Online Linear Programming with Batching

We study Online Linear Programming (OLP) with batching. The planning horizon is cut into $K$ batches, and the decisions on customers arriving within a batch can be delayed to the end of their associated batch. Compared with OLP without batching, the ability to delay decisions brings better operational performance, as measured by regret. Two research questions of interest are: (1) What is a lower bound of the regret as a function of $K$? (2) What algorithms can achieve the regret lower bound? These questions have been analyzed in the literature when the distribution of the reward and the resource consumption of the customers have finite support. By contrast, this paper analyzes these questions when the conditional distribution of the reward given the resource consumption is continuous, and we show the answers are different under this setting. When there is only a single type of resource and the decision maker knows the total number of customers, we propose an algorithm with a $O(\log K)$ regret upper bound and provide a $\Omega(\log K)$ regret lower bound. We also propose algorithms with $O(\log K)$ regret upper bound for the setting in which there are multiple types of resource and the setting in which customers arrive following a Poisson process. All these regret upper and lower bounds are independent of the length of the planning horizon, and all the proposed algorithms delay decisions on customers arriving in only the first and the last batch. We also take customer impatience into consideration and establish a way of selecting an appropriate batch size.

Adam-mini: Use Fewer Learning Rates To Gain More

We propose Adam-mini, an optimizer that achieves on-par or better performance than AdamW with 45% to 50% less memory footprint. Adam-mini reduces memory by cutting down the learning rate resources in Adam (i.e., $1/\sqrt{v}$). We find that $\geq$ 90% of these learning rates in $v$ could be harmlessly removed if we (1) carefully partition the parameters into blocks following our proposed principle on Hessian structure; (2) assign a single but good learning rate to each parameter block. We further find that, for each of these parameter blocks, there exists a single high-quality learning rate that can outperform Adam, provided that sufficient resources are available to search it out. We then provide one cost-effective way to find good learning rates and propose Adam-mini. Empirically, we verify that Adam-mini performs on par or better than AdamW on various language models sized from 125M to 7B for pre-training, supervised fine-tuning, and RLHF. The reduced memory footprint of Adam-mini also alleviates communication overheads among GPUs and CPUs, thereby increasing throughput. For instance, Adam-mini achieves 49.6% higher throughput than AdamW when pre-training Llama2-7B on $2\times$ A800-80GB GPUs, which saves 33% wall-clock time for pre-training.

Diffusion Model for Data-Driven Black-Box Optimization

Generative AI has redefined artificial intelligence, enabling the creation of innovative content and customized solutions that drive business practices into a new era of efficiency and creativity. In this paper, we focus on diffusion models, a powerful generative AI technology, and investigate their potential for black-box optimization over complex structured variables. Consider the practical scenario where one wants to optimize some structured design in a high-dimensional space, based on massive unlabeled data (representing design variables) and a small labeled dataset. We study two practical types of labels: 1) noisy measurements of a real-valued reward function and 2) human preference based on pairwise comparisons. The goal is to generate new designs that are near-optimal and preserve the designed latent structures. Our proposed method reformulates the design optimization problem into a conditional sampling problem, which allows us to leverage the power of diffusion models for modeling complex distributions. In particular, we propose a reward-directed conditional diffusion model, to be trained on the mixed data, for sampling a near-optimal solution conditioned on high predicted rewards. Theoretically, we establish sub-optimality error bounds for the generated designs. The sub-optimality gap nearly matches the optimal guarantee in off-policy bandits, demonstrating the efficiency of reward-directed diffusion models for black-box optimization. Moreover, when the data admits a low-dimensional latent subspace structure, our model efficiently generates high-fidelity designs that closely respect the latent structure. We provide empirical experiments validating our model in decision-making and content-creation tasks.

TriAug: Out-of-Distribution Detection for Imbalanced Breast Lesion in Ultrasound

Different diseases, such as histological subtypes of breast lesions, have severely varying incidence rates. Even trained with substantial amount of in-distribution (ID) data, models often encounter out-of-distribution (OOD) samples belonging to unseen classes in clinical reality. To address this, we propose a novel framework built upon a long-tailed OOD detection task for breast ultrasound images. It is equipped with a triplet state augmentation (TriAug) which improves ID classification accuracy while maintaining a promising OOD detection performance. Meanwhile, we designed a balanced sphere loss to handle the class imbalanced problem. Experimental results show that the model outperforms state-of-art OOD approaches both in ID classification (F1-score=42.12%) and OOD detection (AUROC=78.06%).

Achieving $\tilde{O}(1/ε)$ Sample Complexity for Constrained Markov Decision Process

We consider the reinforcement learning problem for the constrained Markov decision process (CMDP), which plays a central role in satisfying safety or resource constraints in sequential learning and decision-making. In this problem, we are given finite resources and a MDP with unknown transition probabilities. At each stage, we take an action, collecting a reward and consuming some resources, all assumed to be unknown and need to be learned over time. In this work, we take the first step towards deriving optimal problem-dependent guarantees for the CMDP problems. We derive a logarithmic regret bound, which translates into a $O(\frac{\kappa}{\epsilon}\cdot\log^2(1/\epsilon))$ sample complexity bound, with $\kappa$ being a problem-dependent parameter, yet independent of $\epsilon$. Our sample complexity bound improves upon the state-of-art $O(1/\epsilon^2)$ sample complexity for CMDP problems established in the previous literature, in terms of the dependency on $\epsilon$. To achieve this advance, we develop a new framework for analyzing CMDP problems. To be specific, our algorithm operates in the primal space and we resolve the primal LP for the CMDP problem at each period in an online manner, with \textit{adaptive} remaining resource capacities. The key elements of our algorithm are: i). an eliminating procedure that characterizes one optimal basis of the primal LP, and; ii) a resolving procedure that is adaptive to the remaining resources and sticks to the characterized optimal basis.

Decoupling Learning and Decision-Making: Breaking the $\mathcal{O}(\sqrt{T})$ Barrier in Online Resource Allocation with First-Order Methods

Online linear programming plays an important role in both revenue management and resource allocation, and recent research has focused on developing efficient first-order online learning algorithms. Despite the empirical success of first-order methods, they typically achieve a regret no better than $\mathcal{O}(\sqrt{T})$, which is suboptimal compared to the $\mathcal{O}(\log T)$ bound guaranteed by the state-of-the-art linear programming (LP)-based online algorithms. This paper establishes several important facts about online linear programming, which unveils the challenge for first-order-method-based online algorithms to achieve beyond $\mathcal{O}(\sqrt{T})$ regret. To address the challenge, we introduce a new algorithmic framework that decouples learning from decision-making. More importantly, for the first time, we show that first-order methods can attain regret $\mathcal{O}(T^{1/3})$ with this new framework. Lastly, we conduct numerical experiments to validate our theoretical findings.

A Homogenization Approach for Gradient-Dominated Stochastic Optimization

Gradient dominance property is a condition weaker than strong convexity, yet it sufficiently ensures global convergence for first-order methods even in non-convex optimization. This property finds application in various machine learning domains, including matrix decomposition, linear neural networks, and policy-based reinforcement learning (RL). In this paper, we study the stochastic homogeneous second-order descent method (SHSODM) for gradient-dominated optimization with $\alpha \in [1, 2]$ based on a recently proposed homogenization approach. Theoretically, we show that SHSODM achieves a sample complexity of $O(\epsilon^{-7/(2 \alpha) +1})$ for $\alpha \in [1, 3/2)$ and $\tilde{O}(\epsilon^{-2/\alpha})$ for $\alpha \in [3/2, 2]$. We further provide a SHSODM with a variance reduction technique enjoying an improved sample complexity of $O( \epsilon ^{-( 7-3\alpha ) /( 2\alpha )})$ for $\alpha \in [1,3/2)$. Our results match the state-of-the-art sample complexity bounds for stochastic gradient-dominated optimization without \emph{cubic regularization}. Since the homogenization approach only relies on solving extremal eigenvector problems instead of Newton-type systems, our methods gain the advantage of cheaper iterations and robustness in ill-conditioned problems. Numerical experiments on several RL tasks demonstrate the efficiency of SHSODM compared to other off-the-shelf methods.

Pre-trained Mixed Integer Optimization through Multi-variable Cardinality Branching

We propose a new method to accelerate online Mixed Integer Optimization with Pre-trained machine learning models (PreMIO). The key component of PreMIO is a multi-variable cardinality branching procedure that splits the feasible region with data-driven hyperplanes, which can be easily integrated into any MIP solver with two lines of code. Moreover, we incorporate learning theory and concentration inequalities to develop a straightforward and interpretable hyper-parameter selection strategy for our method. We test the performance of PreMIO by applying it to state-of-the-art MIP solvers and running numerical experiments on both classical OR benchmark datasets and real-life instances. The results validate the effectiveness of our proposed method.

Stochastic Dimension-reduced Second-order Methods for Policy Optimization

In this paper, we propose several new stochastic second-order algorithms for policy optimization that only require gradient and Hessian-vector product in each iteration, making them computationally efficient and comparable to policy gradient methods. Specifically, we propose a dimension-reduced second-order method (DR-SOPO) which repeatedly solves a projected two-dimensional trust region subproblem. We show that DR-SOPO obtains an $\mathcal{O}(\epsilon^{-3.5})$ complexity for reaching approximate first-order stationary condition and certain subspace second-order stationary condition. In addition, we present an enhanced algorithm (DVR-SOPO) which further improves the complexity to $\mathcal{O}(\epsilon^{-3})$ based on the variance reduction technique. Preliminary experiments show that our proposed algorithms perform favorably compared with stochastic and variance-reduced policy gradient methods.

Optimal Diagonal Preconditioning: Theory and Practice

Preconditioning has been a staple technique in optimization and machine learning. It often reduces the condition number of the matrix it is applied to, thereby speeding up convergence of optimization algorithms. Although there are many popular preconditioning techniques in practice, most lack theoretical guarantees for reductions in condition number. In this paper, we study the problem of optimal diagonal preconditioning to achieve maximal reduction in the condition number of any full-rank matrix by scaling its rows or columns separately or simultaneously. We first reformulate the problem as a quasi-convex problem and provide a baseline bisection algorithm that is easy to implement in practice, where each iteration consists of an SDP feasibility problem. Then we propose a polynomial time potential reduction algorithm with $O(\log(\frac{1}{\epsilon}))$ iteration complexity, where each iteration consists of a Newton update based on the Nesterov-Todd direction. Our algorithm is based on a formulation of the problem which is a generalized version of the Von Neumann optimal growth problem. Next, we specialize to one-sided optimal diagonal preconditioning problems, and demonstrate that they can be formulated as standard dual SDP problems, to which we apply efficient customized solvers and study the empirical performance of our optimal diagonal preconditioners. Our extensive experiments on large matrices demonstrate the practical appeal of optimal diagonal preconditioners at reducing condition numbers compared to heuristics-based preconditioners.