Between continuous dynamics and discrete decisions. Between simulation and reality. Between what a system predicts and what the world permits.
This space is a notebook for making those boundaries visible—and then moving them.
A map of the questions currently pulling focus.
Learning systems that can adapt under uncertainty without losing sight of the objective—or the cost of being wrong.
Agents that perceive, plan, and act inside worlds that push back—where latency, energy, and safety are first-class constraints.
Understanding the geometry of difficult objectives—and designing algorithms that navigate them with purpose.
Interfaces that turn high-dimensional systems into something a human can see, question, and steer.
Arjun Prakash · Naicheng He · Kaicheng Guo · Saket Tiwari · Ruo Yu Tao · Tyrone Serapio · Amy Greenwald · George Konidaris
As meaningful curvature directions disappear, a network loses the degrees of freedom it needs to learn the next task.
Deep networks can progressively lose plasticity in continual learning, becoming unable to fit new tasks without reinitialization. We identify Hessian spectral collapse at new-task initialization as a precursor to this failure: useful curvature directions vanish and gradient descent becomes ineffective. Rank-based trainability conditions connect this behavior to the loss-weighted Gram matrix and generalized Gauss–Newton curvature, motivating feature-rank preservation and L2 regularization to keep networks adaptable.
A GPU sculpture of rank collapse. One moving particle system loses degrees of freedom continuously—from free 3D motion, to a plane, to a line, to stillness.
A deliberately stylized metaphor for dimensionality—not a numerical reconstruction. The same 4,096 particles keep moving, but each collapse removes a direction permanently until the sequence is reset.
Arjun Prakash · Naicheng He · Denizalp Goktas · Jacob Makar-Limanov · Amy Greenwald
Four update rules carry the same initial states toward cycles, delay, or a Stackelberg equilibrium.
Actor–critic reinforcement learning is naturally a bilevel problem because the actor depends on a critic that is itself learning a best response. BLPO nests critic updates and gives the actor a hypergradient that accounts for how the critic changes. A Nyström approximation makes the required inverse-Hessian computation more stable, while the theory establishes convergence to local strong Stackelberg equilibria under a linear critic parameterization and experiments show competitive control performance.
An exact numerical rendering of Figure 1’s toy MDP dynamics: 25 uniformly spaced initializations evolving under four coupled update rules.
Trajectories use the supplied update equations, clipping, and arc-length resampling. Every trajectory head is a metaball: cycling stretches the field apart while convergence fuses all 25 at equilibrium. A click now injects momentum at that phase-space coordinate; particle repulsion, boundary collisions, and damping physically produce the next 25 initializations, which are then shared across all four rules for a fair replay. In panels (c) and (d), the final 12 display samples ease to the equilibrium exactly as supplied.
Denizalp Goktas · Arjun Prakash · Amy Greenwald
Two policies train inside the paper’s zero-sum reach–avoid experiment: one reaches the goal while the other learns to capture it.
Zero-sum Markov Stackelberg games model sequential decisions in continuous state and action spaces. We develop policy-gradient methods that learn from noisy trajectory estimates and converge in polynomial time for convex-concave games. Reach–avoid control arises naturally inside this framework: one player commits to a policy for reaching a target while the other learns a best response that blocks or captures it, with Stackelberg policies outperforming their Nash counterparts in the paper’s experiments.
Two Dubins cars learn a zero-sum reach–avoid game in self-play. Cyan must enter the goal; coral wins by intercepting it—or simply keeping it out until time expires.
Both two-layer policies are trained from scratch in this tab with REINFORCE, running baselines, and Adam. Alternating mode uses four reach batches followed by four defender batches. Nested mode holds the defender fixed for eight reach-policy inner updates, then takes one defender outer update against the adapted reach policy. Shaped and sparse rewards remain exactly zero-sum. Every reward/schedule combination keeps independent weights, optimizer state, counters, and timeline while this tab is open. The toroidal mathematics still uses shortest wrapped geometry, but only one canonical copy of the world is rendered.
Arjun Prakash · Nick James · Max Menzies · Gilad Francis
A rank statistic detects volatility shifts online, turning one continuous stochastic path into structurally distinct regimes.
Nonstationary financial time series can be simplified into recurring volatility regimes without imposing a parametric switching model. The method first uses change-point detection to partition returns into locally stationary segments, compares the resulting distributions, and clusters them into a learned number of characteristic behaviors. Those historical regimes then support an online trading strategy that matches current market behavior to the past and dynamically manages risk.
An endless Wiener stream changes volatility while a nonparametric Mood test watches its increments through a sliding 620-sample window. Nothing is pre-announced: a regime receives colour only after the rank statistic finds the break.
This uses the paper’s rank-based statistic on every admissible split of the current run. When the maximum crosses the selected threshold, the estimated split is confirmed and the detector restarts there—so the bright detection pulse naturally arrives after the quieter, dashed change-point estimate. Confirmed segments are coloured by their realized RMS volatility σ̂: cyan below 0.78, violet from 0.78 to 1.55, and coral above 1.55. These are legibility thresholds, not the paper’s downstream Wasserstein/spectral clusters. The window then keeps sliding forever. The paper’s full CPM implementation calibrates time-varying thresholds to ARL0 = 10,000.
Every useful system is an argument with reality.
Name the actual system, its state, its constraints, and the thing that would count as progress.
Create a model simple enough to interrogate and rich enough to fail in revealing ways.
Make behavior observable. A metric says what happened; an instrument helps explain why.
Move beyond the nominal case. Perturb, ablate, adversarially probe, and look for phase changes.
Close the loop. Let evidence rewrite the model, the interface, and sometimes the question itself.
A procedural field, rendered on your device. No video. No stock imagery. Every frame is synthesized from time, movement, and position.
WHAT IF THE INTERFACE
THOUGHT IN PUBLIC?
Designed and computed at the edge.
One page. Zero frameworks. Infinite states.