The task of tuning regularization coefficients in regularized regression models with provable guarantees across problem instances still poses a significant challenge in the literature. This paper investigates the sample complexity of tuning regularization parameters in linear and logistic regressions under ℓ1 and ℓ2-constraints in the data-driven setting. For the linear regression problem, by more carefully exploiting the structure of the dual function class, we provide a new upper bound for the pseudo-dimension of the validation loss function class, which significantly improves the best-known results on the problem. Remarkably, we also instantiate the first matching lower bound, proving our results are tight. For tuning the regularization parameters of logistic regression, we introduce a new approach to studying the learning guarantee via an approximation of the validation loss function class. We examine the pseudo-dimension of the approximation class and construct a uniform error bound between the validation loss function class and its approximation, which allows us to instantiate the first learning guarantee for the problem of tuning logistic regression regularization coefficients.

In this work, we investigate an alternative setting for tuning regularization parameters, namely data-driven algorithm design, following the previous line of work by Balcan et al. [

Logistic regression is commonly used for modeling dichotomous outcomes.

Logistic regression is commonly used for modeling dichotomous outcomes. In the classical setting, where the number of observations is much larger than the number of parameters, properties of the maximum likelihood estimator in logistic regression are well understood. Recently, Sur and Candes~\cite{sur2018modern} have studied logistic regression in the high-dimensional regime, where the number of observations and parameters are comparable, and show, among other things, that the maximum likelihood estimator is biased. In the high-dimensional regime the underlying parameter vector is often structured (sparse, block-sparse, finite-alphabet, etc.) and so in this paper we study regularized logistic regression (RLR), where a convex regularizer that encourages the desired structure is added to the negative of the log-likelihood function. An advantage of RLR is that it allows parameter recovery even for instances where the (unconstrained) maximum likelihood estimate does not exist. We provide a precise analysis of the performance of RLR via the solution of a system of six nonlinear equations, through which any performance metric of interest (mean, mean-squared error, probability of support recovery, etc.) can be explicitly computed. Our results generalize those of Sur and Candes and we provide a detailed study for the cases of $\ell_2^2$-RLR and sparse ($\ell_1$-regularized) logistic regression. In both cases, we obtain explicit expressions for various performance metrics and can find the values of the regularizer parameter that optimizes the desired performance. The theory is validated by extensive numerical simulations across a range of parameter values and problem instances.

In this work, we investigate an alternative setting for tuning regularization parameters, namely data-driven algorithm design, following the previous line of work by Balcan et al. [

In this work, we introduce a novel approach for optimizing LLM training by adjusting learning rates across weights of different components in Transformer models. Traditional methods often apply a uniform learning rate across all network layers, potentially overlooking the unique dynamics of each part. Remarkably, our introduced relative learning rates, RLRS, method accelerates the training process by up to $23\%$, particularly in complex models such as Mixture of Experts (MoE). Hyperparameters of RLRS can be efficiently tuned on smaller models and then effectively reused on models up to $27\times$ larger. This simple and effective method results in a substantial reduction in training time and computational resources, offering a practical and scalable solution for optimizing large-scale neural networks.

The probabilistic diffusion model (DM), generating content by inferencing through a recursive chain structure, has emerged as a powerful framework for visual generation. After pre-training on enormous unlabeled data, the model needs to be properly aligned to meet requirements for downstream applications. How to efficiently align the foundation DM is a crucial task. Contemporary methods are either based on Reinforcement Learning (RL) or truncated Backpropagation (BP). However, RL and truncated BP suffer from low sample efficiency and biased gradient estimation respectively, resulting in limited improvement or, even worse, complete training failure. To overcome the challenges, we propose the Recursive Likelihood Ratio (RLR) optimizer, a zeroth-order informed fine-tuning paradigm for DM. The zeroth-order gradient estimator enables the computation graph rearrangement within the recursive diffusive chain, making the RLR's gradient estimator an unbiased one with the lower variance than other methods. We provide theoretical guarantees for the performance of the RLR. Extensive experiments are conducted on image and video generation tasks to validate the superiority of the RLR. Furthermore, we propose a novel prompt technique that is natural for the RLR to achieve a synergistic effect.

The healthcare sector has experienced a rapid accumulation of digital data recently, especially in the form of electronic health records (EHRs). EHRs constitute a precious resource that IS researchers could utilize for clinical applications (e.g., morbidity prediction). Deep learning seems like the obvious choice to exploit this surfeit of data. However, numerous studies have shown that deep learning does not enjoy the same kind of success on EHR data as it has in other domains; simple models like logistic regression are frequently as good as sophisticated deep learning ones. Inspired by this observation, we develop a novel model called rational logistic regression (RLR) that has standard logistic regression (LR) as its special case (and thus inherits LR's inductive bias that aligns with EHR data). RLR has rational series as its theoretical underpinnings, works on longitudinal time-series data, and learns interpretable patterns. Empirical comparisons on real-world clinical tasks demonstrate RLR's efficacy.

Logistic regression is commonly used for modeling dichotomous outcomes. In the classical setting, where the number of observations is much larger than the number of parameters, properties of the maximum likelihood estimator in logistic regression are well understood. Recently, Sur and Candes \cite{sur2018modern} have studied logistic regression in the high-dimensional regime, where the number of observations and parameters are comparable, and show, among other things, that the maximum likelihood estimator is biased. In the high-dimensional regime the underlying parameter vector is often structured (sparse, block-sparse, finite-alphabet, etc.) and so in this paper we study regularized logistic regression (RLR), where a convex regularizer that encourages the desired structure is added to the negative of the log-likelihood function. An advantage of RLR is that it allows parameter recovery even for instances where the (unconstrained) maximum likelihood estimate does not exist.