A link for those interested in reading 🤓
arxiv.org/abs/2512.10825
(4/4)

The "bad" news: Despite being *nearly* optimal, we show for fixed small dimensions that strictly better smoothings exist, approximating the max function more closely and attaining our lower bound. So LogSumExp is only nearly, not exactly, minimax optimal. (3/4)

LogSumExp is within 20% of a lower bound we derive on how good *any* similar smoothing can be. The proof just combines inequalities for smooth convex functions, no heavy machinery needed.

The good news: We aren't leaving much on the table by choosing logSumExp. (2/4)

For those interested in reading 🤓
arxiv.org/pdf/2511.14915

This polynomial characterization opens a lot of new directions in algorithm design. As a 3D printing enthusiast I was quick to want to visualize the set of optimal methods

Below is the region (living in 6 dimensions) of optimal 3-step methods that happens to sit nicely in 3D 4/

Our new work provides a complete description of all minimax-optimal methods. We give a set of polynomial equalities that every optimal method must satisfy ("H invariants") and similarly a needed set of polynomial ineq ("H certificates")

Together these are "if and only if"!! 3/

This is a classic type of problem; fixed points are a broad modelling tool, capturing, for example, gradient descent

In terms of algorithm design (my interest): In recent years the community pinned down an optimal method (Halpern) but showed that infinitely many others exist 2/

It's all performance estimation under the hood :)
That tool does wonders for conceptual framing

Some links for those interested 🤓
Smooth convex: arxiv.org/abs/2412.06731
Adaptive smooth convex: arxiv.org/abs/2510.21617
Nonsmooth convex: arxiv.org/abs/2511.13639

We have done a wide range of numerics for smooth, convex settings where our resulting subgame perfect gradient methods SPGM compete with state-of-the-art L-BFGS methods and beat existing adaptive gradient methods in iter and realtime.

I am excited about the future here :)
4/

In a series of works with the newest showing up on arxiv TODAY, we show that this strengthened standard is surprisingly attainable!

Today we proved a method of Drori and Teboulle 2014 is a subgame perfect subgradient method and designed a new, subgame perfect proximal method 3/

Rather than asking to do the best on the worst-case problem, we should be asking that, as it seems first-order information, our alg updates to do the best against the worst problem **with those gradients**
This demands a dynamic form of optimality, called subgame perfection. 2/

You'll have to read the paper if you want the maths defining these extremal smoothings for any sublinear function and convex cone. I now have a whole family of optimal smoothing Russian nesting dolls living in my office.

Enjoy: arxiv.org/abs/2508.06681

If instead, you wanted the optimal outer smoothings (ie, sets containing K), there is a similar spectrum of optimal smoothings being everything between the minimal and maximal sets shown below.

If we restrict to looking at inner smoothings (ie, subsets of K), it turns out there are infinitely many sets attaining the optimal level of smoothness. Our theory identifies that there is a minimal and maximal such smoothing, shown below (nesting dolls from before).

To do something more nontrivial, consider the exponential cone K={(x,y,z) | z >= y exp(x/y)}, which is foundational to geometric programming. The question: What is the smoothest set differing from this cone by at distance one anywhere?
My 3D print of this cone is below :)

For example, you could invent many smoothings of the two-norm (five given below). In this case, the Moreau envelope gives the optimal outer smoothing. If you wanted the best smoothing of the second-order cone (the epigraph of the two-norm) a different smoothing is optimal.

Oh, that’s so satisfying! I stopped at the 4-norm ball thinking I had the solution as it fits the hole like a pot lid (has a perfect circle as an intersection).