Non-stationary parametric bandits have attracted much attention recently. There are three principled ways to deal with non-stationarity, including sliding-window, weighted, and restart strategies. As many non-stationary environments exhibit gradual drifting patterns, the weighted strategy is commonly adopted in real-world applications. However, previous theoretical studies show that its analysis is more involved and the algorithms are either computationally less efficient or statistically suboptimal. This paper revisits the weighted strategy for non-stationary parametric bandits. In linear bandits (LB), we discover that this undesirable feature is due to an inadequate regret analysis, which results in an overly complex algorithm design. We propose a refined analysis framework, which simplifies the derivation and importantly produces a simpler weight-based algorithm that is as efficient as window/restart-based algorithms while retaining the same regret as previous studies. Furthermore, our new framework can be used to improve regret bounds of other parametric bandits, including Generalized Linear Bandits (GLB) and Self-Concordant Bandits (SCB). For example, we develop a simple weighted GLB algorithm with an $\widetilde{O}(k_\mu^{\frac{5}{4}} c_\mu^{-\frac{3}{4}} d^{\frac{3}{4}} P_T^{\frac{1}{4}}T^{\frac{3}{4}})$ regret, improving the $\widetilde{O}(k_\mu^{2} c_\mu^{-1}d^{\frac{9}{10}} P_T^{\frac{1}{5}}T^{\frac{4}{5}})$ bound in prior work, where $k_\mu$ and $c_\mu$ characterize the reward model's nonlinearity, $P_T$ measures the non-stationarity, $d$ and $T$ denote the dimension and time horizon.
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Price discrimination, which refers to the strategy of setting different prices for different customer groups, has been widely used in online retailing. Although it helps boost the collected revenue for online retailers, it might create serious concerns about fairness, which even violates the regulation and laws. This paper studies the problem of dynamic discriminatory pricing under fairness constraints. In particular, we consider a finite selling horizon of length $T$ for a single product with two groups of customers. Each group of customers has its unknown demand function that needs to be learned. For each selling period, the seller determines the price for each group and observes their purchase behavior. While existing literature mainly focuses on maximizing revenue, ensuring fairness among different customers has not been fully explored in the dynamic pricing literature. This work adopts the fairness notion from Cohen et al. (2022). For price fairness, we propose an optimal dynamic pricing policy regarding regret, which enforces the strict price fairness constraint. In contrast to the standard $\sqrt{T}$-type regret in online learning, we show that the optimal regret in our case is $\tilde{O}(T^{4/5})$. We further extend our algorithm to a more general notion of fairness, which includes demand fairness as a special case. To handle this general class, we propose a soft fairness constraint and develop a dynamic pricing policy that achieves $\tilde{O}(T^{4/5})$ regret. We also demonstrate that our algorithmic techniques can be adapted to more general scenarios such as fairness among multiple groups of customers.
Despite the subject of non-stationary bandit learning having attracted much recent attention, we have yet to identify a formal definition of non-stationarity that can consistently distinguish non-stationary bandits from stationary ones. Prior work has characterized non-stationary bandits as bandits for which the reward distribution changes over time. We demonstrate that this definition can ambiguously classify the same bandit as both stationary and non-stationary; this ambiguity arises in the existing definition's dependence on the latent sequence of reward distributions. Moreover, the definition has given rise to two widely used notions of regret: the dynamic regret and the weak regret. These notions are not indicative of qualitative agent performance in some bandits. Additionally, this definition of non-stationary bandits has led to the design of agents that explore excessively. We introduce a formal definition of non-stationary bandits that resolves these issues. Our new definition provides a unified approach, applicable seamlessly to both Bayesian and frequentist formulations of bandits. Furthermore, our definition ensures consistent classification of two bandits offering agents indistinguishable experiences, categorizing them as either both stationary or both non-stationary. This advancement provides a more robust framework for non-stationary bandit learning.
In a well-calibrated risk prediction model, the average predicted probability is close to the true event rate for any given subgroup. Such models are reliable across heterogeneous populations and satisfy strong notions of algorithmic fairness. However, the task of auditing a model for strong calibration is well-known to be difficult -- particularly for machine learning (ML) algorithms -- due to the sheer number of potential subgroups. As such, common practice is to only assess calibration with respect to a few predefined subgroups. Recent developments in goodness-of-fit testing offer potential solutions but are not designed for settings with weak signal or where the poorly calibrated subgroup is small, as they either overly subdivide the data or fail to divide the data at all. We introduce a new testing procedure based on the following insight: if we can reorder observations by their expected residuals, there should be a change in the association between the predicted and observed residuals along this sequence if a poorly calibrated subgroup exists. This lets us reframe the problem of calibration testing into one of changepoint detection, for which powerful methods already exist. We begin with introducing a sample-splitting procedure where a portion of the data is used to train a suite of candidate models for predicting the residual, and the remaining data are used to perform a score-based cumulative sum (CUSUM) test. To further improve power, we then extend this adaptive CUSUM test to incorporate cross-validation, while maintaining Type I error control under minimal assumptions. Compared to existing methods, the proposed procedure consistently achieved higher power in simulation studies and more than doubled the power when auditing a mortality risk prediction model.
We investigate the fixed-budget best-arm identification (BAI) problem for linear bandits in a potentially non-stationary environment. Given a finite arm set $\mathcal{X}\subset\mathbb{R}^d$, a fixed budget $T$, and an unpredictable sequence of parameters $\left\lbrace\theta_t\right\rbrace_{t=1}^{T}$, an algorithm will aim to correctly identify the best arm $x^* := \arg\max_{x\in\mathcal{X}}x^\top\sum_{t=1}^{T}\theta_t$ with probability as high as possible. Prior work has addressed the stationary setting where $\theta_t = \theta_1$ for all $t$ and demonstrated that the error probability decreases as $\exp(-T /\rho^*)$ for a problem-dependent constant $\rho^*$. But in many real-world $A/B/n$ multivariate testing scenarios that motivate our work, the environment is non-stationary and an algorithm expecting a stationary setting can easily fail. For robust identification, it is well-known that if arms are chosen randomly and non-adaptively from a G-optimal design over $\mathcal{X}$ at each time then the error probability decreases as $\exp(-T\Delta^2_{(1)}/d)$, where $\Delta_{(1)} = \min_{x \neq x^*} (x^* - x)^\top \frac{1}{T}\sum_{t=1}^T \theta_t$. As there exist environments where $\Delta_{(1)}^2/ d \ll 1/ \rho^*$, we are motivated to propose a novel algorithm $\mathsf{P1}$-$\mathsf{RAGE}$ that aims to obtain the best of both worlds: robustness to non-stationarity and fast rates of identification in benign settings. We characterize the error probability of $\mathsf{P1}$-$\mathsf{RAGE}$ and demonstrate empirically that the algorithm indeed never performs worse than G-optimal design but compares favorably to the best algorithms in the stationary setting.
We study the reverse shortest path problem on disk graphs in the plane. In this problem we consider the proximity graph of a set of $n$ disks in the plane of arbitrary radii: In this graph two disks are connected if the distance between them is at most some threshold parameter $r$. The case of intersection graphs is a special case with $r=0$. We give an algorithm that, given a target length $k$, computes the smallest value of $r$ for which there is a path of length at most $k$ between some given pair of disks in the proximity graph. Our algorithm runs in $O^*(n^{5/4})$ randomized expected time, which improves to $O^*(n^{6/5})$ for unit disk graphs, where all the disks have the same radius. Our technique is robust and can be applied to many variants of the problem. One significant variant is the case of weighted proximity graphs, where edges are assigned real weights equal to the distance between the disks or between their centers, and $k$ is replaced by a target weight $w$; that is, we seek a path whose length is at most $w$. In other variants, we want to optimize a parameter different from $r$, such as a scale factor of the radii of the disks. The main technique for the decision version of the problem (determining whether the graph with a given $r$ has the desired property) is based on efficient implementations of BFS (for the unweighted case) and of Dijkstra's algorithm (for the weighted case), using efficient data structures for maintaining the bichromatic closest pair for certain bicliques and several distance functions. The optimization problem is then solved by combining the resulting decision procedure with enhanced variants of the interval shrinking and bifurcation technique of [4].
The utilization of finite field multipliers is pervasive in contemporary digital systems, with hardware implementation for bit parallel operation often necessitating millions of logic gates. However, various digital design issues, whether inherent or stemming from soft errors, can result in gate malfunction, ultimately can cause gates to malfunction, which in turn results in incorrect multiplier outputs. Thus, to prevent susceptibility to error, it is imperative to employ a reliable finite field multiplier implementation that boasts a robust fault detection capability. In order to achieve the best fault detection performance for finite field detection performance for finite field multipliers while maintaining a low-complexity implementation, this study proposes a novel fault detection scheme for a recent bit-parallel polynomial basis over GF(2m). The primary concept behind the proposed approach is centered on the implementation of an efficient BCH decoder that utilize Berlekamp-Rumsey-Solomon (BRS) algorithm and Chien-search method to effectively locate errors with minimal delay. The results of our synthesis indicate that our proposed error detection and correction architecture for a 45-bit multiplier with 5-bit errors achieves a 37% and 49% reduction in critical path delay compared to existing designs. Furthermore, a 45-bit multiplicand with five errors has hardware complexity that is only 80%, which is significantly less complex than the most advanced BCH-based fault recognition techniques, such as TMR, Hamming's single error correction, and LDPC-based methods for finite field multiplication which is desirable in constrained applications, such as smart cards, IoT devices, and implantable medical devices.
We consider the contextual bandit problem where at each time, the agent only has access to a noisy version of the context and the error variance (or an estimator of this variance). This setting is motivated by a wide range of applications where the true context for decision-making is unobserved, and only a prediction of the context by a potentially complex machine learning algorithm is available. When the context error is non-diminishing, classical bandit algorithms fail to achieve sublinear regret. We propose the first online algorithm in this setting with sublinear regret compared to the appropriate benchmark. The key idea is to extend the measurement error model in classical statistics to the online decision-making setting, which is nontrivial due to the policy being dependent on the noisy context observations.
Classic algorithms and machine learning systems like neural networks are both abundant in everyday life. While classic computer science algorithms are suitable for precise execution of exactly defined tasks such as finding the shortest path in a large graph, neural networks allow learning from data to predict the most likely answer in more complex tasks such as image classification, which cannot be reduced to an exact algorithm. To get the best of both worlds, this thesis explores combining both concepts leading to more robust, better performing, more interpretable, more computationally efficient, and more data efficient architectures. The thesis formalizes the idea of algorithmic supervision, which allows a neural network to learn from or in conjunction with an algorithm. When integrating an algorithm into a neural architecture, it is important that the algorithm is differentiable such that the architecture can be trained end-to-end and gradients can be propagated back through the algorithm in a meaningful way. To make algorithms differentiable, this thesis proposes a general method for continuously relaxing algorithms by perturbing variables and approximating the expectation value in closed form, i.e., without sampling. In addition, this thesis proposes differentiable algorithms, such as differentiable sorting networks, differentiable renderers, and differentiable logic gate networks. Finally, this thesis presents alternative training strategies for learning with algorithms.
Since deep neural networks were developed, they have made huge contributions to everyday lives. Machine learning provides more rational advice than humans are capable of in almost every aspect of daily life. However, despite this achievement, the design and training of neural networks are still challenging and unpredictable procedures. To lower the technical thresholds for common users, automated hyper-parameter optimization (HPO) has become a popular topic in both academic and industrial areas. This paper provides a review of the most essential topics on HPO. The first section introduces the key hyper-parameters related to model training and structure, and discusses their importance and methods to define the value range. Then, the research focuses on major optimization algorithms and their applicability, covering their efficiency and accuracy especially for deep learning networks. This study next reviews major services and toolkits for HPO, comparing their support for state-of-the-art searching algorithms, feasibility with major deep learning frameworks, and extensibility for new modules designed by users. The paper concludes with problems that exist when HPO is applied to deep learning, a comparison between optimization algorithms, and prominent approaches for model evaluation with limited computational resources.
Salient object detection is a problem that has been considered in detail and many solutions proposed. In this paper, we argue that work to date has addressed a problem that is relatively ill-posed. Specifically, there is not universal agreement about what constitutes a salient object when multiple observers are queried. This implies that some objects are more likely to be judged salient than others, and implies a relative rank exists on salient objects. The solution presented in this paper solves this more general problem that considers relative rank, and we propose data and metrics suitable to measuring success in a relative objects saliency landscape. A novel deep learning solution is proposed based on a hierarchical representation of relative saliency and stage-wise refinement. We also show that the problem of salient object subitizing can be addressed with the same network, and our approach exceeds performance of any prior work across all metrics considered (both traditional and newly proposed).
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