Shifted Composition IV: Toward Ballistic Acceleration for Log-Concave Sampling

1 Introduction

• Underdamped sampling is motivated by improved dimension and accuracy dependence, while its conjectured √κ scaling would provide ballistic acceleration over the linear-κ rates of existing algorithms.Continuous-time ULD had recently achieved ballistic acceleration, but discrete-time acceleration remained open because discretization error overwhelmed the improvement.Source: §1 · pp. 3–4

1.1.1 Harnack inequalities for the underdamped Langevin diffusion

• The framework proves ULD reverse transport inequalities capturing short-time 1/T^3 degeneracy and long-time 1/T or e^−cT convergence, including a new weakly convex result.Its direct coupling construction is designed to incorporate numerical discretization errors, unlike the implication chain used by [GW12].Source: §1.1.1 · p. 5

1.1.2 KL local error framework for underdamped Langevin discretizations

• The KL local error framework extends elliptic-diffusion analysis to degenerate ULD, combines weak and strong one-step errors, and accumulates discretization error linearly without contractivity.This enables KL guarantees, which imply TV bounds by Pinsker’s inequality, while avoiding exponential accumulation from Wasserstein analyses.Source: §1.1.2 · pp. 5–6

1.1.3 Applications to sampling

• RM–ULMC achieves the first sublinear-in-κ guarantee, using O~(κ^5/6 d^5/3/ε^2/3) gradient evaluations for KL error ε^2.The result applies to strongly log-concave, log-smooth distributions with log χ2(µ0 ∥π)=O~(d).Source: §1.1.3 · p. 7

1.2 Techniques

• The discrete-time method uses shifted composition to obtain long-time KL bounds from short-time estimates while incorporating both weak and strong discretization errors.It derives from shifted Girsanov coupling ideas and is applicable beyond settings where Girsanov’s theorem directly applies.Source: §1.2 · p. 8
• Degeneracy requires more complicated shifts because noise acts only on momentum, so the auxiliary process must interpolate through momentum without introducing a singular position drift.The analysis uses coupling to control the auxiliary process and twisted coordinates for ULD contraction arguments.Source: §1.2 · pp. 8–9
• The continuous-time analysis couples ULD with an auxiliary process in time-varying coordinates, obtaining contraction even in non-convex or low-friction regimes.The effective friction is time-varying, and the low-friction regime is required for the ballistic-acceleration application.Source: §1.2 · p. 9
• The discrete-time framework combines auxiliary interpolation, shifted composition, and time-varying coupling to analyze non-adapted ULD discretizations.It must handle interleaved diffusion and shifting, while compensating for the required h-scale difference between position and momentum errors.Source: §1.2 · pp. 9–10

1.3 Related work

• For general κ-conditioned potentials, the paper obtains a rate of ˜O(κ5/6), resolving whether any κ1−δ acceleration is possible and opening the way to ballistic acceleration.Before this work, acceleration beyond linear dependence on κ was unknown for non-quadratic potentials.Source: §1.3 · p. 10
• Shifted composition removes these barriers by avoiding contractivity, supporting non-adapted randomized midpoint schemes, and recovering sharp continuous-time Harnack inequalities for discretizations.The paper adapts these elliptic-diffusion techniques to degenerate diffusions through new ideas described in §1.2.Source: §1.3 · pp. 10–11
• Prior shifted-divergence analyses could not cover the accelerated low-friction regime, look-ahead discretizations, or nontrivial continuous-time Harnack limits.These limitations arose respectively from reliance on Wasserstein contractivity, inapplicable adapted interpolations, and regularity that vanished with the discretization step.Source: §1.3 · pp. 10–11

• The Wasserstein distance permits a flexible base norm, including the twisted norm where ULD is contractive rather than the standard Euclidean norm.This norm choice is specified whenever used.Source: §2 · p. 11
• Rényi divergence is defined for orders q > 1, with q = 1 yielding KL divergence and q = 2 relating to chi-squared divergence.The preliminaries also state data processing and weak triangle inequalities for Rényi divergences.Source: §2 · pp. 11–12
• The shifted chain rule bounds KL(PY ∥ QY) using KL(PX′ ∥ QX) plus a coupled conditional-divergence term.It extends the standard chain rule by introducing an auxiliary variable X′.Source: §2 · p. 12
• Choosing X′ flexibly enables analysis of two Markov processes with different kernels, specifically to control numerical discretization error for diffusions.This flexibility is the framework’s key application in the paper.Source: §2 · p. 12

3.1 Main result

• Theorem 3.2 gives Harnack inequalities for ULD under α-convexity and β-smoothness, distinguishing high friction γ ≥ √32β from low friction γ < √32β.The theorem applies for arbitrary-sign α; high friction is defined by γ ≳ √β and low friction by γ ≲ √β.Source: §3.1 · p. 13
• Short-time bounds scale as 1/T^3, reflecting the degenerate hypoelliptic nature of ULD, while long-time behavior is exponential for strongly convex potentials, polynomial for weakly convex potentials, and nondecaying in semi-convex or low-friction cases.The 1/T^3 scaling is sharp, and low-friction ULD is noncontractive even when α ≥ 0.Source: §3.1 · p. 13

3.2 Construction of auxiliary process

• The auxiliary-process construction couples a second ULD copy to a process started at the comparison point and forced to hit the target state at time T through diverging shifts and Girsanov’s theorem.The analysis uses time-dependent twisted coordinates that accommodate shifting dynamics, although the coordinates degenerate as t approaches T and require a limiting argument.Source: §3.2 · p. 15

3.3 Warm up: integrated Brownian motion

• The integrated-Brownian-motion warm-up derives optimal shifts for simplified dynamics, verifies the resulting KL bound exactly, and shows that the resulting bounds can be sharp.This example motivates the shift structure used for weakly convex ULD.Source: §3.3 · pp. 15–16

3.4.1 Distance decay identity

• The distance decay identity expresses the coupled dynamics through a time-dependent symmetric matrix M_t, with the intermediate Hessian H_t satisfying αI ⪯ H_t ⪯ βI.The identity follows by coupling the original and auxiliary systems, changing to twisted coordinates, and symmetrizing the resulting quadratic form.Source: §3.4.1 · p. 16

3.4.2 Contraction

• The contraction analysis selects explicit time-dependent shifts to make the Girsanov KL cost small and lower-bounds M_t’s eigenvalues, with different shift scales in contractive, expansive, and low-friction regimes.The shifts diverge near T to enforce interpolation; before the final interval they are exponentially small when ω > 0, of order |ω| when ω < 0, and of order √β under low friction.Source: §3.4.2 · p. 17

3.5 KL bounds via shifted Girsanov

• The proof converts shifted-system contraction into KL Harnack bounds through Girsanov’s theorem, initially on [0,T−δ] and then by handling the coordinate degeneration as δ approaches zero.The section focuses on KL divergence; Rényi extensions follow by the same argument because the shifted system contracts almost surely.Source: §3.2 · p. 14, §3.5 · pp. 18–19
• The shifted-Girsanov proof obtains the desired KL bound by taking δ ↘ 0 and applying weak lower semicontinuity of KL divergence.The limiting relation is (X_{T−δ}, P_{T−δ}) → (X_T, P_T) in L2.Source: §3.5 · p. 19
• Theorem 3.2 is completed by substituting η_t and γ_t, computing the final KL bound, and verifying that the auxiliary process reaches (X_T, P_T).The presented derivation specializes to α = 0 with high friction, while other settings are deferred to §A.3.Source: §3.5 · p. 19
• For the high-friction specialization, the argument requires 2c′c_0 > 3, with c′ = 1/48 allowed and c_0 > 144 sufficient.The resulting comparison controls the squared position and momentum discrepancies as ||x − x̄||^2 + γ_0^-2 ||p − p̄||^2, where γ_0 = γ + 1/T.Source: §3.5 · p. 19

4 KL local error framework for underdamped Langevin discretizations

• The framework extends from elliptic to degenerate diffusions, providing a KL-based analysis of numerical ULD discretizations.It is designed to support different discretization schemes and settings without requiring a closed-form ULD transition kernel.Source: §4 · p. 19, §4.1 · p. 20

4.1 Main result

• Theorem 4.1 bounds final KL error through initialization, local errors in position and momentum, and a cross-regularity term, assuming h ≲ γ^-1 ∧ γ/β.The position-local-error contribution is treated as a factor h smaller than the momentum contribution for the discretizations considered in §5.Source: §4.1 · p. 20
• Cross-regularity is handled by changing the final kernel to ULMC, for which Lemma 4.2 provides a bound under additional restrictions on h and q.This bypasses the difficulty of proving cross-regularity for general discretization kernels.Source: §4.1 · p. 20

4.2 Technical overview

• The analysis introduces a shifted auxiliary process, derives a distance recursion in a time-varying coordinate system, and controls its error terms using local errors between the evolving processes.When initial differences are only in momentum, the dependence is better by a factor of h2 than for initial position differences; the recursion has effective contraction constant Ln < 1.Source: §4.2 · pp. 21–22

4.3 Effective contraction

• Effective contraction requires analyzing diffusion, shifting, and coordinate changes simultaneously; the proof therefore establishes contraction for the “diffuse then shift” process.Modified shifts include a parameter A > 0 that tempers their growth near T and supports the contraction argument.Source: §4.3 · pp. 23, 24, 26

4.4 Distance recursion for auxiliary process

• An auxiliary process yields a distance recursion combining contraction, twisted-coordinate strong errors, and a coupling constant ζ_n.The coupling constant is typically negligible under the stated step-size and parameter conditions.Source: §4.4 · pp. 26–27

4.3 Effective contraction

• The framework uses effective contraction for a “diffuse then shift” process together with changing twisted coordinates.The effective contraction constant is defined for the diffuse-then-shift process, while local errors are measured in twisted coordinates.Source: §4.4 · p. 26

4.5 Computing the bounds via integral estimates

• The analysis converts twisted-coordinate bounds back to position and momentum, then expresses the final KL bound through integrals.These steps assume h ≲ γ^-1 ∧ γ/β and c_0/A ≲ 1 for sufficiently small implied constants, with summations replaced by integrals.Source: §4.5 · pp. 28–29

4.6 Cross-regularity for ULMC

• For ULMC cross-regularity, the proof uses the weak triangle inequality for Rényi divergences instead of Girsanov’s theorem with shifting.The resulting terms are bounded using the conditions of the cross-regularity lemma and Theorem 3.2.Source: §4.6 · p. 29

5 Application to sampling

• The sampling application requires recursively bookkeeping the local-error terms from Theorem 4.1 for the augmented target π = π ⊗ N(0,I).These recursive error estimates are deferred to §C.Source: §5 · p. 30

5.1 Underdamped Langevin Monte Carlo

• ULMC yields a new convergence guarantee under weak log-concavity, whereas its strongly convex rate matches known results and is not new.The framework illustrates local-error analysis for the simple stochastic exponential Euler scheme, while weakly convex rates remain worse than the cited randomized-midpoint overdamped rate.Source: §5.1 · pp. 30–31

5.2 Randomized midpoint discretization

• The double-midpoint RM–ULMC implementation removes the second term from existing rates of the form O~(κ(d/ε^2)^1/3 + κ^7/6(d/ε^2)^1/6).The implementation generates two midpoint discretizations from the same stochastic process, enabling this improvement.Source: §5.2 · p. 31

5.2.1 Log-concavity

• For convex targets, Theorem 5.6 establishes the κd^1/3/ε^2/3 rate for RM–ULMC in KL divergence and therefore in total variation, while weakly convex guarantees are also new.In the constant-accuracy, constant-κ regime, the result gives ε total-variation error with d^1/3 dependence, improving the prior d^5/12 rate cited in the passage.Source: §5.2.1 · p. 33

5.2.2 Discretization bound

• Theorem 5.7 provides a general non-convex RM–ULMC discretization bound that can be combined with any ULD convergence result.The bound controls KL(µ ˆP^{N−1} ˆP′ ∥µP^N) with a factor scaling as O~((1+C_χ2)(1+β^1/2T)β^2dh^3).Source: §5.2.2 · p. 33

5.2.3 Entropic hypocoercivity

• Combining the discretization bound with entropic hypocoercivity yields RM–ULMC guarantees under an LSI assumption, extending beyond strong log-concavity.The analysis uses the decay of a carefully chosen Lyapunov functional for ULD and allows initialization dependence through L(µ) and C_χ2.Source: §5.2.3 · p. 34

5.2.4 Space-time Poincaré inequality

• Theorem 5.11 gives RM–ULMC an o(κ) dependence under a space-time Poincaré inequality, providing the first discretization-based step toward ballistic acceleration for log-concave sampling.With standard strongly log-concave initialization, the rate is κ^5/6d^5/3/ε^2/3 up to logarithmic factors; with a warm start, it becomes κ^5/6d^1/3/ε^2/3.Source: §5.2.4 · p. 35

A.1 Proof of Example 3.7

• The proof starts from two systems evolving from initial points (x,p) and (x̄,p̄), introducing an auxiliary process for their comparison.Source: §A.1 · p. 36
• Only the momentum component may receive an added drift, because altering other components would make the auxiliary path measure singular relative to the original.The added drift η_t is interpreted as a control.Source: §A.1 · p. 36
• The control is chosen by minimizing the Girsanov KL-divergence bound subject to two marginal constraints enforcing position interpolation.Source: §A.1 · p. 36
• Synchronous coupling cancels the Brownian motions, reducing the constrained variational problem to the deterministic relations needed to solve for the optimal shift.Source: §A.1 · p. 36
• After inverting the dynamics for p, p̄, x, and x̄, the proof substitutes these expressions into the optimized control and obtains its explicit form.Source: §A.1 · p. 36
• The optimal shift separates into position and momentum contributions, producing the stated KL-divergence bound.Source: §A.1 · p. 36
• A direct Gaussian calculation shows that the terminal law has the corresponding shifted initial distribution and exactly matches the derived KL bound.Source: §A.1 · p. 36

A.2 Proof of Lemma 3.14

• Lemma 3.14 follows from the identity ˙γ_t = ˙η_t and strict monotonicity of t ↦ η_t.Source: §A.2 · p. 37
• In the high friction case with α ⩾ 0, γ ⩾ √32β ⩾ 2ω = 2α/(3γ), the claim follows since 3γ2 ⩾ 96β ⩾ 2α.Source: §A.2 · p. 37
• For the remaining friction regimes, η_t ⩾ c0|ω| and c0 ⩾ 24 or c0 ⩾ 6 provide the claimed lower bounds.Source: §A.2 · p. 37
• The proof also reduces one lower bound to β ⩽ γ^2, using η̇_t ⩾ 0 and η_t ⩽ γ_t, with γ^2 ⩾ 32β in high friction.Source: §A.2 · p. 37

A.3 Strongly convex and semi-convex settings

• In the strongly convex setting, the proof completes Theorem 3.2 by combining the weakly convex bound for T ⩽ |ω|^-1 with the long-time bound for T ⩾ |ω|^-1.Source: §A.3 · pp. 37–38
• The semi-convex proof concludes by bounding the geometric factor (a^(n+1) − 1)/(a − 1)^(n+1) and using monotonicity of 1/(1 − exp(−c|ω|T)).Source: §A.3 · p. 38
• As δ ↘ 0, (A.1) gives d_(T−δ) = O(δ^(c′c0)), and the auxiliary process converges to (X_T, P_T) in L2.Source: §A.3 · p. 38

• Combining these estimates under h ≲ γ^-1 ∧ γ/β and sufficiently small c_0/A establishes ε_t ≤ 1/2, completing Lemma 4.6.Source: §B.1 · pp. 38–39
• Under sufficiently small c_0/A, the bounds on φ_t ensure ∥φ_t−I∥op can be made at most 1/2 when t ≲ c_0/(Ah).Source: §B.1 · p. 38
• For h ≲ γ^-1 ∧ γ/β and sufficiently small c_0/A, blockwise estimates give ∥N_t∥op ≪ η_t.Source: §B.1 · p. 39
• Using the positivity of M_t and the preceding control of N_t, the proof bounds the first two terms in Lemma B.3, yielding an overall bound β/γ_t + η_t when γ ≳ η_t.Source: §B.1 · p. 39

• The earlier computation in Lemma 4.5 provides expression (4.8) for t ∈ [t−, t+].Source: §B.2 · p. 40
• Applying Lemma B.3 after using Lemmas B.1 and 4.6 yields a coupling constant of O(βh/γt− + hη_p t+).Source: §B.2 · p. 40
• The proof concludes by checking the remaining relation through the differential equation for η_p.Source: §B.2 · p. 40

B.3 Integral estimates

• Lemma B.4 supplies auxiliary distance bounds for convex/high-friction and semi-convex/low-friction regimes, yielding e_T^2 ≲ h^2(Ē_w + Ē_s)^2.The estimates use different time thresholds and decay behaviors across the two regimes.Source: §B.3 · pp. 40–42
• The convex/high-friction proof splits the integrals over [0,T0], [T0,T1], and [T1,T], then combines the three regime-specific bounds.The thresholds are chosen so the relevant decay rates change between the intervals.Source: §B.3 · pp. 41, 43
• The semi-convex/low-friction proof uses the split [0,T0] and [T0,T], with the short-time contribution treated as negligible.The same integral-control strategy yields the overall bound in this regime.Source: §B.3 · pp. 42–43
• Lemma B.5 bounds the main KL term separately for strongly convex/high-friction, weakly convex/high-friction, and semi-convex/low-friction cases.The resulting bounds combine contributions involving Ē_w and Ē_s with coefficients depending on α, β, γ, h, T, and |ω|.Source: §B.3 · p. 43
• Combining Lemmas 4.10, B.4, and B.5 proves Theorem 4.1 after showing the remaining terms are controlled by C(α,β,γ,T) and do not dominate the final bound.The argument checks separate cases according to α and the relation between T and |ω|.Source: §B.3 · p. 44
• Lemma B.6 provides the exponential-integral estimates needed for positive and negative ω and for the late-time η-weighted regime.These estimates are invoked to control the interval contributions in the preceding case analyses.Source: §B.3 · pp. 44–45

C.1 Recursive error control

• Lipschitz conditions let errors along algorithm iterates be controlled by corresponding errors for the auxiliary process.Lemma C.2 then replaces averaged algorithm errors with auxiliary-process errors while preserving Theorem 4.1’s conclusion, up to absorbable extra terms.Source: §C.1 · pp. 45–46
• Change-of-measure and recursive-control lemmas bound auxiliary-process expectations using stationary expectations under bounded Hessian assumptions [AC26].The recursive bounds apply for sufficiently small p and yield error estimates involving dimension d and an initialization-dependent quantity B^2.Source: §C.1 · p. 47

C.1.1 Convex case

• In the convex case, the proof computes local errors, verifies Assumption C.1, and applies Theorem 4.1 through Lemma C.2 over T = Nh.The resulting bounds depend on Ax, Ap, and B, with B^2 containing a term proportional to C(α,β,γ,T)W^2_2(µ,π).Source: §C.1.1 · p. 47
• For the convex final ULMC step, h ≲ 1/β^1/2 ensures the cross-regularity condition, while h ≲ 1/γ supports the accompanying error bound.The Lipschitz constants scale as Lb,x ≲ β^2h^5/2/γ^1/2 and Lb,p ≲ βh^3/2/γ^1/2.Source: §C.1.1 · pp. 47–48
• When β^2h^3/γ ≲ 1, the convex analysis gives h ≲ 1/β^1/2 for γ ≍ √β and h ≲ 1/(β^1/2κ^1/6) for γ ≍ √α.The proof summary consists of checking Assumption C.1, controlling A^2, and deriving the final bound.Source: §C.1.1 · p. 48

C.1.2 Non-convex case

• In the non-convex case α < 0, C(α,β,γ,T) does not vanish as T grows, so the auxiliary process is instead initialized from µ.This shared initialization removes the initial Wasserstein-distance term from B.Source: §C.1.2 · p. 48
• Assuming log(1 + χ^2(µ0 ∥ π)) ≤ Cχ^2d, weak-triangle and data-processing inequalities combine the recursive bounds with the initialization term.For h ≲ γ^1/3/β^2/3, ULMC cross-regularity supplies the discretization bound, which is combined with ULD convergence ν_N → π.Source: §C.1.2 · p. 48

C.2 Proofs for Subsection 5.1

• ULMC satisfies Assumption C.1 in both strongly and weakly convex settings with high friction when h ≲ 1/(β^1/2κ) or h ≲ 1/(βT), respectively.These bounds follow by controlling the dominant Lipschitz-error terms in the two regimes.Source: §C.2 · p. 49
• In the strongly convex case, choosing h ≲ 1/(β^1/2κ^1/2) and T ≳ |ω|^-1 log(αW^2/ε^2) yields 2(µ,π) ⩽ ε^2.The argument applies Theorem 4.1 through Lemmas C.2 and C.6 and discards strong-error terms using Jensen’s inequality.Source: §C.2 · p. 49
• In the weakly convex case, taking T ≍ β^1/2W^2/ε^2 makes the first term at most ε^2, and h is then chosen to make the second term small.Source: §C.2 · p. 49

C.3 Proofs for Subsection 5.2

• Lemma C.7 bounds ULD movement when h ≲ 1/β^1/2 and γ ≲ √β.The resulting gradient-deviation bound is independent of the choice of u.Source: §C.3 · p. 50
• Lemma C.8 provides the midpoint bound for synchronously coupled RM–ULMC and ULD trajectories for every u ∈ [0,1].The same bound applies when u is sampled from the distribution in (5.1).Source: §C.3 · p. 50
• The proof of Lemma 5.5 derives weak and strong one-step errors by comparing RM–ULMC with ULD and invoking Lemmas C.7 and C.8.The weak-error argument uses the midpoint comparison, while the strong-error argument combines both auxiliary bounds.Source: §C.3 · pp. 50–51
• The double-midpoint implementation is justified by examining the momentum coordinate over the horizon [0,h] and matching its integral representation.This design rationale is introduced in Remark C.9 using the ULD momentum equation.Source: §C.3 · p. 51

We seek a discretization of the form

• Good strong error requires c(u) = (1 − e^(-γh))/γ =: c, making the coefficient independent of u.This condition aligns the middle term in the discretization difference under the same integral.Source: We seek a discretization of the form · p. 51
• Good weak error requires averaging c E_u∇V(ˆX++_t) against the corresponding exponentially weighted integral, forcing the law of u in (5.1).Analogous coefficient matching applies to the position coordinate.Source: We seek a discretization of the form · p. 51
• Under the assumptions of Lemma 5.5, Lemma C.10 states that Assumption C.1 holds for RM–ULMC in all cases when h satisfies β^(-2/3)T^(-1/3) ∧ β^(-3/4)T^(-1/2).Source: We seek a discretization of the form · p. 51
• In the strongly convex high-friction case, h ≲ 1/(β^1/2κ^1/3), while the semi-convex case permits h ≲ 1/β^1/2.The weakly convex case requires the additional condition h ≲ β^(-2/3)T^(-1/3) ∧ β^(-3/4)T^(-1/2).Source: We seek a discretization of the form · p. 52

C.3.1 Log-concave case

• Across both regimes, the proofs follow the outlines of earlier appendices while combining Theorem 4.1 with Lemmas C.2 and C.10.The log-concave proof treats strong and weak convexity separately, whereas the non-log-concave proof uses the n > N branch.Source: §C.3.1 · p. 52, §C.3.2 · p. 53
• In the strongly convex log-concave case, Theorem 4.1 yields the required bound when h ≲ 1/(β^1/2κ) and n < N.The proof then substitutes the admissible step size into the main estimate.Source: §C.3.1 · p. 52
• The strongly convex proof concludes with KL(µP_N ∥ π) ≤ ε^2 once T ≳ |ω|^-1 log(αW^2/ε^2), subject to the step-size condition.The restriction on ε arises from requiring h ≲ 1/(β^1/2κ).Source: §C.3.1 · p. 52
• In the weakly convex log-concave case, p ≲ 1 is ensured by h ≲ 1/(β^5/8T^1/4), after which the bound is simplified.The first term is made at most ε^2 by choosing T ≍ β^1/2W^2/ε^2, while h controls the second term under an additional constraint.Source: §C.3.1 · p. 52

C.3.2 Non-log-concave case

• The non-log-concave analysis applies Theorem 4.1 for n > N and obtains its final bound under h ≲ γ^1/3/(β^5/6T^1/3).This step-size condition ensures p ≲ 1, and the result then follows from the corresponding main estimate.Source: §C.3.2 · p. 53
• For the non-log-concave KL guarantee, T is chosen so Lemma 5.8 gives KL(µP_N ∥ π) ≲ ε^2, then Theorem 5.7 transfers this bound to the discretized chain.The argument uses β^1/2T ≳ 1 and concludes with Pinsker’s inequality.Source: §C.3.2 · p. 53
• For χ^2 error, choosing T ≍ α^-1/2 log(χ^2/ε^2) and applying Theorem 5.7 yields KL(µ ˆP_N−1 ˆP′ ∥ π) ≲ ε^2.Here χ^2 abbreviates χ^2(µ ∥ π), β^1/2T ≳ 1, and the final result follows from the weak triangle inequality.Source: §C.3.2 · p. 53