It has been observed that transformers with greater depth (that is, more layers) have more capabilities, but can we establish formally which capabilities are gained? We answer this question with a theoretical proof followed by an empirical study. First, we consider transformers that round to fixed precision except inside attention. We show that this subclass of transformers is expressively equivalent to the programming language $\textsf{C}$-$\textsf{RASP}$ and this equivalence preserves depth. Second, we prove that deeper $\textsf{C}$-$\textsf{RASP}$ programs are more expressive than shallower $\textsf{C}$-$\textsf{RASP}$ programs, implying that deeper transformers are more expressive than shallower transformers (within the subclass mentioned above). The same is also proven for transformers with positional encodings (like RoPE and ALiBi). These results are established by studying a temporal logic with counting operators equivalent to $\textsf{C}$-$\textsf{RASP}$. Finally, we provide empirical evidence that our theory predicts the depth required for transformers without positional encodings to length-generalize on a family of sequential dependency tasks.

Dynamics of interacting systems in engineering, society, and nature often evolve over latent networks that govern which entities can interact. We study the problem of inferring these networks from event-based observations, which arise naturally in finance, seismology, and neuroscience. While there is substantial algorithmic work addressing this important problem, theoretical results are scarce. In this paper we ask the following fundamental question: what is the minimum time that one must observe the dynamics in order to exactly recover the underlying network, as a function of the number $d$ of interacting entities? For a class of stationary Hawkes processes with sparse, weak interactions, we prove that an observation time of order $\log d$ is sufficient and necessary. For the upper bound we construct a two-stage estimator that uses clipped and binned event data for screening, followed by a least-squares refinement, and apply concentration bounds derived from the Poisson cluster representation. For the lower bound we combine Fano's inequality with Jacod's Girsanov formula for point processes on a suitable subclass of networks.

In addition to CYCLIP described in 2, we train two more instantiations of it by keeping either of the two consistency regularizers active in the loss objective (Eq. The instantiation trained by setting λ1 = 0and λ2 = 0.5is termed as C-CYCLIP as only cross-modal consistency regularizer term is added to the loss objective. Similarly, we get I-CYCLIP where only in-modal consistency regularizer is added to the loss by setting λ1 = 0.5 and λ2 = 0. We evaluate C-CYCLIP and I-CYCLIP on most of the experiments discussed in the main text to understand their zero-shot transfer ability on standard datasets and robustness to natural distribution shifts. A.1 Zero-shot Transfer Table 7 presents our results of the zero-shot transfer experiment described in 3.1. We find that CYCLIP outperforms its sub-variants and the CLIP model on the ImageNet1K dataset.

We study learning under regime variation, where the learner, its memory state, and the evaluative conditions may evolve over time. This paper is a foundational and structural contribution: its goal is to define the core learning-theoretic objects required for such settings and to establish their first theorem-supporting consequences. The paper develops a regime-varying framework centered on admissible transport, protected-core preservation, and evaluator-aware learning evolution. It records the immediate closure consequences of admissibility, develops a structural obstruction argument for faithful fixed-ontology reduction in genuinely multi-regime settings, and introduces a protected-stability template together with explicit numerical and symbolic witnesses on controlled subclasses, including convex and deductive settings. It also establishes theorem-layer results on evaluator factorization, morphisms, composition, and partial kernel-level alignment across semantically commensurable layers. A worked two-regime example makes the admissibility certificate, protected evaluative core, and regime-variation cost explicit on a controlled subclass. The symbolic component is deliberately restricted in scope: the paper establishes a first kernel-level compatibility result together with a controlled monotonic deductive witness. The manuscript should therefore be read as introducing a structured learning-theoretic framework for regime-varying learning together with its first theorem-supporting layer, not as a complete quantitative theory of all learning systems.

Decentralized (PO)MDPs provide an expressive framework for sequential decision making in a multiagent system. Given their computational complexity, recent research has focused on tractable yet practical subclasses of Dec-POMDPs. We address such a subclass called CDec-POMDP where the collective behavior of a population of agents affects the joint-reward and environment dynamics. Our main contribution is an actor-critic (AC) reinforcement learning method for optimizing CDec-POMDP policies. Vanilla AC has slow convergence for larger problems. To address this, we show how a particular decomposition of the approximate action-value function over agents leads to effective updates, and also derive a new way to train the critic based on local reward signals. Comparisons on a synthetic benchmark and a real world taxi fleet optimization problem show that our new AC approach provides better quality solutions than previous best approaches.

Most existing approaches for dealing with noisy labels broadly fall into two categories: noise-modeling-based methods and memorization-effects-based methods.