Sparse recovery is among the most well-studied problems in learning theory and high-dimensional statistics. In this work, we investigate the statistical and computational landscapes of sparse recovery with $\ell_\infty$ error guarantees. This variant of the problem is motivated by \emph{variable selection} tasks, where the goal is to estimate the support of a $k$-sparse signal in $\R^d$. Our main contribution is a provable separation between the \emph{oblivious} (“for each”) and \emph{adaptive} (“for all”) models of $\ell_\infty$ sparse recovery. We show that under an oblivious model, the optimal $\ell_\infty$ error is attainable in near-linear time with $\approx k\log d$ samples, whereas in an adaptive model, $\gtrsim k^2$ samples are necessary for any algorithm to achieve this bound. This establishes a surprising contrast with the standard $\ell_2$ setting, where $\approx k \log d$ samples suffice even for adaptive sparse recovery. We conclude with a preliminary examination of a \emph{partially-adaptive} model, where we show nontrivial variable selection guarantees are possible with $\approx k\log d$ measurements.

In list-decodable learning, we are given a set of data points such that an α-fraction of these points come from a “nice” distribution D, for some small α ≪ 1, and the goal is to output a short list of candidate solutions, such that at least one element of this list recovers some non-trivial information about D. By now, there is a large body of work on this topic; however, while many algorithms can achieve optimal list size in terms of α, all known algorithms must incur error which decays, in some cases quite poorly, with 1 / α. In this paper, we ask if this is inherent: is it possible to trade off list size with accuracy in list-decodable learning? More formally, given ε > 0, can we output a slightly larger list in terms of α and ε, but so that one element of this list has error at most ε with the ground truth? We call this problem high-accuracy list-decodable learning.

The prophet secretary problem is a combination of the prophet inequality and the secretary problem, where elements are drawn from known independent distributions and arrive in uniformly random order. In this work, we design 1) a 0.688-competitive algorithm, that breaks the 0.675 barrier of blind strategies (Correa, Saona, Ziliotto, 2021), and 2) a 0.641-competitive algorithm for the prophet secretary matching problem, that breaks the 1 — 1/e ≈ 0.632 barrier for the first time. Our second result also applies to the query-commit model of weighted stochastic matching and improves the state-of-the-art ratio (Derakhshan and Farhadi, 2023).

We initiate the study of Stochastic Online Correlated Selection (SOCS), a family of online rounding algorithms for the general Non-IID model of Stochastic Online Submodular Welfare Maximization and its special cases such as Online Stochastic Matching, Stochastic Ad-Words, and Stochastic Display Ads. At each time step, the algorithm sees the type of an online item and a fractional allocation of the item, then immediately allocates the item to an agent. We propose a metric called the convergence rate that measures the quality of SOCS algorithms in the above special cases. This is cleaner than most metrics in the related Online Correlated Selection (OCS) literature and may be of independent interest. We propose a Type Decomposition framework that reduces the design of SOCS algorithms to the easier special case of two-way SOCS. First, we sample a surrogate type whose fractional allocation is half-integer. The rounding is trivial for a one-way surrogate type fully allocated to one agent. For a two-way surrogate type split equally between two agents, we round it using a two-way SOCS. We design the distribution of surrogate types to get two-way types as often as possible, while respecting the original fractional allocation in expectation. Following this framework, we make progress on nu-merous problems including two open questions related to AdWords.

We study revenue maximization in multi-item auctions, where bidders have subadditive valuations over independent items [48]. Providing a simple mechanism that is approximately revenue-optimal in this setting is a major open problem in mechanism design [20]. In this paper, we present the first simple mechanism whose revenue is at least a constant fraction of the optimal revenue in multi-item auctions with subadditive bidders. Our mechanism is a simultaneous auction that incorporates either a personalized entry fee or a personalized reserve price per item. We prove that for any simultaneous auction that satisfies c-efficiency– a new property we propose, its revenue is at least an $O(c)$-approximation to the optimal revenue. We further show that both the simultaneous first-price and the simultaneous all-pay auction are $\frac{1}{2}$-efficient. Providing revenue guarantees for non-truthful simple mechanisms, e.g., simultaneous auctions, in multi-dimensional environments has been recognized by Roughgarden et al. [47] as an important open question. Prior to our result, the only such revenue guarantees are due to Daskalakis et al. [30] for bidders who have additive valuations over independent items. Our result significantly extends the revenue guarantees of these non-truthful simple auctions to settings where bidders have combinatorial valuations.

Consider Myerson's optimal auction with respect to an inaccurate prior, e.g., estimated from data, which is an underestimation of the true value distribution. Can the auctioneer expect getting at least the optimal revenue w.r.t. the inaccurate prior since the true value distribution is larger? This so-called strong revenue monotonicity is known to be true for single-parameter auctions when the feasible allocations form a matroid. We find that strong revenue monotonicity fails to generalize beyond the matroid setting, and further show that auctions in the matroid setting are the only downward-closed auctions that satisfy strong revenue monotonicity. On the flip side, we recover an approximate version of strong revenue monotonicity that holds for all single-parameter auctions, even without downward-closedness. As applications, we get sample complexity upper bounds for single-parameter auctions under matroid constraints, downward-closed constraints, and general constraints. They improve the state-of-the-art upper bounds and are tight up to logarithmic factors.