Semidefinite programs (SDP's) can be solved in polynomial time by interior point methods, but scalability can be an issue. To address this shortcoming, over a decade ago, Burer and Monteiro proposed to solve SDP's with few equality constraints via rank-restricted, non-convex surrogates. Remarkably, for some applications, local optimization methods seem to converge to global optima of these non-convex surrogates reliably. Although some theory supports this empirical success, a complete explanation of it remains an open question. In this paper, we consider a class of SDP's which includes applications such as max-cut, community detection in the stochastic block model, robust PCA, phase retrieval and synchronization of rotations. We show that the low-rank Burer-Monteiro formulation of SDP's in that class almost never has any spurious local optima.

Such feedback can be stochastic, e.g. the

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EntropySearch (ES) [15],Predictive Entropy Search(PES)[16]and the earlier work of IAGO [46]select queries by maximizing this quantity.

Pre-trained multimodal foundation models have demonstrated remarkable generalizability but pose challenges for deployment due to their large sizes. One effective approach to reducing their sizes is layerwise distillation, wherein small student models are trained to match the hidden representations of large teacher models at each layer. Motivated by our observation that certain architecture components, referred to as modules, contribute more significantly to the student's performance than others, we propose to track the contributions of individual modules by recording the loss decrement after distillation each module and choose the module with a greater contribution to distill more frequently. Such an approach can be naturally formulated as a multi-armed bandit (MAB) problem, where modules and loss decrements are considered as arms and rewards, respectively. We then develop a modified-Thompson sampling algorithm named OPTIMA to address the nonstationarity of module contributions resulting from model updating. Specifically, we leverage the observed contributions in recent history to estimate the changing contribution of each module and select modules based on these estimations to maximize the cumulative contribution. We evaluate the effectiveness of OPTIMA through distillation experiments on various multimodal understanding and image captioning tasks, using the CoCa-Large model \citep{yu2022coca} as the teacher model.

Bayesian optimization (BO) is a popular method for efficiently inferring optima of an expensive black-box function via a sequence of queries. Existing information-theoretic BO procedures aim to make queries that most reduce the uncertainty about optima, where the uncertainty is captured by Shannon entropy. However, an optimal measure of uncertainty would, ideally, factor in how we intend to use the inferred quantity in some downstream procedure. In this paper, we instead consider a generalization of Shannon entropy from work in statistical decision theory (DeGroot 1962, Rao 1984), which contains a broad class of uncertainty measures parameterized by a problem-specific loss function corresponding to a downstream task. We first show that special cases of this entropy lead to popular acquisition functions used in BO procedures such as knowledge gradient, expected improvement, and entropy search. We then show how alternative choices for the loss yield a flexible family of acquisition functions that can be customized for use in novel optimization settings. Additionally, we develop gradient-based methods to efficiently optimize our proposed family of acquisition functions, and demonstrate strong empirical performance on a diverse set of sequential decision making tasks, including variants of top-$k$ optimization, multi-level set estimation, and sequence search.

Non-convex optimization with local search heuristics has been widely used in machine learning, achieving many state-of-art results. It becomes increasingly important to understand why they can work for these NP-hard problems on typical data. The landscape of many objective functions in learning has been conjectured to have the geometric property that ``all local optima are (approximately) global optima'', and thus they can be solved efficiently by local search algorithms. However, establishing such property can be very difficult. In this paper, we analyze the optimization landscape of the random over-complete tensor decomposition problem, which has many applications in unsupervised leaning, especially in learning latent variable models.