Local extrema

While local extrema in one and two dimensions are often depicted as (local) minima/maxima, they “typically” are saddle-points in high dimensions. This is the topic of my second Bachelor thesis where I survey LeCun’s The Loss Surfaces of Multilayer Networks and Auffinger, Ben Arous, Černý’s Random Matrices and Complexity of Spin Glasses. The former paper relates the energy landscape (loss function to be minimised) of a toy model of neural networks to that of certain models dealt with in the mathematical physics literature (the latter paper). This paper even goes a step further by suggesting that the distribution of eigenvalues of the Hessian of critical points is a shifted semicircle. That is, the closer the value at a critical point is to the global minimum, the higher the expected fraction of directions in which our loss function increases (if it increases in all directions, it is a local minimum).

Neural networks

Large (deep) neural networks have lots of counterintuitive properties, of which I will list only few:

  • Adversarial vulnerability for any classifier shows fundamental upper bounds on the robustness of any classifier to perturbations, which provides a baseline to the maximal achievable robustness. When the latent space of the data distribution is high dimensional, our analysis shows that any classifier is vulnerable to very small perturbations.

MCMC algorithms

The analysis of the mixing time of Markov chains (how long do I have to run my MCMC algorithms until they spit out representative samples?) is a beautiful subject combining many different fields in mathematics, such as probability theory (random walks, couplings), geometry (Cheeger’s inequality, Ricci curvature), representation theory, spectral theory, and physics (electrical networks)/computer science (Boolean analysis).

Coming up with decent algorithms remains a particularly hard challenge in cases where the support of the measure we want to sample from is quite sparse; e.g. certain graphical representations of quantum spin systems.

One aspect, however, remains to be overlooked surprisingly frequently: The space of irreversible Markov chains is much bigger than that of reversible Markov chains and often one can find an irreversible one that mixes a lot faster, see e.g. ECMC and Markov Chain Monte Carlo Method without Detailed Balance.


Cauchy-Schwarz is both a very simple and a very powerful bound. I won’t even try to give an extensive list of possible applications, but concentrate instead on a classical one in probability theory.

First notice that Cauchy-Schwarz can simply be interpreted as a fancy way of saying that \(|\cos(x)|\leq 1\) for all real \(x\) by the following identity: \[|\langle a,b\rangle|=|\cos(\theta)||a||b|\leq|a||b|,\] where \(\theta=\angle(a,b)\) is the angle between \(a\) and \(b\).

On \(\mathbb R^n,n\gg1\), how far off is Cauchy–Schwarz “typically”?

  • The intuition is that in higher dimensions it becomes increasingly unlikely for two vectors to be approximately parallel (i.e. θ≈0 or θ≈π in which case Cauchy-Schwarz is sharp) since there are “too many directions”.
  • More rigorously, we may fix \(a=(1,0,\dots,0)\) and sample \(b\) from the Haar measure on the unit sphere.
  • Note that for \(n\gg1\) we can do this by sampling \(n\) iid centered normal variables \((x_i)_{i=1}^n\) with variance \(1\over n\).
    • This ensures that squares have expectation \(\mathbb E x_i^2 = \text{Var}(x_i)-(\mathbb Ex_i)^2 = {1\over n}\) and thus, by linearity of expectation we have that \(\mathbb E\|(x_i)_{i=1}^n\|_2=1\). It is not too hard to see that the variance of \(\|(x_i)_{i=1}^n\|_2\) vanishes as \(n\to\infty\).
  • Thus, we see that each (in particular the first) entry of \(b\) is of order \(O(n^{-{1\over2}})\) and thus \[\mathbb E|\langle a,b\rangle|=O(n^{-{1\over2}}).\]

How tight is \(|\mathbb Ef|^2\leq\mathbb E[|f|^2]\)

  • First note that this inequality can be seen as another manifestation of Cauchy–Schwarz (although the observation that the variance cannot be negative is a quicker way to see it):
    • Every probability measure \(\mathbb P\) induces an inner product on the space of random variables with finite second moment by setting \(\langle f,g\rangle:=\mathbb E f\bar g\).
    • Now apply Cauchy–Schwarz to the constant one function and an arbitrary other function to get \[|\langle 1,f\rangle|^2 = \cos(\theta)\langle f,f\rangle\leq \mathbb E[|f|^2].\]
    • We get another curious identity when trying to convert this additive correction into a multiplicative one by using \(\sin^2+\cos^2=1\): \[|\langle 1,f\rangle|^2=\langle f,f\rangle-\text{Var}f = \langle f,f\rangle\left(1-{\text{Var}f\over\langle f,f\rangle}\right) \Rightarrow {\text{Var}f\over\langle f,f\rangle}=\sin(\theta)^2.\]
  • The takeaway is:

    The smaller the variance (of \(f\)), the tighter the bound \(|\mathbb Ef|^2\leq\mathbb E[|f|^2]\).

  • Using some Poincaré-type inequalities we find another rule of thumb, namely that “all else equal”,

    the smoother \(f\), the smaller the variance of \(f\).