concentration-inequalities #

Shows how exponential Markov turns a tail event P(S≥t) into an optimization over λ of exp(-λt+ψS(λ)). The bottom-left plot animates the exponent f(λ) and highlights the minimizing λ*. The right panel visualizes cumulant-generating functions (ψ) and how independence makes ψ add across summands, while a quadratic (Hoeffding-style) upper bound illustrates the bounded-difference/sub-Gaussian principle that yields exponential tails.

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⏮◀◀▶▶STEP0.25x1xZOOM

t=0s

practical uses #

01.Designing high-probability error bounds for randomized algorithms and Monte Carlo estimates
02.Deriving sample-complexity guarantees in learning theory (generalization bounds)
03.Confidence intervals for sums/averages of bounded independent measurements (A/B tests, sensors)

technical notes #

Pure canvas 2D. Uses time-based sweep to move threshold t and re-optimize λ via coarse-to-fine grid search. CGF uses ψ(λ)=log cosh(aλ) for a bounded Rademacher variable; sum CGF scales with N and demonstrates additivity; quadratic proxy ψ≤λ^2 a^2/2 visualizes Hoeffding-type control. Blocky look via grid snapping and green-on-black palette.

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