Causal Bandits with General Causal Models and Interventions
(2403.00233)Abstract
This paper considers causal bandits (CBs) for the sequential design of interventions in a causal system. The objective is to optimize a reward function via minimizing a measure of cumulative regret with respect to the best sequence of interventions in hindsight. The paper advances the results on CBs in three directions. First, the structural causal models (SCMs) are assumed to be unknown and drawn arbitrarily from a general class $\mathcal{F}$ of Lipschitz-continuous functions. Existing results are often focused on (generalized) linear SCMs. Second, the interventions are assumed to be generalized soft with any desired level of granularity, resulting in an infinite number of possible interventions. The existing literature, in contrast, generally adopts atomic and hard interventions. Third, we provide general upper and lower bounds on regret. The upper bounds subsume (and improve) known bounds for special cases. The lower bounds are generally hitherto unknown. These bounds are characterized as functions of the (i) graph parameters, (ii) eluder dimension of the space of SCMs, denoted by $\operatorname{dim}(\mathcal{F})$, and (iii) the covering number of the function space, denoted by ${\rm cn}(\mathcal{F})$. Specifically, the cumulative achievable regret over horizon $T$ is $\mathcal{O}(K d{L-1}\sqrt{T\operatorname{dim}(\mathcal{F}) \log({\rm cn}(\mathcal{F}))})$, where $K$ is related to the Lipschitz constants, $d$ is the graph's maximum in-degree, and $L$ is the length of the longest causal path. The upper bound is further refined for special classes of SCMs (neural network, polynomial, and linear), and their corresponding lower bounds are provided.
Overview
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The paper introduces a generalized framework for causal bandit algorithms that operates without prior knowledge of the underlying Structural Causal Models (SCMs) and encompasses a broader range of interventions.
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It presents a detailed regret analysis, establishing improved upper and lower bounds that take into account the complexity of the causal graph and the SCM function class.
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Two algorithms, General Causal Bandit Upper Confidence Bound (GCB-UCB) and General Causal Bandit Thompson Sampling (GCB-TS), are proposed for both frequentist and Bayesian settings.
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The research highlights the theoretical and practical implications of its findings, suggesting potential for significantly enhanced causal bandit algorithms for real-world applications.
Understanding Regret in Causal Bandits with General Causal Models and Interventions
Overview
The realm of causal bandit algorithms presents a framework for optimizing interventions in dynamic systems by leveraging causal inference. This framework opens avenues for a wide range of applications, from robotics and gene interaction networks to drug discovery and advertising platforms. Traditional approaches in causal bandits often operate under considerable restrictions, such as known causal structure and simplistic intervention models. In stark contrast, this paper proposes a model that accommodates causal bandits under far more generalized conditions, specifically addressing unknown Structural Causal Models (SCMs), generalized soft interventions, and providing a comprehensive analysis of regret bounds.
Key Contributions
- Generalized Setting: The paper sets the stage by introducing a generalized causal bandit framework (GCB) that operates without prior knowledge of the underlying SCMs. It also embraces a broad spectrum of interventions, ranging from soft to granular, extending beyond the atomic and hard interventions commonly discussed in the literature.
- Regret Analysis: A significant portion of the analysis revolves around establishing both upper and lower bounds on regret, offering insights into the algorithm's performance over time. Notably, the paper improves upon known bounds for special cases while also uncovering new lower bounds for general conditions.
- Function Space Complexity: The paper brings to light how regret bounds are influenced by intrinsic measures of the SCM's class of functions, like the eluder dimension and the covering number. These measures reflect the functional complexity and its potential for over-fitting.
- Graph Structural Parameters: Unlike previous works, the regret bounds delineated in this research exhibit only a logarithmic dependency on the graph's size. Additionally, the results reveal an exponential relationship with the graph's maximum causal path length (L) and a polynomial relation with the maximum in-degree (d).
- Algorithmic Approach: Two algorithms, General Causal Bandit Upper Confidence Bound (GCB-UCB) and General Causal Bandit Thompson Sampling (GCB-TS), are meticulously outlined for both frequentist and Bayesian settings.
Theoretical Implications
The paper's analytical contributions are multidimensional. It provides a robust framework for understanding the interaction between the structure of the causal graph, the complexity of the SCM function class, and the nature of interventions. The regret bounds, characterized by the graph and function space parameters, elucidate the impact of causal depth and intervention granularity. Furthermore, by specializing the results for linear, polynomial, and neural network SCMs, the paper offers tailored insights into specific application settings.
Practical Implications
Practically, this research outlines the path for designing more effective causal bandit algorithms capable of operating in complex environments where the causal structure is unknown or partially known. The detailed regret analysis underscores the importance of considering both the causal graph's structure and the function space's complexity in designing intervention strategies. This implies that in real-world scenarios, where interventions can be costly or consequential, having finely-tuned algorithms that can navigate these complexities efficiently is invaluable.
Future Directions
Looking ahead, several questions remain open. For instance, bridging the gap highlighted between the lower and upper bounds in specific scenarios like linear and neural network SCMs warrants further exploration. Moreover, extending these findings to encompass more complex and high-dimensional causal structures represents a natural progression of this work. Lastly, there is room to explore the application and adaptation of the GCB framework in real-world scenarios across various fields.
Conclusion
In summary, this paper advances our understanding of causal bandits by introducing a more general framework that accommodates unknown structural causal models and a broader definition of interventions. It meticulously explores the impact of these generalizations on regret bounds and outlines algorithms adapted to this broader context. Future research in this direction holds the potential to significantly enhance the capabilities of causal bandit algorithms, paving the way for more nuanced and effective real-world applications.
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