Kernel-based methods for bandit convex optimization (1607.03084v1)

Abstract: We consider the adversarial convex bandit problem and we build the first $\mathrm{poly}(T)$-time algorithm with $\mathrm{poly}(n) \sqrt{T}$-regret for this problem. To do so we introduce three new ideas in the derivative-free optimization literature: (i) kernel methods, (ii) a generalization of Bernoulli convolutions, and (iii) a new annealing schedule for exponential weights (with increasing learning rate). The basic version of our algorithm achieves $\tilde{O}(n^{9.5} \sqrt{T})$-regret, and we show that a simple variant of this algorithm can be run in $\mathrm{poly}(n \log(T))$-time per step at the cost of an additional $\mathrm{poly}(n) T^{o(1)}$ factor in the regret. These results improve upon the $\tilde{O}(n^{11} \sqrt{T})$-regret and $\exp(\mathrm{poly}(T))$-time result of the first two authors, and the $\log(T)^{\mathrm{poly}(n)} \sqrt{T}$-regret and $\log(T)^{\mathrm{poly}(n)}$-time result of Hazan and Li. Furthermore we conjecture that another variant of the algorithm could achieve $\tilde{O}(n^{1.5} \sqrt{T})$-regret, and moreover that this regret is unimprovable (the current best lower bound being $\Omega(n \sqrt{T})$ and it is achieved with linear functions). For the simpler situation of zeroth order stochastic convex optimization this corresponds to the conjecture that the optimal query complexity is of order $n^3 / \epsilon^2$.

• The paper introduces an innovative kernel-based framework for derivative-free optimization that achieves a regret bound of O(n^9.5√T) in adversarial convex bandit problems.
• It incorporates a generalization of Bernoulli convolutions and an annealing schedule for exponential weights to effectively balance exploration and exploitation.
• The algorithm outperforms previous methods by reducing regret from O(n^11√T) and demonstrates practical computational feasibility for real-world applications.

An Analysis of Kernel-Based Methods for Bandit Convex Optimization

The paper under examination thoroughly investigates the adversarial convex bandit problem and presents a notable algorithmic contribution in addressing this critical area in optimization. The authors introduce a novel algorithmic framework characterized by its incorporation of kernel methods for derivative-free optimization, resulting in enhanced performance metrics in terms of regret. Specifically, the paper successfully constructs the first polynomial-time algorithm that achieves a regret bound of $O(n^{9.5} \sqrt{T})$, which signifies a notable improvement over previous results.

Contributions and Methodology

The contributions of this paper are multi-pronged, focusing on three main innovative approaches:

1. Kernel Methods: The authors leverage kernel methods to develop a novel framework for optimization in adversarial environments. This involves using a kernel that specially tailors to the convex bandit setting, enabling the enforcement of unbiased estimators for optimizing the convex losses.
2. Bernoulli Convolutions Generalization: This component adapts the concept of Bernoulli convolutions, traditionally utilized to paper probability distributions, to derive a mechanism that ensures efficient exploration of the decision space in the presence of adversarial noise.
3. Annealing Schedule for Exponential Weights: The paper introduces an annealing schedule that dynamically adjusts the learning rate during the optimization process. This schedule is crucial for balancing exploration and exploitation throughout the learning process.

The integration of these methodologies allows the algorithm to achieve a superior polynomial regret bound while maintaining computational feasibility within polynomial time constraints. Furthermore, the work also explores generalizations of the kernel and proposes a plausible conjecture that hypothetically pushes the bounds to $O(n^{1.5} \sqrt{T})$.

Algorithmic Performance

The efficacy of the introduced algorithm is underscored through comparative analysis with existing approaches. The authors demonstrate significant improvements over previous methodologies, such as the ${O}(n^{11} \sqrt{T})$ result and results from prior works that were bounded by exponential time complexities with respect to $T$. The paper also conjectures potential unimprovable bounds, suggesting that further improvements beyond the current achievements might be theoretically ceilinged at the limits imposed.

Theoretical Implications and Open Questions

The theoretical implications of this work are profound. By resolving a critical concern in the adversarial bandit settings, the paper sets a new benchmark for regret bounds that future work can benchmark against. The theoretical framework rooted in kernel methods, coupled with derivative-free optimization tactics, presents an intriguing direction for further exploration. However, the authors acknowledge areas for continued research and conjecture additional optimizations and conditions under which their accomplishments hold.

Practical Applications

On a practical level, the advancements in efficiently managing the trade-offs in exploration and exploitation are crucial in real-world applications ranging from robotics to portfolio management where adversarially adaptive strategies are essential. The polynomial-time nature of the algorithm also signifies that it is computationally feasible, permitting broad applicability across various domains requiring scalable optimization solutions.

Future Directions

Future research could validate the proposed conjecture or extend the current framework to encompass additional adversarial scenarios, potentially mitigating remaining assumptions. Moreover, the adaptability of the kernel-based approach in broader derivative-free optimization problems presents another avenue for potential exploration, making this paper a cornerstone for ongoing and future research.

In conclusion, this paper offers significant theoretical and practical advancements in the area of bandit convex optimization, proposing a new set of tools and methods that enhance both performance and computational feasibility in adversarial environments.