Ben Grimmer
@profgrimmer.bsky.social
Assistant Professor @JohnsHopkinsAMS, Works in Mathematical Optimization,
Mostly here to share pretty maths/3D prints, sometimes sharing my research
Mostly here to share pretty maths/3D prints, sometimes sharing my research
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/
Below is the region (living in 6 dimensions) of optimal 3-step methods that happens to sit nicely in 3D 4/
November 20, 2025 at 3:46 AM
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/
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/
Together these are "if and only if"!! 3/
November 20, 2025 at 3:46 AM
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/
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/
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/
November 20, 2025 at 3:46 AM
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/
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/
A new paper out with TaeHo Yoon and Ernest Ryu:
We looked at the design of optimal fixed-point algorithms.
That is, seeking to approximately solve T(y)=y using as few evaluations of the operator T() as possible. Maximally efficient methods are "minimax optimal" 1/
We looked at the design of optimal fixed-point algorithms.
That is, seeking to approximately solve T(y)=y using as few evaluations of the operator T() as possible. Maximally efficient methods are "minimax optimal" 1/
November 20, 2025 at 3:46 AM
A new paper out with TaeHo Yoon and Ernest Ryu:
We looked at the design of optimal fixed-point algorithms.
That is, seeking to approximately solve T(y)=y using as few evaluations of the operator T() as possible. Maximally efficient methods are "minimax optimal" 1/
We looked at the design of optimal fixed-point algorithms.
That is, seeking to approximately solve T(y)=y using as few evaluations of the operator T() as possible. Maximally efficient methods are "minimax optimal" 1/
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/
I am excited about the future here :)
4/
November 18, 2025 at 2:59 PM
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/
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/
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/
November 18, 2025 at 2:59 PM
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/
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/
This demands a dynamic form of optimality, called subgame perfection. 2/
November 18, 2025 at 2:59 PM
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/
This demands a dynamic form of optimality, called subgame perfection. 2/
Lately, I have been obsessed with developing theoretically based optimization algorithms that actually attain the best practical performance.
Alas, the classic model of minimax optimal methods is overly conservative; it overfits to tune its worst-case.
We found a path forward 1/
Alas, the classic model of minimax optimal methods is overly conservative; it overfits to tune its worst-case.
We found a path forward 1/
November 18, 2025 at 2:59 PM
Lately, I have been obsessed with developing theoretically based optimization algorithms that actually attain the best practical performance.
Alas, the classic model of minimax optimal methods is overly conservative; it overfits to tune its worst-case.
We found a path forward 1/
Alas, the classic model of minimax optimal methods is overly conservative; it overfits to tune its worst-case.
We found a path forward 1/
Enjoyed being part of the Brin Mathematical Research Center's summer school on Scientific Machine Learning last week. Many very good talks and always nice to visit UMD!
August 14, 2025 at 8:29 PM
Enjoyed being part of the Brin Mathematical Research Center's summer school on Scientific Machine Learning last week. Many very good talks and always nice to visit UMD!
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
Enjoy: arxiv.org/abs/2508.06681
August 12, 2025 at 2:40 PM
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
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.
August 12, 2025 at 2:40 PM
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).
August 12, 2025 at 2:40 PM
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 :)
My 3D print of this cone is below :)
August 12, 2025 at 2:40 PM
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 :)
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.
August 12, 2025 at 2:40 PM
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.
📢 Excited to share a new paper with PhD student Thabo Samakhoana. Nonsmooth optimization often uses smoothings, nearby smooth functions or sets. Often chosen in an ad hoc fashion.
We do away with ad hoc, characterizing optimal smoothings for convex cones and sublinear functions
We do away with ad hoc, characterizing optimal smoothings for convex cones and sublinear functions
August 12, 2025 at 2:40 PM
📢 Excited to share a new paper with PhD student Thabo Samakhoana. Nonsmooth optimization often uses smoothings, nearby smooth functions or sets. Often chosen in an ad hoc fashion.
We do away with ad hoc, characterizing optimal smoothings for convex cones and sublinear functions
We do away with ad hoc, characterizing optimal smoothings for convex cones and sublinear functions
In honor of the fun I've had playing with this puzzle and property, a homemade, ocean-themed, ceramic p=4/3 norm ball. Enjoy!
August 5, 2025 at 2:37 PM
In honor of the fun I've had playing with this puzzle and property, a homemade, ocean-themed, ceramic p=4/3 norm ball. Enjoy!
As a cruel mathematician, I leave the task of verifying that the p=4/3 rotated appropriately fully and perfectly plugs a hole (equivalently has a perfect circle as a shadow) as an exercise to the reader.
The dual of this wonderful property is that the 4-norm hides a circle :)
The dual of this wonderful property is that the 4-norm hides a circle :)
August 5, 2025 at 2:37 PM
As a cruel mathematician, I leave the task of verifying that the p=4/3 rotated appropriately fully and perfectly plugs a hole (equivalently has a perfect circle as a shadow) as an exercise to the reader.
The dual of this wonderful property is that the 4-norm hides a circle :)
The dual of this wonderful property is that the 4-norm hides a circle :)
For good measure, one extra round of this physical verification process, adding a purple ball fully blocks our view of the green ball, entirely plugging the hole and saving our lives yet again.
August 5, 2025 at 2:37 PM
For good measure, one extra round of this physical verification process, adding a purple ball fully blocks our view of the green ball, entirely plugging the hole and saving our lives yet again.
Don't believe me? We can put another green p=4/3-norm ball in the glass. Looking from above, you cannot see any of the blue ball past the green one. It entirely plugs the hole!
August 5, 2025 at 2:37 PM
Don't believe me? We can put another green p=4/3-norm ball in the glass. Looking from above, you cannot see any of the blue ball past the green one. It entirely plugs the hole!
To demonstrate this, suppose my glass is the hole in our boat, we can plug it entirely by placing a blue 4/3-norm ball in the glass.
August 5, 2025 at 2:37 PM
To demonstrate this, suppose my glass is the hole in our boat, we can plug it entirely by placing a blue 4/3-norm ball in the glass.
The solution to cork the hole is surprisingly, radically simple:
Just put the p=4/3 norm ball in the hole.
Appropriately rotated, sending the direction (1,1,1)/sqrt{3} to (0,0,1).
Just put the p=4/3 norm ball in the hole.
Appropriately rotated, sending the direction (1,1,1)/sqrt{3} to (0,0,1).
August 5, 2025 at 2:37 PM
The solution to cork the hole is surprisingly, radically simple:
Just put the p=4/3 norm ball in the hole.
Appropriately rotated, sending the direction (1,1,1)/sqrt{3} to (0,0,1).
Just put the p=4/3 norm ball in the hole.
Appropriately rotated, sending the direction (1,1,1)/sqrt{3} to (0,0,1).
Yesterday I posted a maths puzzle that AIs all failed at (thanks for running the premium versions @xy-han.bsky.social and Ernest Ryu). The puzzle just needs elementary reasoning about p-norm balls (third row on my shelf below).
This thread gives the puzzle, solution, and a 3D printed demo :)
This thread gives the puzzle, solution, and a 3D printed demo :)
August 5, 2025 at 2:37 PM
Yesterday I posted a maths puzzle that AIs all failed at (thanks for running the premium versions @xy-han.bsky.social and Ernest Ryu). The puzzle just needs elementary reasoning about p-norm balls (third row on my shelf below).
This thread gives the puzzle, solution, and a 3D printed demo :)
This thread gives the puzzle, solution, and a 3D printed demo :)
My PhD students are awesome. They gave my fiancee(wife) and I this gorgeous cherry blossom card for our wedding and soon honeymoon in Japan <3
March 15, 2025 at 12:54 PM
My PhD students are awesome. They gave my fiancee(wife) and I this gorgeous cherry blossom card for our wedding and soon honeymoon in Japan <3
From top row to bottom, Figure 0 above has Schatten p-norms, Vector p-norms, Function p-norms, CVAR norms, Their duals, OWL norms, Their duals.
Figure 1 on the other side of my office has induced p->q matrix norm balls. p goes 1 to inf left to right. q goes 1 to inf bottom to top.
Figure 1 on the other side of my office has induced p->q matrix norm balls. p goes 1 to inf left to right. q goes 1 to inf bottom to top.
March 12, 2025 at 7:52 PM
From top row to bottom, Figure 0 above has Schatten p-norms, Vector p-norms, Function p-norms, CVAR norms, Their duals, OWL norms, Their duals.
Figure 1 on the other side of my office has induced p->q matrix norm balls. p goes 1 to inf left to right. q goes 1 to inf bottom to top.
Figure 1 on the other side of my office has induced p->q matrix norm balls. p goes 1 to inf left to right. q goes 1 to inf bottom to top.
As an early wedding present (happening this Saturday!), my dad made me a custom shelf to hold my collection of unit norm balls!
Rockafellar+Wets's thick textbook is included for reference.
Rockafellar+Wets's thick textbook is included for reference.
March 12, 2025 at 7:52 PM
As an early wedding present (happening this Saturday!), my dad made me a custom shelf to hold my collection of unit norm balls!
Rockafellar+Wets's thick textbook is included for reference.
Rockafellar+Wets's thick textbook is included for reference.