207 research outputs found

    Optimal No-regret Learning in Repeated First-price Auctions

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    We study online learning in repeated first-price auctions with censored feedback, where a bidder, only observing the winning bid at the end of each auction, learns to adaptively bid in order to maximize her cumulative payoff. To achieve this goal, the bidder faces a challenging dilemma: if she wins the bid--the only way to achieve positive payoffs--then she is not able to observe the highest bid of the other bidders, which we assume is iid drawn from an unknown distribution. This dilemma, despite being reminiscent of the exploration-exploitation trade-off in contextual bandits, cannot directly be addressed by the existing UCB or Thompson sampling algorithms in that literature, mainly because contrary to the standard bandits setting, when a positive reward is obtained here, nothing about the environment can be learned. In this paper, by exploiting the structural properties of first-price auctions, we develop the first learning algorithm that achieves O(Tlog2T)O(\sqrt{T}\log^2 T) regret bound when the bidder's private values are stochastically generated. We do so by providing an algorithm on a general class of problems, which we call monotone group contextual bandits, where the same regret bound is established under stochastically generated contexts. Further, by a novel lower bound argument, we characterize an Ω(T2/3)\Omega(T^{2/3}) lower bound for the case where the contexts are adversarially generated, thus highlighting the impact of the contexts generation mechanism on the fundamental learning limit. Despite this, we further exploit the structure of first-price auctions and develop a learning algorithm that operates sample-efficiently (and computationally efficiently) in the presence of adversarially generated private values. We establish an O(Tlog3T)O(\sqrt{T}\log^3 T) regret bound for this algorithm, hence providing a complete characterization of optimal learning guarantees for this problem

    Development of Hypoxia Trapping Enhanced BB2R-Targeted Radiopharmaceutics for Prostate Cancer

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    The Gastrin-Releasing Peptide Receptor (BB2r) has been investigated as a diagnostic and therapeutic target for prostate and other cancers due to the high expression level on neoplastic relative to normal tissues. A variety of BB2r-targeted agents have been developed utilizing the bombesin(BBN) peptide, which has shown nanomolar binding affinity to human BB2r. However, as with most of the low-molecular weight, receptor-targeted drugs, a major challenge to clinical translation of BB2r-targeted agents is the low retention at the tumor site due to intrinsically high diffusion and efflux rates. Our laboratory seeks to address this deficiency by developing synthetic approaches to selectively increase retention of BB2r-targeted agents in prostate cancer. Hypoxic regions commonly exist in prostate tumors and many other cancers due to a chaotic vascular architecture which impedes delivery of oxygen. In this dissertation, we explore the incorporation of nitroimidazoles, a hypoxia-selective prodrug which irreversibly binds to intracellular nucleophiles in hypoxic tissues, into the BB2r-targeted agent paradigm. We seek to determine if these agents can increase the long-term retention in the tumor and thereby increase efficacy and clinical potential of BB2r-targeted agents. To that end, we have developed several generations of hypoxia trapping enhanced BBN analogs. Our first in vitro investigation of hypoxia-enhanced 111In-labeled BBN conjugates demonstrated significantly improved retention in hypoxic PC-3 human prostate cancer cells. However, it was determined that the proximity of the 2-nitroimidazole relative to the pharmacophore had a detrimental impact on BB2r binding affinity. To address the problem, our next generation of radioconjugates contained an extended linker to eliminate steric inhibition. The new design demonstrated substantially improved binding affinity and lower clearance rate of the 2-nitroimidazole containing radioconjugates under hypoxic conditions. In vivo biodistribution studies using a PC-3 xenograft mouse model revealed significant tumor retention enhancement. Further work is needed to clarify the mechanisms of cellular retention and to correlate the tumor hypoxia burden with the retention efficacy

    Stochastic Nonsmooth Convex Optimization with Heavy-Tailed Noises

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    Recently, several studies consider the stochastic optimization problem but in a heavy-tailed noise regime, i.e., the difference between the stochastic gradient and the true gradient is assumed to have a finite pp-th moment (say being upper bounded by σp\sigma^{p} for some σ0\sigma\geq0) where p(1,2]p\in(1,2], which not only generalizes the traditional finite variance assumption (p=2p=2) but also has been observed in practice for several different tasks. Under this challenging assumption, lots of new progress has been made for either convex or nonconvex problems, however, most of which only consider smooth objectives. In contrast, people have not fully explored and well understood this problem when functions are nonsmooth. This paper aims to fill this crucial gap by providing a comprehensive analysis of stochastic nonsmooth convex optimization with heavy-tailed noises. We revisit a simple clipping-based algorithm, whereas, which is only proved to converge in expectation but under the additional strong convexity assumption. Under appropriate choices of parameters, for both convex and strongly convex functions, we not only establish the first high-probability rates but also give refined in-expectation bounds compared with existing works. Remarkably, all of our results are optimal (or nearly optimal up to logarithmic factors) with respect to the time horizon TT even when TT is unknown in advance. Additionally, we show how to make the algorithm parameter-free with respect to σ\sigma, in other words, the algorithm can still guarantee convergence without any prior knowledge of σ\sigma

    Revisiting the Last-Iterate Convergence of Stochastic Gradient Methods

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    In the past several years, the last-iterate convergence of the Stochastic Gradient Descent (SGD) algorithm has triggered people's interest due to its good performance in practice but lack of theoretical understanding. For Lipschitz convex functions, different works have established the optimal O(log(1/δ)logT/T)O(\log(1/\delta)\log T/\sqrt{T}) or O(log(1/δ)/T)O(\sqrt{\log(1/\delta)/T}) high-probability convergence rates for the final iterate, where TT is the time horizon and δ\delta is the failure probability. However, to prove these bounds, all the existing works are either limited to compact domains or require almost surely bounded noises. It is natural to ask whether the last iterate of SGD can still guarantee the optimal convergence rate but without these two restrictive assumptions. Besides this important question, there are still lots of theoretical problems lacking an answer. For example, compared with the last-iterate convergence of SGD for non-smooth problems, only few results for smooth optimization have yet been developed. Additionally, the existing results are all limited to a non-composite objective and the standard Euclidean norm. It still remains unclear whether the last-iterate convergence can be provably extended to wider composite optimization and non-Euclidean norms. In this work, to address the issues mentioned above, we revisit the last-iterate convergence of stochastic gradient methods and provide the first unified way to prove the convergence rates both in expectation and in high probability to accommodate general domains, composite objectives, non-Euclidean norms, Lipschitz conditions, smoothness, and (strong) convexity simultaneously. Additionally, we extend our analysis to obtain the last-iterate convergence under heavy-tailed noises.Comment: The preliminary version has been accepted at ICLR 2024. This extended version was finished in November 2023 and revised in March 2024 with fixed typo

    On the Last-Iterate Convergence of Shuffling Gradient Methods

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    Shuffling gradient methods are widely used in modern machine learning tasks and include three popular implementations: Random Reshuffle (RR), Shuffle Once (SO), and Incremental Gradient (IG). Compared to the empirical success, the theoretical guarantee of shuffling gradient methods was not well-understood for a long time. Until recently, the convergence rates had just been established for the average iterate for convex functions and the last iterate for strongly convex problems (using squared distance as the metric). However, when using the function value gap as the convergence criterion, existing theories cannot interpret the good performance of the last iterate in different settings (e.g., constrained optimization). To bridge this gap between practice and theory, we prove the first last-iterate convergence rates for shuffling gradient methods with respect to the objective value even without strong convexity. Our new results either (nearly) match the existing last-iterate lower bounds or are as fast as the previous best upper bounds for the average iterate.Comment: ICML 202
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