Let \(E\) be a real normed vector space and \(f:E\to \mathbb R\) be differentiable everywhere with derivative \(D(x):E\to \mathbb R\). Suppose \(x\mapsto D(x)\) is \(C\)-Lipschitz in operator norm, where \(C\ge 0\). For all \(x,y\in E\), \(|f(y)-f(x)-D(x)(y-x)|\le C\| y-x\| ^2\).

Let \(E\) be a real normed vector space and \(f:E\to \mathbb R\) be differentiable everywhere with derivative \(D(x):E\to \mathbb R\). Suppose \(x\mapsto D(x)\) is \(C\)-Lipschitz in operator norm, where \(C\ge 0\). For \(x,v\in E\) and \(h\in \mathbb R\), \(|(f(x+hv)-f(x))/2-(h/2)D(x)v|\le (C/2)h^2\| v\| ^2\).

For \(n,m\ge 0\) and \(x\in \{ 0,1\} ^{n\times m}\), define \(S_m(x)_k=m^{-1/2}\sum _{l=1}^m(-1)^{x_{kl}}\), interpreting division by zero as zero. If \(x^{ij\leftarrow b}\) replaces bit \((i,j)\) by \(b\), then \(S_m(x^{ij\leftarrow b})=S_m(x)+\bigl((-1)^b-(-1)^{x_{ij}}\bigr)m^{-1/2}e_i\), where \(e_i\) is the \(i\)th coordinate vector.

For \(n,m\ge 0\) and \(x\in \{ 0,1\} ^{n\times m}\), define \(S_m(x)_k=m^{-1/2}\sum _{l=1}^m(-1)^{x_{kl}}\), interpreting division by zero as zero. Let \(f:\mathbb R^n\to \mathbb R\) have derivative family \(D(z)\) that is globally \(C\)-Lipschitz in operator norm for the maximum norm on \(\mathbb R^n\), with \(C\ge 0\). For any bit \((i,j)\), \(\left|\frac{f(S_m(x^{ij\leftarrow 0}))-f(S_m(x^{ij\leftarrow 1}))}{2}-\frac{D(S_m(x))e_i}{\sqrt m}\right|\le 2C/m\).