Maxim Bichuch

Yue Shen, Maxim Bichuch, Enrique Mallada

We consider the problem of learning an inner approximation of the region of attraction (ROA) of an asymptotically stable equilibrium point without an explicit model of the dynamics. Rather than leveraging approximate models with bounded uncertainty to find a (robust) invariant set contained in the ROA, we propose to learn sets that satisfy a more relaxed notion of containment known as recurrence. We define a set to be $τ$-recurrent (resp. $k$-recurrent) if every trajectory that starts within the set, returns to it after at most $τ$ seconds (resp. $k$ steps). We show that under mild assumptions a $τ$-recurrent set containing a stable equilibrium must be a subset of its ROA. We then leverage this property to develop algorithms that compute inner approximations of the ROA using counter-examples of recurrence that are obtained by sampling finite-length trajectories. Our algorithms process samples sequentially, which allow them to continue being executed even after an initial offline training stage. We further provide an upper bound on the number of counter-examples used by the algorithm, and almost sure convergence guarantees.

Maxim Bichuch, Zachary Feinstein

Financial options are fundamental to traditional markets, enabling strategies ranging from hedging to speculating. Yet, while the Automated Market Maker paradigm has revolutionized decentralized spot markets, no equivalent standard has emerged for on-chain options. Typical designs attempt to replicate centralized exchange mechanics, requiring high-frequency oracles and robust liquidation engines which may fail during stress events. This paper presents a design for amortizing perpetual options tailored to the operational and adversarial constraints of blockchain environments. Leveraging this primitive, we introduce a decentralized market framework with minimal consistency requirements. We demonstrate that this contract functions as a foundational risk primitive for DeFi, enabling applications such as endogenous collateralization and explicitly priced de-peg insurance, thereby showing that this design provides a layer for mutualizing tail risk across protocols without reliance on centralized clearing institutions.

Ashwin De Silva, Rahul Ramesh, Lyle Ungar et al.

Learning is a process which can update decision rules, based on past experience, such that future performance improves. Traditionally, machine learning is often evaluated under the assumption that the future will be identical to the past in distribution or change adversarially. But these assumptions can be either too optimistic or pessimistic for many problems in the real world. Real world scenarios evolve over multiple spatiotemporal scales with partially predictable dynamics. Here we reformulate the learning problem to one that centers around this idea of dynamic futures that are partially learnable. We conjecture that certain sequences of tasks are not retrospectively learnable (in which the data distribution is fixed), but are prospectively learnable (in which distributions may be dynamic), suggesting that prospective learning is more difficult in kind than retrospective learning. We argue that prospective learning more accurately characterizes many real world problems that (1) currently stymie existing artificial intelligence solutions and/or (2) lack adequate explanations for how natural intelligences solve them. Thus, studying prospective learning will lead to deeper insights and solutions to currently vexing challenges in both natural and artificial intelligences.