or guiding) functions

$$\psi_t: \mathbb{R}^d \rightarrow (0,\infty), \quad t \geq 1,$$

used to reweight and transform the standard HMM measure. The resulting twisted model employs modified initial and transition densities,

$$\mu^\psi(x_1) = \frac{\mu(x_1)\psi_1(x_1)}{\mu(\psi_1)}, \quad f_t^\psi(x_t \mid x_{t-1}) = \frac{f_t(x_t\mid x_{t-1})\psi_t(x_t)}{Q_t^\psi(x_{t-1})},$$

and corresponding potential functions,

$$g_1^\psi(x_1) = \frac{g_1(y_1\mid x_1)\mu(\psi_1)}{\psi_1(x_1)}\psi_2(x_1), \quad g_t^\psi(x_t) = \frac{g_t(y_t\mid x_t)}{\psi_t(x_t)}\psi_{t+1}(x_t), \quad t \geq 2.$$

The choice of $\psi$ directly regulates importance-weight variance. An infeasible optimal regime, $\psi^\star$, makes incremental weights constant, resulting in zero-variance estimation, but is unavailable in closed form.

Offline CSMC estimates such twisting functions by alternating forward particle propagation (twisted auxiliary particle filtering, $\psi$-APF) and backward least-squares updates on $\log\psi_t$ using particle clouds over the trajectory.

3. Algorithmic Structure: Dual Particle Systems and Rolling Window

To meet real-time and bounded-resource requirements, ORCSMC utilizes two coupled particle systems, both employing the $\psi$-APF within a rolling window of fixed lag $L$:

A. Learning (Control) Particle Filter:

On each rolling window $[t_0, t]$ with $t_0 = \max\{1, t-L+1\}$:

• Particles $\{\widetilde X_s^n, \widetilde W_s^n\}_{n=1}^N$ are initialized with current twisting functions $\psi_{t_0},\dots,\psi_{t-1}$.
• For each of $K$ twist-learning iterations:
  1. Forward pass (propagation and weighting via $\psi$-APF) across the window, yielding weighted particle sets at each $s$.
  2. Backward pass (for $s=t,\dots,t_0$), updating $\psi_s$ by least-squares regression over $\{\widetilde X_s^n,\widetilde W_s^n\}$, typically in a parametric function class (e.g., exponentiated quadratics).

B. Estimation Particle Filter:

After $K$ updates to $\psi_{t_0},\dots,\psi_t$, a single forward $\psi$-APF is performed on $[t_0,t]$ to produce the filtering estimate and unbiased likelihood estimate $Z_t^N \approx p(y_{1:t})$.

A key advantage is the $O(N L K)$ per-step cost and $O(NL)$ memory usage, independent of $t$. This fixed resource profile enables true online operation.

4. Theoretical Properties and Statistical Guarantees

For any fixed twisting sequence $\psi_{1:t+1}$, the marginal likelihood estimator $Z_t^N$ delivered by the $\psi$-APF shows unbiasedness: $\mathbb{E}[Z_t^N] = p(y_{1:t}).$ There exists an optimal, typically infeasible, twist policy $\psi^\star$ that achieves zero-variance weights.

Under strong mixing and boundedness assumptions for the HMM and measurement density, both the twisted particle filter and the entire CSMC/ORCSMC stack admit a Central Limit Theorem for estimators of marginal means.

The rolling window design induces a bias-variance tradeoff: truncating twist function learning to the $L$ most recent steps introduces bias relative to offline CSMC but yiels strong variance reduction compared to the standard bootstrap particle filter.

5. Algorithmic Details and Pseudocode

The ORCSMC method at each time $t$ executes as follows (notation as above):

1. Set rolling window start $t_0 = \max(1, t - L + 1)$.

2. Initialize learning particle system $\widetilde{\mathcal H}_{t_0-1}$.
3. For $k = 1, \dots, K$: a. Forward propagate and weight learning particles via $\psi$-APF over $[t_0, t]$. b. Backward recursion: update $\psi_s$ for $s = t, \dots, t_0$ using weighted particles.
4. Forward $\psi$-APF with final $\psi_{t_0:t}$ for filtering and likelihood estimation.

Within both particle systems, resampling is triggered if the effective sample size falls below a threshold fraction $\kappa N$. Proposals are drawn from the twisted kernel $f_s^\psi$, and second-stage weights utilize $g_s^\psi$.

6. Empirical Performance and Practical Considerations

Numerical experiments on linear-Gaussian models, stochastic volatility models, and neuroscience-inspired binomial models demonstrate that ORCSMC consistently attains substantial variance reduction, stability, and accuracy advantages versus both the bootstrap particle filter and offline CSMC (Xue et al., 1 Aug 2025).

• In high-dimensional Gaussian models, ORCSMC achieves stable $L_1$-errors and RMSE of $\log Z_T^N$ across dimensions $d\in\{2,4,8,16,32,64\}$, greatly outperforming BPF (e.g., BPF with $N=3.2\times 10^5$ vs. ORCSMC with $N=10^3$).
• For univariate stochastic volatility ($T=945$), increasing lag $L$ sharply reduces $\log$-likelihood variance; the method preserves performance in non-Gaussian, heteroskedastic regimes.
• In multichannel binomial models, moderate window sizes maintain effective sample size and stable marginal-likelihood variance even as dimension increases.

The fixed lag $L$ and iterations $K$ permit tuning the bias-variance balance for target applications; larger $L$ trades increased computation for offline CSMC-level accuracy, while small $L$ offers rapid response with bounded effort.

7. Connections, Impact, and Significance

ORCSMC extends the foundational work on controlled SMC and twisted particle filtering to an online, receding-horizon context. The design systematically overcomes the computational scaling and memory limits of full-trajectory CSMC, while preserving its crucial statistical efficiency benefits, particularly variance control, in real-time applications (Xue et al., 1 Aug 2025).

This makes ORCSMC attractive for sequential latent-state inference in complex, high-dimensional, and non-Gaussian state-space models where standard methods such as the bootstrap filter are either unstable or computationally impractical. The use of dual particle systems to learn and exploit twisting functions in real time, combined with rigorous statistical guarantees and demonstrated empirical superiority, establishes ORCSMC as a robust and practically deployable technique for modern sequential inference problems.