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1. Approximate Degree Lower Bounds for Oracle Identification Problems

   Bun, Mark; Voronova, Nadezhda (July 2023, 18th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2023))

   Fawzi, Omar; Walter, Michael (Ed.)

   The approximate degree of a Boolean function is the minimum degree of real polynomial that approximates it pointwise. For any Boolean function, its approximate degree serves as a lower bound on its quantum query complexity, and generically lifts to a quantum communication lower bound for a related function. We introduce a framework for proving approximate degree lower bounds for certain oracle identification problems, where the goal is to recover a hidden binary string x ∈ {0, 1}ⁿ given possibly non-standard oracle access to it. Our lower bounds apply to decision versions of these problems, where the goal is to compute the parity of x. We apply our framework to the ordered search and hidden string problems, proving nearly tight approximate degree lower bounds of Ω(n/log² n) for each. These lower bounds generalize to the weakly unbounded error setting, giving a new quantum query lower bound for the hidden string problem in this regime. Our lower bounds are driven by randomized communication upper bounds for the greater-than and equality functions. 

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2. The Complexity of Making the Gradient Small in Stochastic Convex Optimization

   Foster, Dylan Y; Sekhari, Ayush; Shamir, Ohad; Srebro, Nathan; Sridharan, Karthik; Woodworth, Blake (June 2019, Proceedings of Machine Learning Research)

   We give nearly matching upper and lower bounds on the oracle complexity of finding ϵ-stationary points (∥∇F(x)∥≤ϵ in stochastic convex optimization. We jointly analyze the oracle complexity in both the local stochastic oracle model and the global oracle (or, statistical learning) model. This allows us to decompose the complexity of finding near-stationary points into optimization complexity and sample complexity, and reveals some surprising differences between the complexity of stochastic optimization versus learning. Notably, we show that in the global oracle/statistical learning model, only logarithmic dependence on smoothness is required to find a near-stationary point, whereas polynomial dependence on smoothness is necessary in the local stochastic oracle model. In other words, the separation in complexity between the two models can be exponential, and the folklore understanding that smoothness is required to find stationary points is only weakly true for statistical learning. Our upper bounds are based on extensions of a recent “recursive regularization” technique proposed by Allen-Zhu (2018). We show how to extend the technique to achieve near-optimal rates, and in particular show how to leverage the extra information available in the global oracle model. Our algorithm for the global model can be implemented efficiently through finite sum methods, and suggests an interesting new computational-statistical tradeoff 

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3. The Bayesian Prophet: A Low-Regret Framework for Online Decision Making

   https://doi.org/10.1287/mnsc.2020.3624  

   Vera, Alberto; Banerjee, Siddhartha (March 2021, Management Science)

   null (Ed.)

   We develop a new framework for designing online policies given access to an oracle providing statistical information about an off-line benchmark. Having access to such prediction oracles enables simple and natural Bayesian selection policies and raises the question as to how these policies perform in different settings. Our work makes two important contributions toward this question: First, we develop a general technique we call compensated coupling, which can be used to derive bounds on the expected regret (i.e., additive loss with respect to a benchmark) for any online policy and off-line benchmark. Second, using this technique, we show that a natural greedy policy, which we call the Bayes selector, has constant expected regret (i.e., independent of the number of arrivals and resource levels) for a large class of problems we refer to as “online allocation with finite types,” which includes widely studied online packing and online matching problems. Our results generalize and simplify several existing results for online packing and online matching and suggest a promising pathway for obtaining oracle-driven policies for other online decision-making settings. This paper was accepted by George Shanthikumar, big data analytics. 

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4. A unified approach for maximizing continuous DR-submodular functions

   Mohammad Pedramfar, Christopher John (December 2023, Advances in Neural Information Processing Systems)

   This paper presents a unified approach for maximizing continuous DRsubmodular functions that encompasses a range of settings and oracle access types. Our approach includes a Frank-Wolfe type offline algorithm for both monotone and non-monotone functions, with different restrictions on the general convex set. We consider settings where the oracle provides access to either the gradient of the function or only the function value, and where the oracle access is either deterministic or stochastic. We determine the number of required oracle accesses in all cases. Our approach gives new/improved results for nine out of the sixteen considered cases, avoids computationally expensive projections in three cases, with the proposed framework matching performance of state-of-the-art approaches in the remaining four cases. Notably, our approach for the stochastic function valuebased oracle enables the first regret bounds with bandit feedback for stochastic DR-submodular functions. 

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5. A Unified Approach for Maximizing Continuous DR-submodular Functions

   Pedramfar, Mohammad; Quinn, Christopher; Aggarwal, Vaneet (December 2023, Curran Associates, Inc.)

   Oh, A; Naumann, T; Globerson, A; Saenko, K; Hardt, M; Levine, S (Ed.)

   This paper presents a unified approach for maximizing continuous DR-submodular functions that encompasses a range of settings and oracle access types. Our approach includes a Frank-Wolfe type offline algorithm for both monotone and non-monotone functions, with different restrictions on the general convex set. We consider settings where the oracle provides access to either the gradient of the function or only the function value, and where the oracle access is either deterministic or stochastic. We determine the number of required oracle accesses in all cases. Our approach gives new/improved results for nine out of the sixteen considered cases, avoids computationally expensive projections in three cases, with the proposed framework matching performance of state-of-the-art approaches in the remaining four cases. Notably, our approach for the stochastic function value-based oracle enables the first regret bounds with bandit feedback for stochastic DR-submodular functions. 

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