We study hypothesis testing (aka state certification) in the non-identically distributed setting. A recent work (Garg et al. 2023) considered the classical case, in which one is given (independent) samples from $T$ unknown probability distributions $p_1, \dots, p_T$ on $[d] = \{1, 2, \dots, d\}$, and one wishes to accept/reject the hypothesis that their average $p_{\mathrm{avg}}$ equals a known hypothesis distribution $q$. Garg et al. showed that if one has just $c = 2$ samples from each $p_i$, and provided $T \gg \frac{\sqrt{d}}{ε^2} + \frac{1}{ε^4}$, one can (whp) distinguish $p_{\mathrm{avg}} = q$ from $d_{\mathrm{TV}}(p_{\mathrm{avg}},q) > ε$. This nearly matches the optimal result for the classical iid setting (namely, $T \gg \frac{\sqrt{d}}{ε^2}$). Besides optimally improving this result (and generalizing to tolerant testing with more stringent distance measures), we study the analogous problem of hypothesis testing for non-identical quantum states. Here we uncover an unexpected phenomenon: for any $d$-dimensional hypothesis state $σ$, and given just a single copy ($c = 1$) of each state $ρ_1, \dots, ρ_T$, one can distinguish $ρ_{\mathrm{avg}} = σ$ from $D_{\mathrm{tr}}(ρ_{\mathrm{avg}},σ) > ε$ provided $T \gg d/ε^2$. (Again, we generalize to tolerant testing with more stringent distance measures.) This matches the optimal result for the iid case, which is surprising because doing this with $c = 1$ is provably impossible in the classical case. We also show that the analogous phenomenon happens for the non-iid extension of identity testing between unknown states. A technical tool we introduce may be of independent interest: an Efron-Stein inequality, and more generally an Efron-Stein decomposition, in the quantum setting.

We study a collaborative learning problem where $m$ agents estimate a vector $\mu\in\mathbb{R}^d$ by collecting samples from normal distributions, with each agent $i$ incurring a cost $c_{i,k} \in (0, \infty]$ to sample from the $k^{\text{th}}$ distribution $\mathcal{N}(\mu_k, \sigma^2)$. Instead of working on their own, agents can collect data that is cheap to them, and share it with others in exchange for data that is expensive or even inaccessible to them, thereby simultaneously reducing data collection costs and estimation error. However, when agents have different collection costs, we need to first decide how to fairly divide the work of data collection so as to benefit all agents. Moreover, in naive sharing protocols, strategic agents may under-collect and/or fabricate data, leading to socially undesirable outcomes. Our mechanism addresses these challenges by combining ideas from cooperative and non-cooperative game theory. We use ideas from axiomatic bargaining to divide the cost of data collection. Given such a solution, we develop a Nash incentive-compatible (NIC) mechanism to enforce truthful reporting. We achieve a $\mathcal{O}(\sqrt{m})$ approximation to the minimum social penalty (sum of agent estimation errors and data collection costs) in the worst case, and a $\mathcal{O}(1)$ approximation under favorable conditions. We complement this with a hardness result, showing that $\Omega(\sqrt{m})$ is unavoidable in any NIC mechanism.

Agents' learning from feedback shapes economic outcomes, and many economic decision-makers today employ learning algorithms to make consequential choices. This note shows that a widely used learning algorithm, $\varepsilon$-Greedy, exhibits emergent risk aversion: it prefers actions with lower variance. When presented with actions of the same expectation, under a wide range of conditions, $\varepsilon$-Greedy chooses the lower-variance action with probability approaching one. This emergent preference can have wide-ranging consequences, ranging from concerns about fairness to homogenization, and holds transiently even when the riskier action has a strictly higher expected payoff. We discuss two methods to correct this bias. The first method requires the algorithm to reweight data as a function of how likely the actions were to be chosen. The second requires the algorithm to have optimistic estimates of actions for which it has not collected much data. We show that risk-neutrality is restored with these corrections.

High training costs of generative models and the need to fine-tune them for specific tasks have created a strong interest in model reuse and composition. A key challenge in composing iterative generative processes, such as GFlowNets and diffusion models, is that to realize the desired target distribution, all steps of the generative process need to be coordinated, and satisfy delicate balance conditions. In this work, we propose Compositional Sculpting: a general approach for defining compositions of iterative generative processes. We then introduce a method for sampling from these compositions built on classifier guidance. We showcase ways to accomplish compositional sculpting in both GFlowNets and diffusion models. We highlight two binary operations $\unicode{x2014}$ the harmonic mean ($p_1 \otimes p_2$) and the contrast ($p_1 \unicode{x25D1}\,p_2$) between pairs, and the generalization of these operations to multiple component distributions. We offer empirical results on image and molecular generation tasks.

The concept of median/consensus has been widely investigated in order to provide a statistical summary of ranking data, i.e. realizations of a random permutation $\Sigma$ of a finite set, $\{1,\; \ldots,\; n\}$ with $n\geq 1$ say. As it sheds light onto only one aspect of $\Sigma$'s distribution $P$, it may neglect other informative features. It is the purpose of this paper to define analogs of quantiles, ranks and statistical procedures based on such quantities for the analysis of ranking data by means of a metric-based notion of depth function on the symmetric group. Overcoming the absence of vector space structure on $\mathfrak{S}_n$, the latter defines a center-outward ordering of the permutations in the support of $P$ and extends the classic metric-based formulation of consensus ranking (medians corresponding then to the deepest permutations). The axiomatic properties that ranking depths should ideally possess are listed, while computational and generalization issues are studied at length. Beyond the theoretical analysis carried out, the relevance of the novel concepts and methods introduced for a wide variety of statistical tasks are also supported by numerous numerical experiments.

From scientific experiments to online A/B testing, the previously observed data often affects how future experiments are performed, which in turn affects which data will be collected. Such adaptivity introduces complex correlations between the data and the collection procedure. In this paper, we prove that when the data collection procedure satisfies natural conditions, then sample means of the data have systematic \emph{negative} biases. As an example, consider an adaptive clinical trial where additional data points are more likely to be tested for treatments that show initial promise. Our surprising result implies that the average observed treatment effects would underestimate the true effects of each treatment. We quantitatively analyze the magnitude and behavior of this negative bias in a variety of settings. We also propose a novel debiasing algorithm based on selective inference techniques. In experiments, our method can effectively reduce bias and estimation error.