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Precise Regret Bounds for Log-loss via a Truncated Bayesian Algorithm
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Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use smooth truncation with a global sequential α-cover | Sequential minimax regret for a general expert class · General expert class, sequential design | sequential minimax regret upper bound: 2αT + log |G_α| | Helped |
| Use a Lipschitz parameter-space cover | Lipschitz parametric classes · Lipschitz parametric class, sequential design | sequential minimax regret upper bound: d log(2RLT/d + 1) + 2d | Helped |
| Restrict only to Lipschitz regularity, without stronger curvature assumptions | Lipschitz parametric classes · Lipschitz parametric class, sequential design | sequential minimax regret lower bound: d log(RLT/d) - d log 64 - d log log(RLT) | Mixed |
| Assume the Hessian of log f is uniformly bounded | Parametric prediction classes · Bounded-Hessian class, sequential design | sequential minimax regret upper bound: (d/2) log(2CR²T/d + 2) + d/2 + log 2 | Helped |
| Use generalized linear functions with f'(0) ≠ 0 over ℓ_s balls | Generalized linear functions f(⟨w,x⟩) · Generalized linear class under ℓ_s balls | sequential minimax regret lower bound: (d/2) log(T/d^((s+2)/s)) - O(d) | Mixed |
| Use sequential fat-shattering to bound global sequential cover size | Classes with sequential fat-shattering number of order α^-2 · Linear absolute-value class under ℓ_2 balls, d ≥ T | sequential minimax regret upper bound: Tilde O(T^(2/3)) | Helped |
| Compare the absolute-value linear class with the theorem's lower-bound construction | Linear absolute-value class under ℓ_2 balls · Linear absolute-value class under ℓ_2 balls, d ≥ T | sequential minimax regret: Ω(T^(2/3)) ≤ r^a_T(H_f) ≤ Tilde O(T^(2/3)) | Mixed |
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