arXiv subjects

Jonathan Scarlett

Publications and source records attributed to Jonathan Scarlett.

At least 55 records · Page 3Linked to original sources

arXiv2023

Benefits of Monotonicity in Safe Exploration with Gaussian Processes

We consider the problem of sequentially maximising an unknown function over a set of actions while ensuring that every sampled point has a function value below a given safety threshold. We model the function using kernel-based and Gaussian process methods, while differing from previous works in our assumption that the function is monotonically increasing with respect to a \emph{safety variable}. This assumption is motivated by various practical applications such as adaptive clinical trial design and robotics. Taking inspiration from the \textsc{\sffamily GP-UCB} and \textsc{\sffamily SafeOpt} algorithms, we propose an algorithm, monotone safe {\sffamily UCB} (\textsc{\sffamily M-SafeUCB}) for this task. We show that \textsc{\sffamily M-SafeUCB} enjoys theoretical guarantees in terms of safety, a suitably-defined regret notion, and approximately finding the entire safe boundary. In addition, we illustrate that the monotonicity assumption yields significant benefits in terms of the guarantees obtained, as well as algorithmic simplicity and efficiency. We support our theoretical findings by performing empirical evaluations on a variety of functions, including a simulated clinical trial experiment.

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arXiv2023

Communication-Constrained Bandits under Additive Gaussian Noise

We study a distributed stochastic multi-armed bandit where a client supplies the learner with communication-constrained feedback based on the rewards for the corresponding arm pulls. In our setup, the client must encode the rewards such that the second moment of the encoded rewards is no more than $P$, and this encoded reward is further corrupted by additive Gaussian noise of variance $σ^2$; the learner only has access to this corrupted reward. For this setting, we derive an information-theoretic lower bound of $Ω\left(\sqrt{\frac{KT}{\mathtt{SNR} \wedge1}} \right)$ on the minimax regret of any scheme, where $ \mathtt{SNR} := \frac{P}{σ^2}$, and $K$ and $T$ are the number of arms and time horizon, respectively. Furthermore, we propose a multi-phase bandit algorithm, $\mathtt{UE\text{-}UCB++}$, which matches this lower bound to a minor additive factor. $\mathtt{UE\text{-}UCB++}$ performs uniform exploration in its initial phases and then utilizes the {\em upper confidence bound }(UCB) bandit algorithm in its final phase. An interesting feature of $\mathtt{UE\text{-}UCB++}$ is that the coarser estimates of the mean rewards formed during a uniform exploration phase help to refine the encoding protocol in the next phase, leading to more accurate mean estimates of the rewards in the subsequent phase. This positive reinforcement cycle is critical to reducing the number of uniform exploration rounds and closely matching our lower bound.

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arXiv2023

Maxflow-Based Bounds for Low-Rate Information Propagation over Noisy Networks

We study error exponents for the problem of low-rate communication over a directed graph, where each edge in the graph represents a noisy communication channel, and there is a single source and destination. We derive maxflow-based achievability and converse bounds on the error exponent that match when there are two messages and all channels satisfy a symmetry condition called pairwise reversibility. More generally, we show that the upper and lower bounds match to within a factor of 4. We also show that with three messages there are cases where the maxflow-based error exponent is strictly suboptimal, thus showing that our tightness result cannot be extended beyond two messages without further assumptions.

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arXiv2023

Multi-Bit Relaying over a Tandem of Channels

We study error exponents for the problem of relaying a message over a tandem of two channels sharing the same transition law, in particular moving beyond the 1-bit setting studied in recent related works. Our main results show that the 1-hop and 2-hop exponents coincide in both of the following settings: (i) the number of messages is fixed, and the channel law satisfies a condition called pairwise reversibility, or (ii) the channel is arbitrary, and a zero-rate limit is taken from above. In addition, we provide various extensions of our results that relax the assumptions of pairwise reversibility and/or the two channels having identical transition laws, and we provide an example for which the 2-hop exponent is strictly below the 1-hop exponent.

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arXiv2023

For One and All: Individual and Group Fairness in the Allocation of Indivisible Goods

Fair allocation of indivisible goods is a well-explored problem. Traditionally, research focused on individual fairness - are individual agents satisfied with their allotted share? - and group fairness - are groups of agents treated fairly? In this paper, we explore the coexistence of individual envy-freeness (i-EF) and its group counterpart, group weighted envy-freeness (g-WEF), in the allocation of indivisible goods. We propose several polynomial-time algorithms that provably achieve i-EF and g-WEF simultaneously in various degrees of approximation under three different conditions on the agents' (i) when agents have identical additive valuation functions, i-EFX and i-WEF1 can be achieved simultaneously; (ii) when agents within a group share a common valuation function, an allocation satisfying both i-EF1 and g-WEF1 exists; and (iii) when agents' valuations for goods within a group differ, we show that while maintaining i-EF1, we can achieve a 1/3-approximation to ex-ante g-WEF1. Our results thus provide a first step towards connecting individual and group fairness in the allocation of indivisible goods, in hopes of its useful application to domains requiring the reconciliation of diversity with individual demands.

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arXiv2023

On Average-Case Error Bounds for Kernel-Based Bayesian Quadrature

In this paper, we study error bounds for {\em Bayesian quadrature} (BQ), with an emphasis on noisy settings, randomized algorithms, and average-case performance measures. We seek to approximate the integral of functions in a {\em Reproducing Kernel Hilbert Space} (RKHS), particularly focusing on the Matérn-$ν$ and squared exponential (SE) kernels, with samples from the function potentially being corrupted by Gaussian noise. We provide a two-step meta-algorithm that serves as a general tool for relating the average-case quadrature error with the $L^2$-function approximation error. When specialized to the Matérn kernel, we recover an existing near-optimal error rate while avoiding the existing method of repeatedly sampling points. When specialized to other settings, we obtain new average-case results for settings including the SE kernel with noise and the Matérn kernel with misspecification. Finally, we present algorithm-independent lower bounds that have greater generality and/or give distinct proofs compared to existing ones.

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arXiv2023

Max-Quantile Grouped Infinite-Arm Bandits

In this paper, we consider a bandit problem in which there are a number of groups each consisting of infinitely many arms. Whenever a new arm is requested from a given group, its mean reward is drawn from an unknown reservoir distribution (different for each group), and the uncertainty in the arm's mean reward can only be reduced via subsequent pulls of the arm. The goal is to identify the infinite-arm group whose reservoir distribution has the highest $(1-α)$-quantile (e.g., median if $α= \frac{1}{2}$), using as few total arm pulls as possible. We introduce a two-step algorithm that first requests a fixed number of arms from each group and then runs a finite-arm grouped max-quantile bandit algorithm. We characterize both the instance-dependent and worst-case regret, and provide a matching lower bound for the latter, while discussing various strengths, weaknesses, algorithmic improvements, and potential lower bounds associated with our instance-dependent upper bounds.

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arXiv2023

Theoretical Perspectives on Deep Learning Methods in Inverse Problems

In recent years, there have been significant advances in the use of deep learning methods in inverse problems such as denoising, compressive sensing, inpainting, and super-resolution. While this line of works has predominantly been driven by practical algorithms and experiments, it has also given rise to a variety of intriguing theoretical problems. In this paper, we survey some of the prominent theoretical developments in this line of works, focusing in particular on generative priors, untrained neural network priors, and unfolding algorithms. In addition to summarizing existing results in these topics, we highlight several ongoing challenges and open problems.

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arXiv2022

Mismatched Rate-Distortion Theory: Ensembles, Bounds, and General Alphabets

In this paper, we consider the mismatched rate-distortion problem, in which the encoding is done using a codebook, and the encoder chooses the minimum-distortion codeword according to a mismatched distortion function that differs from the true one. For the case of discrete memoryless sources, we establish achievable rate-distortion bounds using multi-user coding techniques, namely, superposition coding and expurgated parallel coding. We study examples where these attain the matched rate-distortion trade-off but a standard ensemble with independent codewords fails to do so. On the other hand, in contrast with the channel coding counterpart, we show that there are cases where structured random codebooks can perform worse than their unstructured counterparts. In addition, in view of the difficulties in adapting the existing and above-mentioned results to general alphabets, we consider a simpler i.i.d. random coding ensemble, and establish its achievable rate-distortion bounds for general alphabets.

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arXiv2022

Model-Based and Graph-Based Priors for Group Testing

The goal of the group testing problem is to identify a set of defective items within a larger set of items, using suitably-designed tests whose outcomes indicate whether any defective item is present. In this paper, we study how the number of tests can be significantly decreased by leveraging the structural dependencies between the items, i.e., by incorporating prior information. To do so, we pursue two different perspectives: (i) As a generalization of the uniform combinatorial prior, we consider the case that the defective set is uniform over a \emph{subset} of all possible sets of a given size, and study how this impacts the information-theoretic limits on the number of tests for approximate recovery; (ii) As a generalization of the i.i.d.~prior, we introduce a new class of priors based on the Ising model, where the associated graph represents interactions between items. We show that this naturally leads to an Integer Quadratic Program decoder, which can be converted to an Integer Linear Program and/or relaxed to a non-integer variant for improved computational complexity, while maintaining strong empirical recovery performance.

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arXiv2022

Optimal Rates of Teaching and Learning Under Uncertainty

In this paper, we consider a recently-proposed model of teaching and learning under uncertainty, in which a teacher receives independent observations of a single bit corrupted by binary symmetric noise, and sequentially transmits to a student through another binary symmetric channel based on the bits observed so far. After a given number $n$ of transmissions, the student outputs an estimate of the unknown bit, and we are interested in the exponential decay rate of the error probability as $n$ increases. We propose a novel block-structured teaching strategy in which the teacher encodes the number of 1s received in each block, and show that the resulting error exponent is the binary relative entropy $D\big(\frac{1}{2}\|\max(p,q)\big)$, where $p$ and $q$ are the noise parameters. This matches a trivial converse result based on the data processing inequality, and settles two conjectures of [Jog and Loh, 2021] and [Huleihel, Polyanskiy, and Shayevitz, 2019]. In addition, we show that the computation time required by the teacher and student is linear in $n$. We also study a more general setting in which the binary symmetric channels are replaced by general binary-input discrete memoryless channels. We provide an achievability bound and a converse bound, and show that the two coincide in certain cases, including (i) when the two channels are identical, and (ii) when the student-teacher channel is a binary symmetric channel. More generally, we give sufficient conditions under which our learning rate is the best possible for block-structured protocols.

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arXiv2022

Simple Coding Techniques for Many-Hop Relaying

In this paper, we study the problem of relaying a single bit of information across a series of binary symmetric channels, and the associated trade-off between the number of hops $m$, the transmission time $n$, and the error probability. We introduce a simple, efficient, and deterministic protocol that attains positive information velocity (i.e., a non-vanishing ratio $\frac{m}{n}$ and small error probability) and is significantly simpler than existing protocols that do so. In addition, we characterize the optimal low-noise and high-noise scaling laws of the information velocity, and we adapt our 1-bit protocol to transmit $k$ bits over $m$ hops with $O(m+k)$ transmission time.

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arXiv2022

Fast Splitting Algorithms for Sparsity-Constrained and Noisy 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 at least one defective item is present. This problem is relevant in areas such as medical testing, DNA sequencing, communication protocols, and many more. In this paper, we study (i) a sparsity-constrained version of the problem, in which the testing procedure is subjected 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; and (ii) a noisy version of the problem, where each test outcome is independently flipped with some constant probability. Under each of these settings, considering the for-each recovery guarantee with asymptotically vanishing error probability, we introduce a fast splitting algorithm and establish its near-optimality not only in terms of the number of tests, but also in terms of the decoding time. While the most basic formulations of our algorithms require $Ω(n)$ storage for each algorithm, we also provide low-storage variants based on hashing, with similar recovery guarantees.

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arXiv2022

Performance Bounds for Group Testing With Doubly-Regular Designs

In the group testing problem, 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, DNA sequencing, and communications. In this paper, we study a doubly-regular design in which the number of tests-per-item and the number of items-per-test are fixed. We analyze the performance of this test design alongside the Definite Defectives (DD) decoding algorithm in several settings, namely, (i) the sub-linear regime $k=o(n)$ with exact recovery, (ii) the linear regime $k=Θ(n)$ with approximate recovery, and (iii) the size-constrained setting, where the number of items per test is constrained. Under setting (i), we show that our design together with the DD algorithm, matches an existing achievability result for the DD algorithm with the near-constant tests-per-item design, which is known to be asymptotically optimal in broad scaling regimes. Under setting (ii), we provide novel approximate recovery bounds that complement a hardness result regarding exact recovery. Lastly, under setting (iii), we improve on the best known upper and lower bounds in scaling regimes where the maximum test size grows with the total number of items.

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arXiv2022

Generative Principal Component Analysis

In this paper, we study the problem of principal component analysis with generative modeling assumptions, adopting a general model for the observed matrix that encompasses notable special cases, including spiked matrix recovery and phase retrieval. The key assumption is that the underlying signal lies near the range of an $L$-Lipschitz continuous generative model with bounded $k$-dimensional inputs. We propose a quadratic estimator, and show that it enjoys a statistical rate of order $\sqrt{\frac{k\log L}{m}}$, where $m$ is the number of samples. We also provide a near-matching algorithm-independent lower bound. Moreover, we provide a variant of the classic power method, which projects the calculated data onto the range of the generative model during each iteration. We show that under suitable conditions, this method converges exponentially fast to a point achieving the above-mentioned statistical rate. We perform experiments on various image datasets for spiked matrix and phase retrieval models, and illustrate performance gains of our method to the classic power method and the truncated power method devised for sparse principal component analysis.

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arXiv2022

Improved Convergence Rates for Sparse Approximation Methods in Kernel-Based Learning

Kernel-based models such as kernel ridge regression and Gaussian processes are ubiquitous in machine learning applications for regression and optimization. It is well known that a major downside for kernel-based models is the high computational cost; given a dataset of $n$ samples, the cost grows as $\mathcal{O}(n^3)$. Existing sparse approximation methods can yield a significant reduction in the computational cost, effectively reducing the actual cost down to as low as $\mathcal{O}(n)$ in certain cases. Despite this remarkable empirical success, significant gaps remain in the existing results for the analytical bounds on the error due to approximation. In this work, we provide novel confidence intervals for the Nyström method and the sparse variational Gaussian process approximation method, which we establish using novel interpretations of the approximate (surrogate) posterior variance of the models. Our confidence intervals lead to improved performance bounds in both regression and optimization problems.

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arXiv2022

Adversarial Attacks on Gaussian Process Bandits

Gaussian processes (GP) are a widely-adopted tool used to sequentially optimize black-box functions, where evaluations are costly and potentially noisy. Recent works on GP bandits have proposed to move beyond random noise and devise algorithms robust to adversarial attacks. This paper studies this problem from the attacker's perspective, proposing various adversarial attack methods with differing assumptions on the attacker's strength and prior information. Our goal is to understand adversarial attacks on GP bandits from theoretical and practical perspectives. We focus primarily on targeted attacks on the popular GP-UCB algorithm and a related elimination-based algorithm, based on adversarially perturbing the function $f$ to produce another function $\tilde{f}$ whose optima are in some target region $\mathcal{R}_{\rm target}$. Based on our theoretical analysis, we devise both white-box attacks (known $f$) and black-box attacks (unknown $f$), with the former including a Subtraction attack and Clipping attack, and the latter including an Aggressive subtraction attack. We demonstrate that adversarial attacks on GP bandits can succeed in forcing the algorithm towards $\mathcal{R}_{\rm target}$ even with a low attack budget, and we test our attacks' effectiveness on a diverse range of objective functions.

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arXiv2022

Universal 1-Bit Compressive Sensing for Bounded Dynamic Range Signals

A {\em universal 1-bit compressive sensing (CS)} scheme consists of a measurement matrix $A$ such that all signals $x$ belonging to a particular class can be approximately recovered from $\textrm{sign}(Ax)$. 1-bit CS models extreme quantization effects where only one bit of information is revealed per measurement. We focus on universal support recovery for 1-bit CS in the case of {\em sparse} signals with bounded {\em dynamic range}. Specifically, a vector $x \in \mathbb{R}^n$ is said to have sparsity $k$ if it has at most $k$ nonzero entries, and dynamic range $R$ if the ratio between its largest and smallest nonzero entries is at most $R$ in magnitude. Our main result shows that if the entries of the measurement matrix $A$ are i.i.d.~Gaussians, then under mild assumptions on the scaling of $k$ and $R$, the number of measurements needs to be $\tildeΩ(Rk^{3/2})$ to recover the support of $k$-sparse signals with dynamic range $R$ using $1$-bit CS. In addition, we show that a near-matching $O(R k^{3/2} \log n)$ upper bound follows as a simple corollary of known results. The $k^{3/2}$ scaling contrasts with the known lower bound of $\tildeΩ(k^2 \log n)$ for the number of measurements to recover the support of arbitrary $k$-sparse signals.

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