Improved Best-of-Both-Worlds Regret for Bandits with Delayed Feedback

We study the multi-armed bandit problem with adversarially chosen delays in the Best-of-Both-Worlds (BoBW) framework, which aims to achieve near-optimal performance in both stochastic and adversarial environments. While prior work has made progress toward this goal, existing algorithms suffer from significant gaps to the known lower bounds, especially in the stochastic settings. Our main contribution is a new algorithm that, up to logarithmic factors, matches the known lower bounds in each setting individually. In the adversarial case, our algorithm achieves regret of $\widetilde{O}(\sqrt{KT} + \sqrt{D})$, which is optimal up to logarithmic terms, where $T$ is the number of rounds, $K$ is the number of arms, and $D$ is the cumulative delay. In the stochastic case, we provide a regret bound which scale as $\sum_{i:\Delta_i>0}\left(\log T/\Delta_i\right) + \frac{1}{K}\sum \Delta_i \sigma_{max}$, where $\Delta_i$ is the sub-optimality gap of arm $i$ and $\sigma_{\max}$ is the maximum number of missing observations. To the best of our knowledge, this is the first BoBW algorithm to simultaneously match the lower bounds in both stochastic and adversarial regimes in delayed environment. Moreover, even beyond the BoBW setting, our stochastic regret bound is the first to match the known lower bound under adversarial delays, improving the second term over the best known result by a factor of $K$.

The impact of allocation strategies in subset learning on the expressive power of neural networks

In traditional machine learning, models are defined by a set of parameters, which are optimized to perform specific tasks. In neural networks, these parameters correspond to the synaptic weights. However, in reality, it is often infeasible to control or update all weights. This challenge is not limited to artificial networks but extends to biological networks, such as the brain, where the extent of distributed synaptic weight modification during learning remains unclear. Motivated by these insights, we theoretically investigate how different allocations of a fixed number of learnable weights influence the capacity of neural networks. Using a teacher-student setup, we introduce a benchmark to quantify the expressivity associated with each allocation. We establish conditions under which allocations have maximal or minimal expressive power in linear recurrent neural networks and linear multi-layer feedforward networks. For suboptimal allocations, we propose heuristic principles to estimate their expressivity. These principles extend to shallow ReLU networks as well. Finally, we validate our theoretical findings with empirical experiments. Our results emphasize the critical role of strategically distributing learnable weights across the network, showing that a more widespread allocation generally enhances the network's expressive power.

Delay as Payoff in MAB

In this paper, we investigate a variant of the classical stochastic Multi-armed Bandit (MAB) problem, where the payoff received by an agent (either cost or reward) is both delayed, and directly corresponds to the magnitude of the delay. This setting models faithfully many real world scenarios such as the time it takes for a data packet to traverse a network given a choice of route (where delay serves as the agent's cost); or a user's time spent on a web page given a choice of content (where delay serves as the agent's reward). Our main contributions are tight upper and lower bounds for both the cost and reward settings. For the case that delays serve as costs, which we are the first to consider, we prove optimal regret that scales as $\sum_{i:\Delta_i > 0}\frac{\log T}{\Delta_i} + d^*$, where $T$ is the maximal number of steps, $\Delta_i$ are the sub-optimality gaps and $d^*$ is the minimal expected delay amongst arms. For the case that delays serves as rewards, we show optimal regret of $\sum_{i:\Delta_i > 0}\frac{\log T}{\Delta_i} + \bar{d}$, where $\bar d$ is the second maximal expected delay. These improve over the regret in the general delay-dependent payoff setting, which scales as $\sum_{i:\Delta_i > 0}\frac{\log T}{\Delta_i} + D$, where $D$ is the maximum possible delay. Our regret bounds highlight the difference between the cost and reward scenarios, showing that the improvement in the cost scenario is more significant than for the reward. Finally, we accompany our theoretical results with an empirical evaluation.