In online marketplaces, dynamic assortment selection and pricing for sequentially arriving buyers presents a challenge for online learning. Since the preferences of buyers are varying and uncertain, adaptive strategies are essential to meet their needs and maximize the effectiveness of offers. To address this problem, we investigate the application of online learning techniques for contextual assortment selection and pricing. Assortment selection involves the seller choosing a subset of items from a vast catalog to present to buyers, and dynamically assigning prices to the offered items. The overall goal is to maximize revenue over the course of repeated interactions. Dynamic assortment selection and pricing strategies are deployed in a variety of online sectors including e-commerce (e.g., Amazon), food delivery (e.g., Uber Eats), and hospitality (e.g., Airbnb). With similar systems becoming ubiquitous in our daily lives, there is a growing opportunity to deliver tailored product recommendations and pricing adjustments. Therefore, it is crucial to consider data-driven approaches that can enhance user experiences and boost profitability in today's highly competitive digital industry.

Large language models (LLMs) have revolutionized machine learning due to their ability to capture complex interactions between input features. Popular post-hoc explanation methods like SHAP provide marginal feature attributions, while their extensions to interaction importances only scale to small input lengths ($\approx 20$). We propose Spectral Explainer (SPEX), a model-agnostic interaction attribution algorithm that efficiently scales to large input lengths ($\approx 1000)$. SPEX exploits underlying natural sparsity among interactions -- common in real-world data -- and applies a sparse Fourier transform using a channel decoding algorithm to efficiently identify important interactions. We perform experiments across three difficult long-context datasets that require LLMs to utilize interactions between inputs to complete the task. For large inputs, SPEX outperforms marginal attribution methods by up to 20% in terms of faithfully reconstructing LLM outputs. Further, SPEX successfully identifies key features and interactions that strongly influence model output. For one of our datasets, HotpotQA, SPEX provides interactions that align with human annotations. Finally, we use our model-agnostic approach to generate explanations to demonstrate abstract reasoning in closed-source LLMs (GPT-4o mini) and compositional reasoning in vision-language models.

With the rapid growth of large-scale machine learning models in genomics, Shapley values have emerged as a popular method for model explanations due to their theoretical guarantees. While Shapley values explain model predictions locally for an individual input query sequence, extracting biological knowledge requires global explanation across thousands of input sequences. This demands exponential model evaluations per sequence, resulting in significant computational cost and carbon footprint. Herein, we develop SHAP zero, a method that estimates Shapley values and interactions with a near-zero marginal cost for future queried sequences after paying a one-time fee for model sketching. SHAP zero achieves this by establishing a surprisingly underexplored connection between the Shapley values and interactions and the Fourier transform of the model. Explaining two genomic models, one trained to predict guide RNA binding and the other to predict DNA repair outcome, we demonstrate that SHAP zero achieves orders of magnitude reduction in amortized computational cost compared to state-of-the-art algorithms, revealing almost all predictive motifs -- a finding previously inaccessible due to the combinatorial space of possible interactions.

Fourier transformations of pseudo-Boolean functions are popular tools for analyzing functions of binary sequences. Real-world functions often have structures that manifest in a sparse Fourier transform, and previous works have shown that under the assumption of sparsity the transform can be computed efficiently. But what if we want to compute the Fourier transform of functions defined over a $q$-ary alphabet? These types of functions arise naturally in many areas including biology. A typical workaround is to encode the $q$-ary sequence in binary, however, this approach is computationally inefficient and fundamentally incompatible with the existing sparse Fourier transform techniques. Herein, we develop a sparse Fourier transform algorithm specifically for $q$-ary functions of length $n$ sequences, dubbed $q$-SFT, which provably computes an $S$-sparse transform with vanishing error as $q^n \rightarrow \infty$ in $O(Sn)$ function evaluations and $O(S n^2 \log q)$ computations, where $S = q^{n\delta}$ for some $\delta < 1$. Under certain assumptions, we show that for fixed $q$, a robust version of $q$-SFT has a sample complexity of $O(Sn^2)$ and a computational complexity of $O(Sn^3)$ with the same asymptotic guarantees. We present numerical simulations on synthetic and real-world RNA data, demonstrating the scalability of $q$-SFT to massively high dimensional $q$-ary functions.

Large-scale online recommendation systems must facilitate the allocation of a limited number of items among competing users while learning their preferences from user feedback. As a principled way of incorporating market constraints and user incentives in the design, we consider our objectives to be two-fold: maximal social welfare with minimal instability. To maximize social welfare, our proposed framework enhances the quality of recommendations by exploring allocations that optimistically maximize the rewards. To minimize instability, a measure of users' incentives to deviate from recommended allocations, the algorithm prices the items based on a scheme derived from the Walrasian equilibria. Though it is known that these equilibria yield stable prices for markets with known user preferences, our approach accounts for the inherent uncertainty in the preferences and further ensures that the users accept their recommendations under offered prices. To the best of our knowledge, our approach is the first to integrate techniques from combinatorial bandits, optimal resource allocation, and collaborative filtering to obtain an algorithm that achieves sub-linear social welfare regret as well as sub-linear instability. Empirical studies on synthetic and real-world data also demonstrate the efficacy of our strategy compared to approaches that do not fully incorporate all these aspects.

Recommendation systems when employed in markets play a dual role: they assist users in selecting their most desired items from a large pool and they help in allocating a limited number of items to the users who desire them the most. Despite the prevalence of capacity constraints on allocations in many real-world recommendation settings, a principled way of incorporating them in the design of these systems has been lacking. Motivated by this, we propose an interactive framework where the system provider can enhance the quality of recommendations to the users by opportunistically exploring allocations that maximize user rewards and respect the capacity constraints using appropriate pricing mechanisms. We model the problem as an instance of a low-rank combinatorial multi-armed bandit problem with selection constraints on the arms. We employ an integrated approach using techniques from collaborative filtering, combinatorial bandits, and optimal resource allocation to provide an algorithm that provably achieves sub-linear regret, namely $\tilde{\mathcal{O}} ( \sqrt{N M (N+M) RT} )$ in $T$ rounds for a problem with $N$ users, $M$ items and rank $R$ mean reward matrix. Empirical studies on synthetic and real-world data also demonstrate the effectiveness and performance of our approach.