This introduces a Online advertising, a multi-billion dollar industry, relies on realtime challenging exploration-exploitation dilemma: the advertiser must auctions to connect advertisers with users. These auctions, balance exploring different bids to estimate impression values with triggered by user queries or website visits, allow advertisers to exploiting current knowledge to bid effectively. To address this, we bid for advertising slots, such as prominent placements on search propose a novel Upper Confidence Bound (UCB)-style algorithm engine results pages or in social media feeds. Advertisers aim to that carefully manages this trade-off. Via a rigorous theoretical maximize their returns, measured in conversions or other relevant analysis, we prove that our algorithm achieves ( log(|B|)) metrics, by carefully determining their bids while adhering to regret and constraint violation, where is the number of bidding budget constraints and desired return-on-spend (RoS) targets. To rounds and B is the domain of possible bids. This establishes the achieve this, a wide array of bidding strategies have been developed, first optimal regret and constraint violation bounds for bidding in leveraging techniques from optimization, online learning, and game the online setting with unknown impression values. Moreover, our theory to maximize advertiser utility [2, 7, 11, 23, 29, 30, 36, 47, 52].

Imitation learning has emerged as a powerful tool in machine learning, enabling agents to learn complex behaviors by imitating expert demonstrations acquired either from a human demonstrator or a policy computed offline [3, 11, 12, 13]. Despite its significant success, imitation learning often suffers from a compounding error problem: Successive evaluations of the approximate policy could accumulate error, resulting in out-of-distribution failures [3]. Recent results in imitation learning [31, 32, 34] have identified smoothness (i.e., Lipschitzness of the derivative of the optimal controller with respect to the initial state) and stability of the expert as two key properties that circumvent this issue, thereby allowing for end-to-end performance guarantees for the final learned controller. In this paper, our focus is on enabling such guarantees when the expert being imitated is a Model Predictive Controller (MPC), a powerful class of control algorithms based on solving an optimization problem over a receding prediction horizon [23]. In some cases, the solution to this multiparametric optimization problem, known as the explicit MPC representation [6], can be pre-computed. For instance, in our setup -- linear systems with polytopic constraints -- the optimal control input is a piecewise affine (and, hence, highly non-smooth) function of the state [6].

Algorithms for bilevel optimization often encounter Hessian computations, which are prohibitive in high dimensions. While recent works offer first-order methods for unconstrained bilevel problems, the constrained setting remains relatively underexplored. We present first-order linearly constrained optimization methods with finite-time hypergradient stationarity guarantees. For linear equality constraints, we attain $\epsilon$-stationarity in $\widetilde{O}(\epsilon^{-2})$ gradient oracle calls, which is nearly-optimal. For linear inequality constraints, we attain $(\delta,\epsilon)$-Goldstein stationarity in $\widetilde{O}(d{\delta^{-1} \epsilon^{-3}})$ gradient oracle calls, where $d$ is the upper-level dimension. Finally, we obtain for the linear inequality setting dimension-free rates of $\widetilde{O}({\delta^{-1} \epsilon^{-4}})$ oracle complexity under the additional assumption of oracle access to the optimal dual variable. Along the way, we develop new nonsmooth nonconvex optimization methods with inexact oracles. We verify these guarantees with preliminary numerical experiments.

Recent works in dimensionality reduction for regression tasks have introduced the notion of sensitivity, an estimate of the importance of a specific datapoint in a dataset, offering provable guarantees on the quality of the approximation after removing low-sensitivity datapoints via subsampling. However, fast algorithms for approximating $\ell_p$ sensitivities, which we show is equivalent to approximate $\ell_p$ regression, are known for only the $\ell_2$ setting, in which they are termed leverage scores. In this work, we provide efficient algorithms for approximating $\ell_p$ sensitivities and related summary statistics of a given matrix. In particular, for a given $n \times d$ matrix, we compute $\alpha$-approximation to its $\ell_1$ sensitivities at the cost of $O(n/\alpha)$ sensitivity computations. For estimating the total $\ell_p$ sensitivity (i.e. the sum of $\ell_p$ sensitivities), we provide an algorithm based on importance sampling of $\ell_p$ Lewis weights, which computes a constant factor approximation to the total sensitivity at the cost of roughly $O(\sqrt{d})$ sensitivity computations. Furthermore, we estimate the maximum $\ell_1$ sensitivity, up to a $\sqrt{d}$ factor, using $O(d)$ sensitivity computations. We generalize all these results to $\ell_p$ norms for $p > 1$. Lastly, we experimentally show that for a wide class of matrices in real-world datasets, the total sensitivity can be quickly approximated and is significantly smaller than the theoretical prediction, demonstrating that real-world datasets have low intrinsic effective dimensionality.

Online advertising has recently grown into a highly competitive and complex multi-billion-dollar industry, with advertisers bidding for ad slots at large scales and high frequencies. This has resulted in a growing need for efficient "auto-bidding" algorithms that determine the bids for incoming queries to maximize advertisers' targets subject to their specified constraints. This work explores efficient online algorithms for a single value-maximizing advertiser under an increasingly popular constraint: Return-on-Spend (RoS). We quantify efficiency in terms of regret relative to the optimal algorithm, which knows all queries a priori. We contribute a simple online algorithm that achieves near-optimal regret in expectation while always respecting the specified RoS constraint when the input sequence of queries are i.i.d. samples from some distribution. We also integrate our results with the previous work of Balseiro, Lu, and Mirrokni [BLM20] to achieve near-optimal regret while respecting both RoS and fixed budget constraints. Our algorithm follows the primal-dual framework and uses online mirror descent (OMD) for the dual updates. However, we need to use a non-canonical setup of OMD, and therefore the classic low-regret guarantee of OMD, which is for the adversarial setting in online learning, no longer holds. Nonetheless, in our case and more generally where low-regret dynamics are applied in algorithm design, the gradients encountered by OMD can be far from adversarial but influenced by our algorithmic choices. We exploit this key insight to show our OMD setup achieves low regret in the realm of our algorithm.

Within machine learning, NNLS problems arise whenever having negative labels is not meaningful, for example, when labels represent quantities like prices, age, pixel intensities, chemical concentrations, or frequency counts. NNLS is also widely used as a subroutine in nonnegative matrix factorization to extract sparse features in applications like image processing, computational biology, clustering, collaborative filtering, and community detection [Gil14]. From a statistical perspective, NNLS problems can be shown to possess a regularization property that enforces sparsity similar to LASSO [Tib96], while being comparatively simpler, without the need to tune a regularization parameter or perform cross-validation [SH14, BEZ08, FK14]. From an algorithmic standpoint, the nonnegativity constraint in NNLS problems is typically viewed as an obstacle: most NNLS algorithms need to perform additional work to handle it, and the problem is considered harder than unconstrained least squares. However, in many applications that use NNLS, the data is also nonnegative. This is true, for example, in problems arising in image processing, computational genomics, functional MRI, and in applications traditionally addressed using nonnegative matrix factorization. We argue in this paper that when the data for NNLS is nonnegative, it is in fact possible to obtain stronger guarantees than for traditional least squares problems.