We initiate a resource-aware theory of \textit{language generation in the limit} under the minimal constraint of space efficiency. In our framework, a learner observes an adversarial positive stream from a target language $K$ and must eventually output a hallucination-free hypothesis language $L \subseteq K$ while omitting at most $\Delta$ strings of $K$. We focus on $\mathcal{C}_{s,k}$, the collection of languages recognized by DFAs with at most $s$ states over an alphabet of size $k$, as the natural hypothesis class for memory-bounded learners. In the exponential-space regime, we prove that a learner can exactly identify the target $K$. Under a stricter memory budget, we characterize the strongest possible generation guarantees. In particular, we present a streaming algorithm using $\mathrm{poly}(s,k)$ space that converges to a hypothesis with generation gap $\Delta = O(k^{2s-2})$. Moreover, the learned hypothesis captures every string in $K$ of length at least $2s-1$. We complement this result with a near-matching lower bound through a reduction from a standard communication complexity problem. Specifically, achieving generation gap $\Delta \le k^{(1-\varepsilon)s}$ requires $k^{\Omega(\varepsilon s)}$ memory. Together, these results reveal a sharp transition between polynomial-space generation and exponential-space exact identification.

This paper studies the problem of learning computable functions in the limit by extending Gold’s inductive inference framework to incorporate \textit{computational observations} and \textit{restricted input sources}. Complimentary to the traditional Input-Output Observations, we introduce Time-Bound Observations, and Policy-Trajectory Observations to study the learnability of general recursive functions under more realistic constraints. While input-output observations do not suffice for learning the class of general recursive functions in the limit, we overcome this learning barrier by imposing computational complexity constraints or supplementing with approximate time-bound observations. Further, we build a formal framework around observations of \textit{computational agents} and show that learning computable functions from policy trajectories reduces to learning rational functions from input and output, thereby revealing interesting connections to finite-state transducer inference. On the negative side, we show that computable or polynomial-mass characteristic sets cannot exist for the class of linear-time computable functions even for policy-trajectory observations.

In this work, we investigate the effect of momentum on the optimisation trajectory of gradient descent. We leverage a continuous-time approach in the analysis of momentum gradient descent with step size $\gamma$ and momentum parameter $\beta$ that allows us to identify an intrinsic quantity $\lambda = \frac{ \gamma }{ (1 - \beta)^2 }$ which uniquely defines the optimisation path and provides a simple acceleration rule. When training a $2$-layer diagonal linear network in an overparametrised regression setting, we characterise the recovered solution through an implicit regularisation problem. We then prove that small values of $\lambda$ help to recover sparse solutions. Finally, we give similar but weaker results for stochastic momentum gradient descent. We provide numerical experiments which support our claims.