Our Results:Arbitrary Distribution ·CSP instanceΦ=(V,C)with solution set∑ Each variable v has distribution Du overo Individual falsifying probability:p=max Pr[c is False] D is uniform cEc →x=2andy>0.145 ·Degree::△=max#{c'∈C I vbl(c)nvbl(c')≠o} Already beats 0.142 from [JPV'20] Smoothness:K=max 2,maxmax Dv(q) VEV q.q'En Du(q') +ln(K+1)-V1n2(x+1)+61ln(k+1) ·lfpY·△≤1/x,then we can r= 9 ·sampleo∈∑under distributionΠDo ·in expected O(n log n) is a solution]Our Results: Arbitrary Distribution • CSP instance Φ = 𝑉, 𝐶 with solution set Σ • Each variable 𝑣 has distribution 𝐷𝑣 over Ω𝑣 • Individual falsifying probability: 𝑝 = max 𝑐∈𝐶 Pr[𝑐 is 𝐅𝐚𝐥𝐬𝐞] • Degree: Δ = max 𝑐∈𝐶 # 𝑐 ′ ∈ 𝐶 vbl 𝑐 ∩ vbl 𝑐 ′ ≠ ∅ • Smoothness: 𝜅 = max 2, max 𝑣∈𝑉 max 𝑞,𝑞 ′∈Ω𝑣 𝐷𝑣 𝑞 𝐷𝑣 𝑞 ′ • If 𝑝 𝛾 ⋅ Δ ≲ 1/𝜅, then we can • sample 𝜎 ∈ Σ under distribution ς𝐷𝑣 • in expected 𝑂 𝑛 log 𝑛 𝛾 = 3 + ln(𝜅 + 1) − ln2(𝜅 + 1) + 6 ln 𝜅 + 1 9 Pr 𝜎∼ς𝐷𝑣 [⋅∣ 𝜎 is a solution] 𝐷𝑣 is uniform ⇒ 𝜅 = 2 and 𝛾 > 0.145 Already beats 0.142 from [JPV’20]