The setting described is summarized in Box A.

We consider contextual bandit problems with knapsacks [CBwK], a problem where at each round, a scalar reward is obtained and vector-valued costs are suffered. The learner aims to maximize the cumulative rewards while ensuring that the cumulative costs are lower than some predetermined cost constraints. We assume that contexts come from a continuous set, that costs can be signed, and that the expected reward and cost functions, while unknown, may be uniformly estimated -- a typical assumption in the literature. In this setting, total cost constraints had so far to be at least of order $T^{3/4}$, where $T$ is the number of rounds, and were even typically assumed to depend linearly on $T$. We are however motivated to use CBwK to impose a fairness constraint of equalized average costs between groups: the budget associated with the corresponding cost constraints should be as close as possible to the natural deviations, of order $\sqrt{T}$. To that end, we introduce a dual strategy based on projected-gradient-descent updates, that is able to deal with total-cost constraints of the order of $\sqrt{T}$ up to poly-logarithmic terms. This strategy is more direct and simpler than existing strategies in the literature. It relies on a careful, adaptive, tuning of the step size.

This work shows that a diverse collection of linear optimization methods, when run on general data, fail to overfit, despite lacking any explicit constraints or regularization: with high probability, their trajectories stay near the curve of optimal constrained solutions over the population distribution. This analysis is powered by an elementary but flexible proof scheme which can handle many settings, summarized as follows. Firstly, the data can be general: unlike other implicit bias works, it need not satisfy large margin or other structural conditions, and moreover can arrive sequentially IID, sequentially following a Markov chain, as a batch, and lastly it can have heavy tails. Secondly, while the main analysis is for mirror descent, rates are also provided for the Temporal-Difference fixed-point method from reinforcement learning; all prior high probability analyses in these settings required bounded iterates, bounded updates, bounded noise, or some equivalent. Thirdly, the losses are general, and for instance the logistic and squared losses can be handled simultaneously, unlike other implicit bias works. In all of these settings, not only is low population error guaranteed with high probability, but moreover low sample complexity is guaranteed so long as there exists any low-complexity near-optimal solution, even if the global problem structure and in particular global optima have high complexity.

We apply proof-theoretic techniques in answer Set Programming. The main results include: 1. A characterization of continuity properties of Gelfond-Lifschitz operator for logic program. 2. A propositional characterization of stable models of logic programs (without referring to loop formulas.