An increasing number of studies have focused on stochastic first-order methods (SFOMs) under heavy-tailed gradient noises, which have been observed in the training of practical deep learning models. In this paper, we focus on two types of gradient noises: one is sub-Weibull noise, and the other is noise under the assumption that it has a bounded $p$-th central moment ($p$-BCM) with $p\in (1, 2]$. The latter is more challenging due to the occurrence of infinite variance when $p\in (1, 2)$. Under these two gradient noise assumptions, the in-expectation and high-probability convergence of SFOMs have been extensively studied in the contexts of convex optimization and standard smooth optimization. However, for weakly convex objectives-a class that includes all Lipschitz-continuous convex objectives and smooth objectives-our understanding of the in-expectation and high-probability convergence of SFOMs under these two types of noises remains incomplete. We investigate the high-probability convergence of the vanilla stochastic subgradient descent (SsGD) method under sub-Weibull noises, as well as the high-probability and in-expectation convergence of clipped SsGD under the $p$-BCM noises. Both analyses are conducted in the context of weakly convex optimization. For weakly convex objectives that may be non-convex and non-smooth, our results demonstrate that the theoretical dependence of vanilla SsGD on the failure probability and number of iterations under sub-Weibull noises does not degrade compared to the case of smooth objectives. Under $p$-BCM noises, our findings indicate that the non-smoothness and non-convexity of weakly convex objectives do not impact the theoretical dependence of clipped SGD on the failure probability relative to the smooth case; however, the sample complexity we derived is worse than a well-known lower bound for smooth optimization.

The Sliced Gromov-Wasserstein (SGW) distance, aiming to relieve the computational cost of solving a non-convex quadratic program that is the Gromov-Wasserstein distance, utilizes projecting directions sampled uniformly from unit hyperspheres. This slicing mechanism incurs unnecessary computational costs due to uninformative directions, which also affects the representative power of the distance. However, finding a more appropriate distribution over the projecting directions (slicing distribution) is often an optimization problem in itself that comes with its own computational cost. In addition, with more intricate distributions, the sampling itself may be expensive. As a remedy, we propose an optimization-free slicing distribution that provides fast sampling for the Monte Carlo approximation. We do so by introducing the Relation-Aware Projecting Direction (RAPD), effectively capturing the pairwise association of each of two pairs of random vectors, each following their ambient law. This enables us to derive the Relation-Aware Slicing Distribution (RASD), a location-scale law corresponding to sampled RAPDs. Finally, we introduce the RASGW distance and its variants, e.g., IWRASGW (Importance Weighted RASGW), which overcome the shortcomings experienced by SGW. We theoretically analyze its properties and substantiate its empirical prowess using extensive experiments on various alignment tasks.

Following the recent push for trustworthy AI, there has been an increasing interest in developing contrastive explanation techniques for optimisation, especially concerning the solution of specific decision-making processes formalised as MILPs. Along these lines, we propose X-MILP, a domain-agnostic approach for building contrastive explanations for MILPs based on constraint reasoning techniques. First, we show how to encode the queries a user makes about the solution of an MILP problem as additional constraints. Then, we determine the reasons that constitute the answer to the user's query by computing the Irreducible Infeasible Subsystem (IIS) of the newly obtained set of constraints. Finally, we represent our explanation as a "graph of reasons" constructed from the IIS, which helps the user understand the structure among the reasons that answer their query. We test our method on instances of well-known optimisation problems to evaluate the empirical hardness of computing explanations.

Characterizing the convergence of real-valued or vector-v alued sequences is a key theoretical problem in data science, where the sequence index typically correspon ds to the number of iterations of an iterative algorithm (such as in optimization and signal processing) o r the number of observations (as in statistics and machine learning). This characterization can be done in mostly two ways, asymptotically or non-asymptotically. In an asymptotic analysis, an asymptotic e quivalent of the sequence is identified, which readily allows comparisons with other algorithms; however, without further analysis, the behavior at any finite time cannot be controlled. This is exactly what non-as ymptotic analysis aims to achieve, by providing bounds that are valid even for a finite index, but then only pro viding bounds that cannot always be compared. While the two approaches have their own merits, in this paper, we focus on asymptotic analysis and sequences that tend to their limit at a sub-exponential r ate that is a power of the sequence index. The main goal of this paper is to show how a classical tool from signal processing, control theory, and electrical engineering ( Oppenheim et al., 1996), the z -transform method ( Jury, 1964), can be used in this context with a striking efficiency at obtaining asymptotic eq uivalents for the class of algorithms that can be seen as iterations of potentially random linear operators i n a Hilbert space. This includes gradient descent for quadratic optimization problems as well as its accelera ted and stochastic variants ( Nesterov, 2018), 1 Landweber iterations in inverse problems ( Benning and Burger, 2018), or gossip algorithms in distributed computing ( Boyd et al., 2006).

Reinforcement Learning's high sensitivity to hyperparameters is a source of instability and inefficiency, creating significant challenges for practitioners. Hyperparameter Optimization (HPO) algorithms have been developed to address this issue, among them Population-Based Training (PBT) stands out for its ability to generate hyperparameters schedules instead of fixed configurations. PBT trains a population of agents, each with its own hyperparameters, frequently ranking them and replacing the worst performers with mutations of the best agents. These intermediate selection steps can cause PBT to focus on short-term improvements, leading it to get stuck in local optima and eventually fall behind vanilla Random Search over longer timescales. This paper studies how this greediness issue is connected to the choice of evolution frequency, the rate at which the selection is done. We propose Multiple-Frequencies Population-Based Training (MF-PBT), a novel HPO algorithm that addresses greediness by employing sub-populations, each evolving at distinct frequencies. MF-PBT introduces a migration process to transfer information between sub-populations, with an asymmetric design to balance short and long-term optimization. Extensive experiments on the Brax suite demonstrate that MF-PBT improves sample efficiency and long-term performance, even without actually tuning hyperparameters.

Fairness has emerged as one of the key challenges in federated learning. In horizontal federated settings, data heterogeneity often leads to substantial performance disparities across clients, raising concerns about equitable model behavior. To address this issue, we propose FedGA, a fairness-aware federated learning algorithm. We first employ the Gini coefficient to measure the performance disparity among clients. Based on this, we establish a relationship between the Gini coefficient $G$ and the update scale of the global model ${U_s}$, and use this relationship to adaptively determine the timing of fairness intervention. Subsequently, we dynamically adjust the aggregation weights according to the system's real-time fairness status, enabling the global model to better incorporate information from clients with relatively poor performance.We conduct extensive experiments on the Office-Caltech-10, CIFAR-10, and Synthetic datasets. The results show that FedGA effectively improves fairness metrics such as variance and the Gini coefficient, while maintaining strong overall performance, demonstrating the effectiveness of our approach.

We consider the problem of personalizing audio to maximize user experience. Briefly, we aim to find a filter $h^*$, which applied to any music or speech, will maximize the user's satisfaction. This is a black-box optimization problem since the user's satisfaction function is unknown. Substantive work has been done on this topic where the key idea is to play audio samples to the user, each shaped by a different filter $h_i$, and query the user for their satisfaction scores $f(h_i)$. A family of ``surrogate" functions is then designed to fit these scores and the optimization method gradually refines these functions to arrive at the filter $\hat{h}^*$ that maximizes satisfaction. In certain applications, we observe that a second type of querying is possible where users can tell us the individual elements $h^*[j]$ of the optimal filter $h^*$. Consider an analogy from cooking where the goal is to cook a recipe that maximizes user satisfaction. A user can be asked to score various cooked recipes (e.g., tofu fried rice) or to score individual ingredients (say, salt, sugar, rice, chicken, etc.). Given a budget of $B$ queries, where a query can be of either type, our goal is to find the recipe that will maximize this user's satisfaction. Our proposal builds on Sparse Gaussian Process Regression (GPR) and shows how a hybrid approach can outperform any one type of querying. Our results are validated through simulations and real world experiments, where volunteers gave feedback on music/speech audio and were able to achieve high satisfaction levels. We believe this idea of hybrid querying opens new problems in black-box optimization and solutions can benefit other applications beyond audio personalization.

Optimization has been widely used to generate smooth trajectories for motion planning. However, existing trajectory optimization methods show weakness when dealing with large-scale long trajectories. Recent advances in parallel computing have accelerated optimization in some fields, but how to efficiently solve trajectory optimization via parallelism remains an open question. In this paper, we propose a novel trajectory optimization framework based on the Consensus Alternating Direction Method of Multipliers (CADMM) algorithm, which decomposes the trajectory into multiple segments and solves the subproblems in parallel. The proposed framework reduces the time complexity to O(1) per iteration to the number of segments, compared to O(N) of the state-of-the-art (SOTA) approaches. Furthermore, we introduce a closed-form solution that integrates convex linear and quadratic constraints to speed up the optimization, and we also present numerical solutions for general inequality constraints. A series of simulations and experiments demonstrate that our approach outperforms the SOTA approach in terms of efficiency and smoothness. Especially for a large-scale trajectory, with one hundred segments, achieving over a tenfold speedup. To fully explore the potential of our algorithm on modern parallel computing architectures, we deploy our framework on a GPU and show high performance with thousands of segments.

Fund allocation has been an increasingly important problem in the financial domain. In reality, we aim to allocate the funds to buy certain assets within a certain future period. Naive solutions such as prediction-only or Predict-then-Optimize approaches suffer from goal mismatch. Additionally, the introduction of the SOTA time series forecasting model inevitably introduces additional uncertainty in the predicted result. To solve both problems mentioned above, we introduce a Risk-aware Time-Series Predict-and-Allocate (RTS-PnO) framework, which holds no prior assumption on the forecasting models. Such a framework contains three features: (i) end-to-end training with objective alignment measurement, (ii) adaptive forecasting uncertainty calibration, and (iii) agnostic towards forecasting models. The evaluation of RTS-PnO is conducted over both online and offline experiments. For offline experiments, eight datasets from three categories of financial applications are used: Currency, Stock, and Cryptos. RTS-PnO consistently outperforms other competitive baselines. The online experiment is conducted on the Cross-Border Payment business at FiT, Tencent, and an 8.4\% decrease in regret is witnessed when compared with the product-line approach. The code for the offline experiment is available at https://github.com/fuyuanlyu/RTS-PnO.

Quantum Approximate Optimization Algorithm (QAOA) is one of the most promising candidates to achieve the quantum advantage in solving combinatorial optimization problems. The process of finding a good set of variational parameters in the QAOA circuit has proven to be challenging due to multiple factors, such as barren plateaus. As a result, there is growing interest in exploiting parameter transferability, where parameter sets optimized for one problem instance are transferred to another that could be more complex either to estimate the solution or to serve as a warm start for further optimization. But can we transfer parameters from one class of problems to another? Leveraging parameter sets learned from a well-studied class of problems could help navigate the less studied one, reducing optimization overhead and mitigating performance pitfalls. In this paper, we study whether pretrained QAOA parameters of MaxCut can be used as is or to warm start the Maximum Independent Set (MIS) circuits. Specifically, we design machine learning models to find good donor candidates optimized on MaxCut and apply their parameters to MIS acceptors. Our experimental results show that such parameter transfer can significantly reduce the number of optimization iterations required while achieving comparable approximation ratios.