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Double-Loop Unadjusted Langevin Algorithm
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Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use a double-loop geometrically decaying step-size schedule instead of standard ULA | ULA · Smooth unconstrained log-concave sampling | iteration complexity (KL): \wt{O}\l(Ld^3\epsilon^{-2} \r) -> \tilde{\mathcal{O}}(Ld^3\epsilon^{-\frac{3}{2}}) | Helped |
| Use a double-loop geometrically decaying step-size schedule instead of the cited TV baseline | ULA · Smooth unconstrained log-concave sampling | iteration complexity (TV): \wt{O} \l(Ld^3\epsilon^{-4} \r) -> \wt{\mathcal{O}}(Ld^3\epsilon^{-3}) | Mixed |
| Use a double-loop schedule and projection | ULA · Smooth unconstrained log-concave sampling | iteration complexity (W2): \text{no prior result in Table 1} -> \tilde{\mathcal{O}}(Ld^9\epsilon^{-6}) | Helped |
| Apply the double-loop schedule and decrease the penalty parameter across outer loops | MYULA · Constrained log-concave sampling | iteration complexity (TV): d^{5} \wt{O} \l(\epsilon^{-6} \r) -> \mathcal{O}\left(d^{3.5} \epsilon^{-5} \right) | Helped |
| Decrease the penalty parameter across outer loops | MYULA · Constrained log-concave sampling | asymptotic convergence | Helped |
| Add a projection step after each outer loop | DL-ULA · Smooth unconstrained log-concave sampling | tail control for W2 analysis | Helped |
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