We investigate the low rank matrix completion problem in an online setting with ${M}$ users, ${N}$ items, ${T}$ rounds, and an unknown rank-$r$ reward matrix ${R}\in \mathbb{R}^{{M}\times {N}}$. This problem has been well-studied in the literature and has several applications in practice. In each round, we recommend ${S}$ carefully chosen distinct items to every user and observe noisy rewards. In the regime where ${M},{N} >> {T}$, we propose two distinct computationally efficient algorithms for recommending items to users and analyze them under the benign \emph{hott items} assumption.1) First, for ${S}=1$, under additional incoherence/smoothness assumptions on ${R}$, we propose the phased algorithm \textsc{PhasedClusterElim}. Our algorithm obtains a near-optimal per-user regret of $\tilde{O}({N}{M}^{-1}(\Delta^{-1}+\Delta_{{hott}}^{-2}))$ where $\Delta_{{hott}},\Delta$ are problem-dependent gap parameters with $\Delta_{{hott}} >> \Delta$ almost always. 2) Second, we consider a simplified setting with ${S}=r$ where we make significantly milder assumptions on ${R}$. Here, we introduce another phased algorithm, \textsc{DeterminantElim}, to derive a regret guarantee of $\widetilde{O}({N}{M}^{-1/r}\Delta_{det}^{-1}))$ where $\Delta_{{det}}$ is another problem-dependent gap. Both algorithms crucially use collaboration among users to jointly eliminate sub-optimal items for groups of users successively in phases, but with distinctive and novel approaches.

Ask any American what their favorite part of a Christmas dinner is and you'll here a wide range of answers. Many families see the turkey or roast ham as the true centerpiece of the meal, while others drool over side dishes like stuffing, mashed potatoes or green beans. And the star of the show was rather surprising. Ask any American what their favorite part of a Christmas dinner is and you'll here a wide range of answers. But we didn't ask any American, we asked AI In most cognitive tests, Bard outperforms GPT-4, which powers ChatGPT. Microsoft's bot can tell users when an omelet is cooked, suggest the best design for a paper airplane or help a football player improve their skills.

It's a meal that many of us look forward to all year. But what exactly are the best items in a British Christmas Dinner? While many of us see the Roast Turkey, Goose or Ham as the main event, others prefer the trimmings, whether it's pigs in blankets, stuffing, or even Brussels Sprouts. With just 10 days to go before we get to devour our Christmas Dinner, MailOnline asked ChatGPT to rank the elements on the meal. So, do you agree with the AI chatbot's ranking? To get to the bottom of the Christmas Dinner ranking, MailOnline simply asked ChatGPT: 'How do you rank the elements of a British Christmas dinner?' Within seconds, the AI bot began to reply, diplomatically stating that'the ranking of elements can vary based on personal preferences and regional traditions.'

But despite dating back to the 14th century, diners still can't agree on the best items on a Full English Breakfast. While many enjoy the meaty elements, others argue that a Full English just isn't complete without a healthy serving of grilled mushrooms. To settle the debate once and for all this Full English Breakfast Day, MailOnline has enlisted the help of ChatGPT. So, do you agree with the AI bot's rankings? But despite dating back to the 14th century, diners still can't agree on the best items on a Full English Breakfast Today is Full English Breakfast Day, which is celebrated by millions of people internationally, according to the English Breakfast Society.

Ranking alternatives is a natural way for humans to explain their preferences. It is used in many settings, such as school choice, course allocations and residency matches. Without having any information on the underlying cardinal utilities, arguing about the fairness of allocations requires extending the ordinal item ranking to ordinal bundle ranking. The most commonly used such extension is stochastic dominance (SD), where a bundle X is preferred over a bundle Y if its score is better according to all additive score functions. SD is a very conservative extension, by which few allocations are necessarily fair while many allocations are possibly fair. We propose to make a natural assumption on the underlying cardinal utilities of the players, namely that the difference between two items at the top is larger than the difference between two items at the bottom. This assumption implies a preference extension which we call diminishing differences (DD), where X is preferred over Y if its score is better according to all additive score functions satisfying the DD assumption. We give a full characterization of allocations that are necessarily-proportional or possibly-proportional according to this assumption. Based on this characterization, we present a polynomial-time algorithm for finding a necessarily-DD-proportional allocation whenever it exists. Using simulations, we compare the various fairness criteria in terms of their probability of existence, and their probability of being fair by the underlying cardinal valuations. We find that necessary-DD-proportionality fares well in both measures. We also consider envy-freeness and Pareto optimality under diminishing-differences, as well as chore allocation under the analogous condition --- increasing-differences.

Random utility models (RUMs) are a popular and well-established framework for studying behavioral choices by individuals and groups Thurstone [1927]. In a RUM with finite alternatives or items, a distribution on the preferred alternative(s) is assumed to arise from a random utility drawn from a distribution for each item, followed by rank ordering the items according to their utilities. Perhaps the most widely known RUM is the Plackett-Luce or multinomial logit model Plackett [1975], Luce [2012] which results when each item's utility is sampled from an additive model with a Gumbel-distributed perturbation. It is unique in the sense of enjoying the property of independence of irrelevant attributes (IIA), which is often key in permitting efficient inference of Plackett-Luce models from data Khetan and Oh [2016]. Other well-known RUMs include the probit model Bliss [1934] featuring random Gaussian perturbations to the intrinsic utilities, mixed logit, nested logit, etc. A long line of work in statistics and machine learning focuses on estimating RUM properties from observed data Soufiani et al. [2014], Zhao et al. [2018], Soufiani et al. [2013].

We consider PAC learning for identifying a good item from subset-wise samples in \pl\, probability models, with instance-dependent sample complexity performance. For the setting where subsets of a fixed size can be tested and top-ranked feedback is made available to the learner each time, we give the first $(\epsilon,\delta)$-PAC best item algorithm with an instance-dependent sample complexity bound. The algorithm relies on a wrapper that uses a weaker PAC algorithm with worst-case performance guarantees to adapt to the hardness of the input instance. The sample complexity is shown to be multiplicatively better depending on the length of rank-ordered feedback available in each subset play. We also give a new fixed-budget best-item algorithm for the \pl\, model along with an error bound. Numerical results of simulations of the algorithms are reported.

We introduce the probably approximately correct (PAC) version of the problem of {Battling-bandits} with the Plackett-Luce (PL) model -- an online learning framework where in each trial, the learner chooses a subset of $k \le n$ arms from a pool of fixed set of $n$ arms, and subsequently observes a stochastic feedback indicating preference information over the items in the chosen subset; e.g., the most preferred item or ranking of the top $m$ most preferred items etc. The objective is to recover an `approximate-best' item of the underlying PL model with high probability. This framework is motivated by practical settings such as recommendation systems and information retrieval, where it is easier and more efficient to collect relative feedback for multiple arms at once. Our framework can be seen as a generalization of the well-studied PAC-{Dueling-Bandit} problem over set of $n$ arms. We propose two different feedback models: just the winner information (WI), and ranking of top-$m$ items (TR), for any $2\le m \le k$. We show that with just the winner information (WI), one cannot recover the `approximate-best' item with sample complexity lesser than $\Omega\bigg( \frac{n}{\epsilon^2} \ln \frac{1}{\delta}\bigg)$, which is independent of $k$, and same as the one required for standard dueling bandit setting ($k=2$). However with top-$m$ ranking (TR) feedback, our lower analysis proves an improved sample complexity guarantee of $\Omega\bigg( \frac{n}{m\epsilon^2} \ln \frac{1}{\delta}\bigg)$, which shows a relative improvement of $\frac{1}{m}$ factor compared to WI feedback, rightfully justifying the additional information gain due to the knowledge of ranking of topmost $m$ items. We also provide algorithms for each of the above feedback models, our theoretical analyses proves the {optimality} of their sample complexities which matches the derived lower bounds (upto logarithmic factors).

ABSTRACT Quality assurance remains a key topic in human computation research. Prior work indicates that majority voting is effective for low difficulty tasks, but has limitations for harder tasks. This paper explores two methods of addressing this problem: tournament selection and elimination selection, which exploit 2-, 3-and 4-way comparisons between different answers to human computation tasks. Our experimental results and statistical analyses show that both methods produce the correct answer in noisy human computation environment more often than majority voting. Furthermore, we find that the use of 4-way comparisons can significantly reduce the cost of quality assurance relative to the use of 2-way comparisons. INTRODUCTION Human computation is a growing research field that holds promise of humans and computers working seamlessly together to implement powerful systems. Algorithmically aggregating outputs from human computation workers is the key to such an integrated human-computer system (Little & Sun 2011).