Testing the Feasibility of Linear Programs with Bandit Feedback

While the recent literature has seen a surge in the study of constrained bandit problems, all existing methods for these begin by assuming the feasibility of the underlying problem. We initiate the study of testing such feasibility assumptions, and in particular address the problem in the linear bandit setting, thus characterising the costs of feasibility testing for an unknown linear program using bandit feedback. Concretely, we test if $\exists x: Ax \ge 0$ for an unknown $A \in \mathbb{R}^{m \times d}$, by playing a sequence of actions $x_t\in \mathbb{R}^d$, and observing $Ax_t + \mathrm{noise}$ in response. By identifying the hypothesis as determining the sign of the value of a minimax game, we construct a novel test based on low-regret algorithms and a nonasymptotic law of iterated logarithms. We prove that this test is reliable, and adapts to the `signal level,' $\Gamma,$ of any instance, with mean sample costs scaling as $\widetilde{O}(d^2/\Gamma^2)$. We complement this by a minimax lower bound of $\Omega(d/\Gamma^2)$ for sample costs of reliable tests, dominating prior asymptotic lower bounds by capturing the dependence on $d$, and thus elucidating a basic insight missing in the extant literature on such problems.

Counterfactually Comparing Abstaining Classifiers

Abstaining classifiers have the option to abstain from making predictions on inputs that they are unsure about. These classifiers are becoming increasingly popular in high-stake decision-making problems, as they can withhold uncertain predictions to improve their reliability and safety. When evaluating black-box abstaining classifier(s), however, we lack a principled approach that accounts for what the classifier would have predicted on its abstentions. These missing predictions are crucial when, e.g., a radiologist is unsure of their diagnosis or when a driver is inattentive in a self-driving car. In this paper, we introduce a novel approach and perspective to the problem of evaluating and comparing abstaining classifiers by treating abstentions as missing data. Our evaluation approach is centered around defining the counterfactual score of an abstaining classifier, defined as the expected performance of the classifier had it not been allowed to abstain. We specify the conditions under which the counterfactual score is identifiable: if the abstentions are stochastic, and if the evaluation data is independent of the training data (ensuring that the predictions are missing at random), then the score is identifiable. Note that, if abstentions are deterministic, then the score is unidentifiable because the classifier can perform arbitrarily poorly on its abstentions. Leveraging tools from observational causal inference, we then develop nonparametric and doubly robust methods to efficiently estimate this quantity under identification. Our approach is examined in both simulated and real data experiments.

A Doubly Optimistic Strategy for Safe Linear Bandits

We propose a \underline{d}oubly \underline{o}ptimistic strategy for the \underline{s}afe-\underline{l}inear-\underline{b}andit problem, DOSLB. The safe linear bandit problem is to optimise an unknown linear reward whilst satisfying unknown round-wise safety constraints on actions, using stochastic bandit feedback of reward and safety-risks of actions. In contrast to prior work on aggregated resource constraints, our formulation explicitly demands control on roundwise safety risks. Unlike existing optimistic-pessimistic paradigms for safe bandits, DOSLB exercises supreme optimism, using optimistic estimates of reward and safety scores to select actions. Yet, and surprisingly, we show that DOSLB rarely takes risky actions, and obtains $\tilde{O}(d \sqrt{T})$ regret, where our notion of regret accounts for both inefficiency and lack of safety of actions. Specialising to polytopal domains, we first notably show that the $\sqrt{T}$-regret bound cannot be improved even with large gaps, and then identify a slackened notion of regret for which we show tight instance-dependent $O(\log^2 T)$ bounds. We further argue that in such domains, the number of times an overly risky action is played is also bounded as $O(\log^2T)$.

Strategies for Safe Multi-Armed Bandits with Logarithmic Regret and Risk

We investigate a natural but surprisingly unstudied approach to the multi-armed bandit problem under safety risk constraints. Each arm is associated with an unknown law on safety risks and rewards, and the learner's goal is to maximise reward whilst not playing unsafe arms, as determined by a given threshold on the mean risk. We formulate a pseudo-regret for this setting that enforces this safety constraint in a per-round way by softly penalising any violation, regardless of the gain in reward due to the same. This has practical relevance to scenarios such as clinical trials, where one must maintain safety for each round rather than in an aggregated sense. We describe doubly optimistic strategies for this scenario, which maintain optimistic indices for both safety risk and reward. We show that schema based on both frequentist and Bayesian indices satisfy tight gap-dependent logarithmic regret bounds, and further that these play unsafe arms only logarithmically many times in total. This theoretical analysis is complemented by simulation studies demonstrating the effectiveness of the proposed schema, and probing the domains in which their use is appropriate.

Online Selective Classification with Limited Feedback

Motivated by applications to resource-limited and safety-critical domains, we study selective classification in the online learning model, wherein a predictor may abstain from classifying an instance. For example, this may model an adaptive decision to invoke more resources on this instance. Two salient aspects of the setting we consider are that the data may be non-realisable, due to which abstention may be a valid long-term action, and that feedback is only received when the learner abstains, which models the fact that reliable labels are only available when the resource intensive processing is invoked. Within this framework, we explore strategies that make few mistakes, while not abstaining too many times more than the best-in-hindsight error-free classifier from a given class. That is, the one that makes no mistakes, while abstaining the fewest number of times. We construct simple versioning-based schemes for any $\mu \in (0,1],$ that make most $T^\mu$ mistakes while incurring \smash{$\tilde{O}(T^{1-\mu})$} excess abstention against adaptive adversaries. We further show that this dependence on $T$ is tight, and provide illustrative experiments on realistic datasets.

Selective Classification via One-Sided Prediction

We propose a novel method for selective classification (SC), a problem which allows a classifier to abstain from predicting some instances, thus trading off accuracy against coverage (the fraction of instances predicted). In contrast to prior gating or confidence-set based work, our proposed method optimises a collection of class-wise decoupled one-sided empirical risks, and is in essence a method for explicitly finding the largest decision sets for each class that have few false positives. This one-sided prediction (OSP) based relaxation yields an SC scheme that attains near-optimal coverage in the practically relevant high target accuracy regime, and further admits efficient implementation, leading to a flexible and principled method for SC. We theoretically derive generalization bounds for SC and OSP, and empirically we show that our scheme strongly outperforms state of the art methods in coverage at small error levels.

Piecewise Linear Regression via a Difference of Convex Functions

We present a new piecewise linear regression methodology that utilizes fitting a difference of convex functions (DC functions) to the data. These are functions $f$ that may be represented as the difference $\phi_1 - \phi_2$ for a choice of convex functions $\phi_1, \phi_2$. The method proceeds by estimating piecewise-liner convex functions, in a manner similar to max-affine regression, whose difference approximates the data. The choice of the function is regularised by a new seminorm over the class of DC functions that controls the $\ell_\infty$ Lipschitz constant of the estimate. The resulting methodology can be efficiently implemented via Quadratic programming even in high dimensions, and is shown to have close to minimax statistical risk. We empirically validate the method, showing it to be practically implementable, and to have comparable performance to existing regression/classification methods on real-world datasets.

Budget Learning via Bracketing

Conventional machine learning applications in the mobile/IoT setting transmit data to a cloud-server for predictions. Due to cost considerations (power, latency, monetary), it is desirable to minimise device-to-server transmissions. The budget learning (BL) problem poses the learner's goal as minimising use of the cloud while suffering no discernible loss in accuracy, under the constraint that the methods employed be edge-implementable. We propose a new formulation for the BL problem via the concept of bracketings. Concretely, we propose to sandwich the cloud's prediction, $g,$ via functions $h^-, h^+$ from a `simple' class so that $h^- \le g \le h^+$ nearly always. On an instance $x$, if $h^+(x)=h^-(x)$, we leverage local processing, and bypass the cloud. We explore theoretical aspects of this formulation, providing PAC-style learnability definitions; associating the notion of budget learnability to approximability via brackets; and giving VC-theoretic analyses of their properties. We empirically validate our theory on real-world datasets, demonstrating improved performance over prior gating based methods.

Testing Changes in Communities for the Stochastic Block Model

We introduce the problems of goodness-of-fit and two-sample testing of the latent community structure in a 2-community, symmetric, stochastic block model (SBM), in the regime where recovery of the structure is difficult. The latter problem may be described as follows: let $x,y$ be two latent community partitions. Given graphs $G,H$ drawn according to SBMs with partitions $x,y$, respectively, we wish to test the hypothesis $x = y$ against $d(x,y) \ge s,$ for a given Hamming distortion parameter $s \ll n$. Prior work showed that `partial' recovery of these partitions up to distortion $s$ with vanishing error probability requires that the signal-to-noise ratio $(\mathrm{SNR})$ is $\gtrsim C \log (n/s).$ We prove by constructing simple schemes that if $s \gg \sqrt{n \log n},$ then these testing problems can be solved even if $\mathrm{SNR} = O(1).$ For $s = o(\sqrt{n}),$ and constant order degrees, we show via an information-theoretic lower bound that both testing problems require $\mathrm{SNR} = \Omega(\log(n)),$ and thus at this scale the na\"{i}ve scheme of learning the communities and comparing them is minimax optimal up to constant factors. These results are augmented by simulations of goodness-of-fit and two-sample testing for standard SBMs as well as for Gaussian Markov random fields with underlying SBM structure.

Lower Bounds for Two-Sample Structural Change Detection in Ising and Gaussian Models

The change detection problem is to determine if the Markov network structures of two Markov random fields differ from one another given two sets of samples drawn from the respective underlying distributions. We study the trade-off between the sample sizes and the reliability of change detection, measured as a minimax risk, for the important cases of the Ising models and the Gaussian Markov random fields restricted to the models which have network structures with $p$ nodes and degree at most $d$, and obtain information-theoretic lower bounds for reliable change detection over these models. We show that for the Ising model, $\Omega\left(\frac{d^2}{(\log d)^2}\log p\right)$ samples are required from each dataset to detect even the sparsest possible changes, and that for the Gaussian, $\Omega\left( \gamma^{-2} \log(p)\right)$ samples are required from each dataset to detect change, where $\gamma$ is the smallest ratio of off-diagonal to diagonal terms in the precision matrices of the distributions. These bounds are compared to the corresponding results in structure learning, and closely match them under mild conditions on the model parameters. Thus, our change detection bounds inherit partial tightness from the structure learning schemes in previous literature, demonstrating that in certain parameter regimes, the naive structure learning based approach to change detection is minimax optimal up to constant factors.