Lecture 2: Matrix Chernoff bounds

Lecturer : David P . Williamson Scribe : Faisal Alkaabneh

Today we will look at a matrix analog of the standard scalar Chernoff bounds. This matrix analog will be used in the next lecture when we talk about graph sparsification. While we’re more interested in the application of the theorem than its proof, it’s still useful to see the similarities and the differences of moving from the proof of the result for scalars to the same result for matrices. Sc...

Lecture 4 : Concentration and Matrix Multiplication

Today, we will continue with our discussion of scalar and matrix concentration, with a discussion of the matrix analogues of Markov’s, Chebychev’s, and Chernoff’s Inequalities. Then, we will return to bounding the error for our approximating matrix multiplication algorithm. We will start with using Hoeffding-Azuma bounds from last class to get improved Frobenius norm bounds, and then (next time...

Lecture 15: Chernoff bounds and Sequential detection

1 Chernoff Bounds 1.1 Bayesian Hypothesis Test A test using log-likelihood ratio statistic has the form, T (Y ) = logL(Y ) T τ. (1) Bound-1: The probability of error Pe is bounded as, Pe ≤ (π0 + π1e )eμT,0(s0)−s0τ , (2) where μT,0(s) = logE0[e ], and μ ′ T,0(s0) = τ . Bound-2: ∀ s ∈ [0, 1], Pe ≤ max(π0, π1e )eμT,0(s)−sτ . (3) Derivation of the above bound: Consider, Pe = π0P0(Γ1) + π1P1(Γ0), = ...

Lecture 03 : Chernoff Bounds and Intro to Spectral Graph Theory 3 1 . 1 Hoeffding ’ s Inequality

Orie 6334 Spectral Graph Theory Lecture 21

Just like matrix Chernoff bounds were a generalization of scalar Chernoff bounds, the multiplicative weights algorithm can be generalized to matrices. Recall that in the setup for the multiplicative weight update algorithm, we had a sequence of time steps t = 1, . . . , T ; in each time step t, we made a decision i ∈ {1...N} and got a value vt(i) ∈ [0, 1]. After we made a decision in time step ...