arXiv subjects

Jonathan Scarlett

Publications and source records attributed to Jonathan Scarlett.

At least 73 records · Page 4Linked to original sources

arXiv2022

A Robust Phased Elimination Algorithm for Corruption-Tolerant Gaussian Process Bandits

We consider the sequential optimization of an unknown, continuous, and expensive to evaluate reward function, from noisy and adversarially corrupted observed rewards. When the corruption attacks are subject to a suitable budget $C$ and the function lives in a Reproducing Kernel Hilbert Space (RKHS), the problem can be posed as corrupted Gaussian process (GP) bandit optimization. We propose a novel robust elimination-type algorithm that runs in epochs, combines exploration with infrequent switching to select a small subset of actions, and plays each action for multiple time instants. Our algorithm, Robust GP Phased Elimination (RGP-PE), successfully balances robustness to corruptions with exploration and exploitation such that its performance degrades minimally in the presence (or absence) of adversarial corruptions. When $T$ is the number of samples and $γ_T$ is the maximal information gain, the corruption-dependent term in our regret bound is $O(C γ_T^{3/2})$, which is significantly tighter than the existing $O(C \sqrt{T γ_T})$ for several commonly-considered kernels. We perform the first empirical study of robustness in the corrupted GP bandit setting, and show that our algorithm is robust against a variety of adversarial attacks.

stat.ML↗

arXiv2022

Max-Min Grouped Bandits

In this paper, we introduce a multi-armed bandit problem termed max-min grouped bandits, in which the arms are arranged in possibly-overlapping groups, and the goal is to find the group whose worst arm has the highest mean reward. This problem is of interest in applications such as recommendation systems and resource allocation, and is also closely related to widely-studied robust optimization problems. We present two algorithms based successive elimination and robust optimization, and derive upper bounds on the number of samples to guarantee finding a max-min optimal or near-optimal group, as well as an algorithm-independent lower bound. We discuss the degree of tightness of our bounds in various cases of interest, and the difficulties in deriving uniformly tight bounds.

stat.ML↗

arXiv2022

Gaussian Process Bandit Optimization with Few Batches

In this paper, we consider the problem of black-box optimization using Gaussian Process (GP) bandit optimization with a small number of batches. Assuming the unknown function has a low norm in the Reproducing Kernel Hilbert Space (RKHS), we introduce a batch algorithm inspired by batched finite-arm bandit algorithms, and show that it achieves the cumulative regret upper bound $O^\ast(\sqrt{Tγ_T})$ using $O(\log\log T)$ batches within time horizon $T$, where the $O^\ast(\cdot)$ notation hides dimension-independent logarithmic factors and $γ_T$ is the maximum information gain associated with the kernel. This bound is near-optimal for several kernels of interest and improves on the typical $O^\ast(\sqrt{T}γ_T)$ bound, and our approach is arguably the simplest among algorithms attaining this improvement. In addition, in the case of a constant number of batches (not depending on $T$), we propose a modified version of our algorithm, and characterize how the regret is impacted by the number of batches, focusing on the squared exponential and Matérn kernels. The algorithmic upper bounds are shown to be nearly minimax optimal via analogous algorithm-independent lower bounds.

stat.ML↗

arXiv2022

Tight Regret Bounds for Noisy Optimization of a Brownian Motion

We consider the problem of Bayesian optimization of a one-dimensional Brownian motion in which the $T$ adaptively chosen observations are corrupted by Gaussian noise. We show that as the smallest possible expected cumulative regret and the smallest possible expected simple regret scale as $Ω(σ\sqrt{T / \log (T)}) \cap \mathcal{O}(σ\sqrt{T} \cdot \log T)$ and $Ω(σ/ \sqrt{T \log (T)}) \cap \mathcal{O}(σ\log T / \sqrt{T})$ respectively, where $σ^2$ is the noise variance. Thus, our upper and lower bounds are tight up to a factor of $\mathcal{O}( (\log T)^{1.5} )$. The upper bound uses an algorithm based on confidence bounds and the Markov property of Brownian motion (among other useful properties), and the lower bound is based on a reduction to binary hypothesis testing.

cs.LG↗

arXiv2021

Noisy Non-Adaptive Group Testing: A (Near-)Definite Defectives Approach

The group testing problem consists of determining a small set of defective items from a larger set of items based on a number of possibly-noisy tests, and is relevant in applications such as medical testing, communication protocols, pattern matching, and more. We study the noisy version of this problem, where the outcome of each standard noiseless group test is subject to independent noise, corresponding to passing the noiseless result through a binary channel. We introduce a class of algorithms that we refer to as Near-Definite Defectives (NDD), and study bounds on the required number of tests for asymptotically vanishing error probability under Bernoulli random test designs. In addition, we study algorithm-independent converse results, giving lower bounds on the required number of tests under Bernoulli test designs. Under reverse Z-channel noise, the achievable rates and converse results match in a broad range of sparsity regimes, and under Z-channel noise, the two match in a narrower range of dense/low-noise regimes. We observe that although these two channels have the same Shannon capacity when viewed as a communication channel, they can behave quite differently when it comes to group testing. Finally, we extend our analysis of these noise models to a general binary noise model (including symmetric noise), and show improvements over known existing bounds in broad scaling regimes.

cs.IT↗

arXiv2021

Near optimal sparsity-constrained group testing: improved bounds and algorithms

Recent advances in noiseless non-adaptive group testing have led to a precise asymptotic characterization of the number of tests required for high-probability recovery in the sublinear regime $k = n^θ$ (with $θ\in (0,1)$), with $n$ individuals among which $k$ are infected. However, the required number of tests may increase substantially under real-world practical constraints, notably including bounds on the maximum number $Δ$ of tests an individual can be placed in, or the maximum number $Γ$ of individuals in a given test. While previous works have given recovery guarantees for these settings, significant gaps remain between the achievability and converse bounds. In this paper, we substantially or completely close several of the most prominent gaps. In the case of $Δ$-divisible items, we show that the definite defectives (DD) algorithm coupled with a random regular design is asymptotically optimal in dense scaling regimes, and optimal to within a factor of $\eul$ more generally; we establish this by strengthening both the best known achievability and converse bounds. In the case of $Γ$-sized tests, we provide a comprehensive analysis of the regime $Γ= Θ(1)$, and again establish a precise threshold proving the asymptotic optimality of SCOMP (a slight refinement of DD) equipped with a tailored pooling scheme. Finally, for each of these two settings, we provide near-optimal adaptive algorithms based on sequential splitting, and provably demonstrate gaps between the performance of optimal adaptive and non-adaptive algorithms.

cs.DS↗

arXiv2021

Noisy Adaptive Group Testing via Noisy Binary Search

The group testing problem consists of determining a small set of defective items from a larger set of items based on a number of possibly-noisy tests, and has numerous practical applications. One of the defining features of group testing is whether the tests are adaptive (i.e., a given test can be chosen based on all previous outcomes) or non-adaptive (i.e., all tests must be chosen in advance). In this paper, building on the success of binary splitting techniques in noiseless group testing (Hwang, 1972), we introduce noisy group testing algorithms that apply noisy binary search as a subroutine. We provide three variations of this approach with increasing complexity, culminating in an algorithm that succeeds using a number of tests that matches the best known previously (Scarlett, 2019), while overcoming fundamental practical limitations of the existing approach, and more precisely capturing the dependence of the number of tests on the error probability. We provide numerical experiments demonstrating that adaptive group testing strategies based on noisy binary search can be highly effective in practice, using significantly fewer tests compared to state-of-the-art non-adaptive strategies.

cs.IT↗

arXiv2021

Open Problem: Tight Online Confidence Intervals for RKHS Elements

Confidence intervals are a crucial building block in the analysis of various online learning problems. The analysis of kernel based bandit and reinforcement learning problems utilize confidence intervals applicable to the elements of a reproducing kernel Hilbert space (RKHS). However, the existing confidence bounds do not appear to be tight, resulting in suboptimal regret bounds. In fact, the existing regret bounds for several kernelized bandit algorithms (e.g., GP-UCB, GP-TS, and their variants) may fail to even be sublinear. It is unclear whether the suboptimal regret bound is a fundamental shortcoming of these algorithms or an artifact of the proof, and the main challenge seems to stem from the online (sequential) nature of the observation points. We formalize the question of online confidence intervals in the RKHS setting and overview the existing results.

stat.ML↗

arXiv2021

Towards Sample-Optimal Compressive Phase Retrieval with Sparse and Generative Priors

Compressive phase retrieval is a popular variant of the standard compressive sensing problem in which the measurements only contain magnitude information. In this paper, motivated by recent advances in deep generative models, we provide recovery guarantees with near-optimal sample complexity for phase retrieval with generative priors. We first show that when using i.i.d. Gaussian measurements and an $L$-Lipschitz continuous generative model with bounded $k$-dimensional inputs, roughly $O(k \log L)$ samples suffice to guarantee that any signal minimizing an amplitude-based empirical loss function is close to the true signal. Attaining this sample complexity with a practical algorithm remains a difficult challenge, and finding a good initialization for gradient-based methods has been observed to pose a major bottleneck. To partially address this, we further show that roughly $O(k \log L)$ samples ensure sufficient closeness between the underlying signal and any {\em globally optimal} solution to an optimization problem designed for spectral initialization (though finding such a solution may still be challenging). We also adapt this result to sparse phase retrieval, and show that $O(s \log n)$ samples are sufficient for a similar guarantee when the underlying signal is $s$-sparse and $n$-dimensional, matching an information-theoretic lower bound. While these guarantees do not directly correspond to a practical algorithm, we propose a practical spectral initialization method motivated by our findings, and experimentally observe performance gains over various existing spectral initialization methods for sparse phase retrieval.

stat.ML↗

arXiv2021

Robust 1-bit Compressive Sensing with Partial Gaussian Circulant Matrices and Generative Priors

In 1-bit compressive sensing, each measurement is quantized to a single bit, namely the sign of a linear function of an unknown vector, and the goal is to accurately recover the vector. While it is most popular to assume a standard Gaussian sensing matrix for 1-bit compressive sensing, using structured sensing matrices such as partial Gaussian circulant matrices is of significant practical importance due to their faster matrix operations. In this paper, we provide recovery guarantees for a correlation-based optimization algorithm for robust 1-bit compressive sensing with randomly signed partial Gaussian circulant matrices and generative models. Under suitable assumptions, we match guarantees that were previously only known to hold for i.i.d.~Gaussian matrices that require significantly more computation. We make use of a practical iterative algorithm, and perform numerical experiments on image datasets to corroborate our theoretical results.

cs.LG↗

arXiv2021

Optimal Non-Adaptive Probabilistic Group Testing in General Sparsity Regimes

In this paper, we consider the problem of noiseless non-adaptive probabilistic group testing, in which the goal is high-probability recovery of the defective set. We show that in the case of $n$ items among which $k$ are defective, the smallest possible number of tests equals $\min\{ C_{k,n} k \log n, n\}$ up to lower-order asymptotic terms, where $C_{k,n}$ is a uniformly bounded constant (varying depending on the scaling of $k$ with respect to $n$) with a simple explicit expression. The algorithmic upper bound follows from a minor adaptation of an existing analysis of the Definite Defectives (DD) algorithm, and the algorithm-independent lower bound builds on existing works for the regimes $k \le n^{1-Ω(1)}$ and $k = Θ(n)$. In sufficiently sparse regimes (including $k = o\big( \frac{n}{\log n} \big)$), our main result generalizes that of Coja-Oghlan {\em et al.} (2020) by avoiding the assumption $k \le n^{1-Ω(1)}$, whereas in sufficiently dense regimes (including $k = ω\big( \frac{n}{\log n} \big)$), our main result shows that individual testing is asymptotically optimal for any non-zero target success probability, thus strengthening an existing result of Aldridge (2019) in terms of both the error probability and the assumed scaling of $k$.

cs.IT↗

arXiv2021

Lenient Regret and Good-Action Identification in Gaussian Process Bandits

In this paper, we study the problem of Gaussian process (GP) bandits under relaxed optimization criteria stating that any function value above a certain threshold is "good enough". On the theoretical side, we study various {\em lenient regret} notions in which all near-optimal actions incur zero penalty, and provide upper bounds on the lenient regret for GP-UCB and an elimination algorithm, circumventing the usual $O(\sqrt{T})$ term (with time horizon $T$) resulting from zooming extremely close towards the function maximum. In addition, we complement these upper bounds with algorithm-independent lower bounds. On the practical side, we consider the problem of finding a single "good action" according to a known pre-specified threshold, and introduce several good-action identification algorithms that exploit knowledge of the threshold. We experimentally find that such algorithms can often find a good action faster than standard optimization-based approaches.

stat.ML↗

arXiv2021

On Lower Bounds for Standard and Robust Gaussian Process Bandit Optimization

In this paper, we consider algorithm-independent lower bounds for the problem of black-box optimization of functions having a bounded norm is some Reproducing Kernel Hilbert Space (RKHS), which can be viewed as a non-Bayesian Gaussian process bandit problem. In the standard noisy setting, we provide a novel proof technique for deriving lower bounds on the regret, with benefits including simplicity, versatility, and an improved dependence on the error probability. In a robust setting in which every sampled point may be perturbed by a suitably-constrained adversary, we provide a novel lower bound for deterministic strategies, demonstrating an inevitable joint dependence of the cumulative regret on the corruption level and the time horizon, in contrast with existing lower bounds that only characterize the individual dependencies. Furthermore, in a distinct robust setting in which the final point is perturbed by an adversary, we strengthen an existing lower bound that only holds for target success probabilities very close to one, by allowing for arbitrary success probabilities above $\frac{2}{3}$.

stat.ML↗

arXiv2021

On the All-Or-Nothing Behavior of Bernoulli Group Testing

In this paper, we study the problem of non-adaptive group testing, in which one seeks to identify which items are defective given a set of suitably-designed tests whose outcomes indicate whether or not at least one defective item was included in the test. The most widespread recovery criterion seeks to exactly recover the entire defective set, and relaxed criteria such as approximate recovery and list decoding have also been considered. In this paper, we study the fundamental limits of group testing under the significantly relaxed {\em weak recovery} criterion, which only seeks to identify a small fraction (e.g., $0.01$) of the defective items. Given the near-optimality of i.i.d.~Bernoulli testing for exact recovery in sufficiently sparse scaling regimes, it is natural to ask whether this design additionally succeeds with much fewer tests under weak recovery. Our main negative result shows that this is not the case, and in fact, under i.i.d.~Bernoulli random testing in the sufficiently sparse regime, an {\em all-or-nothing} phenomenon occurs: When the number of tests is slightly below a threshold, weak recovery is impossible, whereas when the number of tests is slightly above the same threshold, high-probability exact recovery is possible. In establishing this result, we additionally prove similar negative results under Bernoulli designs for the weak detection problem (distinguishing between the group testing model vs.~completely random outcomes) and the problem of identifying a single item that is definitely defective. On the positive side, we show that all three relaxed recovery criteria can be attained using considerably fewer tests under suitably-chosen non-Bernoulli designs.

cs.IT↗

arXiv2020

High-Dimensional Bayesian Optimization via Tree-Structured Additive Models

Bayesian Optimization (BO) has shown significant success in tackling expensive low-dimensional black-box optimization problems. Many optimization problems of interest are high-dimensional, and scaling BO to such settings remains an important challenge. In this paper, we consider generalized additive models in which low-dimensional functions with overlapping subsets of variables are composed to model a high-dimensional target function. Our goal is to lower the computational resources required and facilitate faster model learning by reducing the model complexity while retaining the sample-efficiency of existing methods. Specifically, we constrain the underlying dependency graphs to tree structures in order to facilitate both the structure learning and optimization of the acquisition function. For the former, we propose a hybrid graph learning algorithm based on Gibbs sampling and mutation. In addition, we propose a novel zooming-based algorithm that permits generalized additive models to be employed more efficiently in the case of continuous domains. We demonstrate and discuss the efficacy of our approach via a range of experiments on synthetic functions and real-world datasets.

stat.ML↗

arXiv2020

Improved Bounds and Algorithms for Sparsity-Constrained Group Testing

In group testing, the goal is to identify a subset of defective items within a larger set of items based on tests whose outcomes indicate whether any defective item is present. This problem is relevant in areas such as medical testing, data science, communications, and many more. Motivated by physical considerations, we consider a sparsity-based constrained setting (Gandikota et al., 2019) in which the testing procedure is subject to one of the following two constraints: items are finitely divisible and thus may participate in at most $γ$ tests; or tests are size-constrained to pool no more than $ρ$ items per test. While information-theoretic limits and algorithms are known for the non-adaptive setting, relatively little is known in the adaptive setting. We address this gap by providing an information-theoretic converse that holds even in the adaptive setting, as well as a near-optimal noiseless adaptive algorithm for $γ$-divisible items. In broad scaling regimes, our upper and lower bounds asymptotically match up to a factor of $e$. We also present a simple asymptotically optimal adaptive algorithm for $ρ$-sized tests. In addition, in the non-adaptive setting with $γ$-divisible items, we use the Definite Defectives (DD) decoding algorithm and study bounds on the required number of tests for vanishing error probability under the random near-constant test-per-item design. We show that the number of tests required can be significantly less than the Combinatorial Orthogonal Matching Pursuit (COMP) decoding algorithm, and is asymptotically optimal in broad scaling regimes.

cs.IT↗

arXiv2020

Stochastic Linear Bandits Robust to Adversarial Attacks

We consider a stochastic linear bandit problem in which the rewards are not only subject to random noise, but also adversarial attacks subject to a suitable budget $C$ (i.e., an upper bound on the sum of corruption magnitudes across the time horizon). We provide two variants of a Robust Phased Elimination algorithm, one that knows $C$ and one that does not. Both variants are shown to attain near-optimal regret in the non-corrupted case $C = 0$, while incurring additional additive terms respectively having a linear and quadratic dependency on $C$ in general. We present algorithm independent lower bounds showing that these additive terms are near-optimal. In addition, in a contextual setting, we revisit a setup of diverse contexts, and show that a simple greedy algorithm is provably robust with a near-optimal additive regret term, despite performing no explicit exploration and not knowing $C$.

stat.ML↗

arXiv2020

The Generalized Lasso with Nonlinear Observations and Generative Priors

In this paper, we study the problem of signal estimation from noisy non-linear measurements when the unknown $n$-dimensional signal is in the range of an $L$-Lipschitz continuous generative model with bounded $k$-dimensional inputs. We make the assumption of sub-Gaussian measurements, which is satisfied by a wide range of measurement models, such as linear, logistic, 1-bit, and other quantized models. In addition, we consider the impact of adversarial corruptions on these measurements. Our analysis is based on a generalized Lasso approach (Plan and Vershynin, 2016). We first provide a non-uniform recovery guarantee, which states that under i.i.d.~Gaussian measurements, roughly $O\left(\frac{k}{ε^2}\log L\right)$ samples suffice for recovery with an $\ell_2$-error of $ε$, and that this scheme is robust to adversarial noise. Then, we apply this result to neural network generative models, and discuss various extensions to other models and non-i.i.d.~measurements. Moreover, we show that our result can be extended to the uniform recovery guarantee under the assumption of a so-called local embedding property, which is satisfied by the 1-bit and censored Tobit models.

stat.ML↗