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Register a new userOptimal Gradient-based Algorithms for Non-concave Bandit Optimization
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Bandit problems with linear or concave reward have been extensively studied, but relatively few works have studied bandits with non-concave reward. This work considers a large family of bandit problems where the unknown underlying reward function is non-concave, including the low-rank generalized linear bandit problems and two-layer neural network with polynomial activation bandit problem. For the low-rank generalized linear bandit problem, we provide a minimax-optimal algorithm in the dimension, refuting both conjectures in [LMT21, JWWN19]. Our algorithms are based on a unified zeroth-order optimization paradigm that applies in great generality and attains optimal rates in several structured polynomial settings (in the dimension). We further demonstrate the applicability of our algorithms in RL in the generative model setting, resulting in improved sample complexity over prior approaches. Finally, we show that the standard optimistic algorithms (e.g., UCB) are sub-optimal by dimension factors. In the neural net setting (with polynomial activation functions) with noiseless reward, we provide a bandit algorithm with sample complexity equal to the intrinsic algebraic dimension. Again, we show that optimistic approaches have worse sample complexity, polynomial in the extrinsic dimension (which could be exponentially worse in the polynomial degree).
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Bandit Convex Optimization (BCO) is a fundamental framework for modeling sequential decision-making with partial information, where the only feedback available to the player is the one-point or two-point function values. In this paper, we investigate BCO in non-stationary environments and choose the emph{dynamic regret} as the performance measure, which is defined as the difference between the cumulative loss incurred by the algorithm and that of any feasible comparator sequence. Let $T$ be the time horizon and $P_T$ be the path-length of the comparator sequence that reflects the non-stationarity of environments. We propose a novel algorithm that achieves $O(T^{3/4}(1+P_T)^{1/2})$ and $O(T^{1/2}(1+P_T)^{1/2})$ dynamic regret respectively for the one-point and two-point feedback models. The latter result is optimal, matching the $Omega(T^{1/2}(1+P_T)^{1/2})$ lower bound established in this paper. Notably, our algorithm is more adaptive to non-stationary environments since it does not require prior knowledge of the path-length $P_T$ ahead of time, which is generally unknown.
We study the problem of corralling stochastic bandit algorithms, that is combining multiple bandit algorithms designed for a stochastic environment, with the goal of devising a corralling algorithm that performs almost as well as the best base algorithm. We give two general algorithms for this setting, which we show benefit from favorable regret guarantees. We show that the regret of the corralling algorithms is no worse than that of the best algorithm containing the arm with the highest reward, and depends on the gap between the highest reward and other rewards.
Performance of machine learning algorithms depends critically on identifying a good set of hyperparameters. While recent approaches use Bayesian optimization to adaptively select configurations, we focus on speeding up random search through adaptive resource allocation and early-stopping. We formulate hyperparameter optimization as a pure-exploration non-stochastic infinite-armed bandit problem where a predefined resource like iterations, data samples, or features is allocated to randomly sampled configurations. We introduce a novel algorithm, Hyperband, for this framework and analyze its theoretical properties, providing several desirable guarantees. Furthermore, we compare Hyperband with popular Bayesian optimization methods on a suite of hyperparameter optimization problems. We observe that Hyperband can provide over an order-of-magnitude speedup over our competitor set on a variety of deep-learning and kernel-based learning problems.
Many applications require a learner to make sequential decisions given uncertainty regarding both the systems payoff function and safety constraints. In safety-critical systems, it is paramount that the learners actions do not violate the safety constraints at any stage of the learning process. In this paper, we study a stochastic bandit optimization problem where the unknown payoff and constraint functions are sampled from Gaussian Processes (GPs) first considered in [Srinivas et al., 2010]. We develop a safe variant of GP-UCB called SGP-UCB, with necessary modifications to respect safety constraints at every round. The algorithm has two distinct phases. The first phase seeks to estimate the set of safe actions in the decision set, while the second phase follows the GP-UCB decision rule. Our main contribution is to derive the first sub-linear regret bounds for this problem. We numerically compare SGP-UCB against existing safe Bayesian GP optimization algorithms.
Most bandit policies are designed to either minimize regret in any problem instance, making very few assumptions about the underlying environment, or in a Bayesian sense, assuming a prior distribution over environment parameters. The former are often too conservative in practical settings, while the latter require assumptions that are hard to verify in practice. We study bandit problems that fall between these two extremes, where the learning agent has access to sampled bandit instances from an unknown prior distribution $mathcal{P}$ and aims to achieve high reward on average over the bandit instances drawn from $mathcal{P}$. This setting is of a particular importance because it lays foundations for meta-learning of bandit policies and reflects more realistic assumptions in many practical domains. We propose the use of parameterized bandit policies that are differentiable and can be optimized using policy gradients. This provides a broadly applicable framework that is easy to implement. We derive reward gradients that reflect the structure of bandit problems and policies, for both non-contextual and contextual settings, and propose a number of interesting policies that are both differentiable and have low regret. Our algorithmic and theoretical contributions are supported by extensive experiments that show the importance of baseline subtraction, learned biases, and the practicality of our approach on a range problems.
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