Which LLM to Play? Convergence-Aware Online Model Selection with Time-Increasing Bandits

Web-based applications such as chatbots, search engines and news recommendations continue to grow in scale and complexity with the recent surge in the adoption of LLMs. Online model selection has thus garnered increasing attention due to the need to choose the best model among a diverse set while balancing task reward and exploration cost. Organizations faces decisions like whether to employ a costly API-based LLM or a locally finetuned small LLM, weighing cost against performance. Traditional selection methods often evaluate every candidate model before choosing one, which are becoming impractical given the rising costs of training and finetuning LLMs. Moreover, it is undesirable to allocate excessive resources towards exploring poor-performing models. While some recent works leverage online bandit algorithm to manage such exploration-exploitation trade-off in model selection, they tend to overlook the increasing-then-converging trend in model performances as the model is iteratively finetuned, leading to less accurate predictions and suboptimal model selections. In this paper, we propose a time-increasing bandit algorithm TI-UCB, which effectively predicts the increase of model performances due to finetuning and efficiently balances exploration and exploitation in model selection. To further capture the converging points of models, we develop a change detection mechanism by comparing consecutive increase predictions. We theoretically prove that our algorithm achieves a logarithmic regret upper bound in a typical increasing bandit setting, which implies a fast convergence rate. The advantage of our method is also empirically validated through extensive experiments on classification model selection and online selection of LLMs. Our results highlight the importance of utilizing increasing-then-converging pattern for more efficient and economic model selection in the deployment of LLMs.

Improved Bandits in Many-to-one Matching Markets with Incentive Compatibility

Two-sided matching markets have been widely studied in the literature due to their rich applications. Since participants are usually uncertain about their preferences, online algorithms have recently been adopted to learn them through iterative interactions. \citet{wang2022bandit} initiate the study of this problem in a many-to-one setting with \textit{responsiveness}. However, their results are far from optimal and lack guarantees of incentive compatibility. An extension of \citet{kong2023player} to this more general setting achieves a near-optimal bound for player-optimal regret. Nevertheless, due to the substantial requirement for collaboration, a single player's deviation could lead to a huge increase in its own cumulative rewards and an $O(T)$ regret for others. In this paper, we aim to enhance the regret bound in many-to-one markets while ensuring incentive compatibility. We first propose the adaptively explore-then-deferred-acceptance (AETDA) algorithm for responsiveness setting and derive an $O(N\min\left\{N,K\right\}C\log T/\Delta^2)$ upper bound for player-optimal stable regret while demonstrating its guarantee of incentive compatibility, where $N$ represents the number of players, $K$ is the number of arms, $T$ denotes the time horizon, $C$ is arms' total capacities and $\Delta$ signifies the minimum preference gap among players. This result is a significant improvement over \citet{wang2022bandit}. And to the best of our knowledge, it constitutes the first player-optimal guarantee in matching markets that offers such robust assurances. We also consider broader \textit{substitutable} preferences, one of the most general conditions to ensure the existence of a stable matching and cover responsiveness. We devise an online DA (ODA) algorithm and establish an $O(NK\log T/\Delta^2)$ player-pessimal stable regret bound for this setting.

Player-optimal Stable Regret for Bandit Learning in Matching Markets

The problem of matching markets has been studied for a long time in the literature due to its wide range of applications. Finding a stable matching is a common equilibrium objective in this problem. Since market participants are usually uncertain of their preferences, a rich line of recent works study the online setting where one-side participants (players) learn their unknown preferences from iterative interactions with the other side (arms). Most previous works in this line are only able to derive theoretical guarantees for player-pessimal stable regret, which is defined compared with the players' least-preferred stable matching. However, under the pessimal stable matching, players only obtain the least reward among all stable matchings. To maximize players' profits, player-optimal stable matching would be the most desirable. Though \citet{basu21beyond} successfully bring an upper bound for player-optimal stable regret, their result can be exponentially large if players' preference gap is small. Whether a polynomial guarantee for this regret exists is a significant but still open problem. In this work, we provide a new algorithm named explore-then-Gale-Shapley (ETGS) and show that the optimal stable regret of each player can be upper bounded by $O(K\log T/\Delta^2)$ where $K$ is the number of arms, $T$ is the horizon and $\Delta$ is the players' minimum preference gap among the first $N+1$-ranked arms. This result significantly improves previous works which either have a weaker player-pessimal stable matching objective or apply only to markets with special assumptions. When the preferences of participants satisfy some special conditions, our regret upper bound also matches the previously derived lower bound.

Online Influence Maximization under Decreasing Cascade Model

We study online influence maximization (OIM) under a new model of decreasing cascade (DC). This model is a generalization of the independent cascade (IC) model by considering the common phenomenon of market saturation. In DC, the chance of an influence attempt being successful reduces with previous failures. The effect is neglected by previous OIM works under IC and linear threshold models. We propose the DC-UCB algorithm to solve this problem, which achieves a regret bound of the same order as the state-of-the-art works on the IC model. Extensive experiments on both synthetic and real datasets show the effectiveness of our algorithm.

Best-of-three-worlds Analysis for Linear Bandits with Follow-the-regularized-leader Algorithm

The linear bandit problem has been studied for many years in both stochastic and adversarial settings. Designing an algorithm that can optimize the environment without knowing the loss type attracts lots of interest. \citet{LeeLWZ021} propose an algorithm that actively detects the loss type and then switches between different algorithms specially designed for different settings. However, such an approach requires meticulous designs to perform well in all settings. Follow-the-regularized-leader (FTRL) is another popular algorithm type that can adapt to different environments. This algorithm is of simple design and the regret bounds are shown to be optimal in traditional multi-armed bandit problems compared with the detect-switch type algorithms. Designing an FTRL-type algorithm for linear bandits is an important question that has been open for a long time. In this paper, we prove that the FTRL-type algorithm with a negative entropy regularizer can achieve the best-of-three-world results for the linear bandit problem with the tacit cooperation between the choice of the learning rate and the specially designed self-bounding inequality.

Improved Regret Bounds for Linear Adversarial MDPs via Linear Optimization

Learning Markov decision processes (MDP) in an adversarial environment has been a challenging problem. The problem becomes even more challenging with function approximation, since the underlying structure of the loss function and transition kernel are especially hard to estimate in a varying environment. In fact, the state-of-the-art results for linear adversarial MDP achieve a regret of $\tilde{O}(K^{6/7})$ ($K$ denotes the number of episodes), which admits a large room for improvement. In this paper, we investigate the problem with a new view, which reduces linear MDP into linear optimization by subtly setting the feature maps of the bandit arms of linear optimization. This new technique, under an exploratory assumption, yields an improved bound of $\tilde{O}(K^{4/5})$ for linear adversarial MDP without access to a transition simulator. The new view could be of independent interest for solving other MDP problems that possess a linear structure.

Simultaneously Learning Stochastic and Adversarial Bandits with General Graph Feedback

The problem of online learning with graph feedback has been extensively studied in the literature due to its generality and potential to model various learning tasks. Existing works mainly study the adversarial and stochastic feedback separately. If the prior knowledge of the feedback mechanism is unavailable or wrong, such specially designed algorithms could suffer great loss. To avoid this problem, \citet{erez2021towards} try to optimize for both environments. However, they assume the feedback graphs are undirected and each vertex has a self-loop, which compromises the generality of the framework and may not be satisfied in applications. With a general feedback graph, the observation of an arm may not be available when this arm is pulled, which makes the exploration more expensive and the algorithms more challenging to perform optimally in both environments. In this work, we overcome this difficulty by a new trade-off mechanism with a carefully-designed proportion for exploration and exploitation. We prove the proposed algorithm simultaneously achieves $\mathrm{poly} \log T$ regret in the stochastic setting and minimax-optimal regret of $\tilde{O}(T^{2/3})$ in the adversarial setting where $T$ is the horizon and $\tilde{O}$ hides parameters independent of $T$ as well as logarithmic terms. To our knowledge, this is the first best-of-both-worlds result for general feedback graphs.

Thompson Sampling for Bandit Learning in Matching Markets

The problem of two-sided matching markets has a wide range of real-world applications and has been extensively studied in the literature. A line of recent works have focused on the problem setting where the preferences of one-side market participants are unknown \emph{a priori} and are learned by iteratively interacting with the other side of participants. All these works are based on explore-then-commit (ETC) and upper confidence bound (UCB) algorithms, two common strategies in multi-armed bandits (MAB). Thompson sampling (TS) is another popular approach, which attracts lots of attention due to its easier implementation and better empirical performances. In many problems, even when UCB and ETC-type algorithms have already been analyzed, researchers are still trying to study TS for its benefits. However, the convergence analysis of TS is much more challenging and remains open in many problem settings. In this paper, we provide the first regret analysis for TS in the new setting of iterative matching markets. Extensive experiments demonstrate the practical advantages of the TS-type algorithm over the ETC and UCB-type baselines.

The Hardness Analysis of Thompson Sampling for Combinatorial Semi-bandits with Greedy Oracle

Thompson sampling (TS) has attracted a lot of interest in the bandit area. It was introduced in the 1930s but has not been theoretically proven until recent years. All of its analysis in the combinatorial multi-armed bandit (CMAB) setting requires an exact oracle to provide optimal solutions with any input. However, such an oracle is usually not feasible since many combinatorial optimization problems are NP-hard and only approximation oracles are available. An example (Wang and Chen, 2018) has shown the failure of TS to learn with an approximation oracle. However, this oracle is uncommon and is designed only for a specific problem instance. It is still an open question whether the convergence analysis of TS can be extended beyond the exact oracle in CMAB. In this paper, we study this question under the greedy oracle, which is a common (approximation) oracle with theoretical guarantees to solve many (offline) combinatorial optimization problems. We provide a problem-dependent regret lower bound of order $\Omega(\log T/\Delta^2)$ to quantify the hardness of TS to solve CMAB problems with greedy oracle, where $T$ is the time horizon and $\Delta$ is some reward gap. We also provide an almost matching regret upper bound. These are the first theoretical results for TS to solve CMAB with a common approximation oracle and break the misconception that TS cannot work with approximation oracles.