论文检索

Theory · Optimization

Lichuan Xiang, Quan Nguyen-Tri, Lan-Cuong Nguyen, Hoang Pham, Khoat Than, Long Tran-Thanh, Hongkai Wen

Pruning at Initialization (PaI) is a technique in neural network optimization characterized by the proactive elimination of weights before the network's training on designated tasks. This innovative strategy potentially reduces the costs for training and inference, significantly advancing computational efficiency. A key factor leading to PaI's effectiveness is that it considers the saliency of weights in an untrained network, and prioritizes the trainability and optimization potential of the pruned subnetworks. Recent methods can effectively prevent the formation of hard-to-optimize networks, e.g. through iterative adjustments at each network layer. However, this way often results in large-scale discrete optimization problems, which could make PaI further challenging. This paper introduces a novel method, called DPaI, that involves a differentiable optimization of the pruning mask. DPaI adopts a dynamic and adaptable pruning process, allowing easier optimization processes and better solutions. More importantly, our differentiable formulation enables readily use of the existing rich body of efficient gradient-based methods for PaI. Our empirical results demonstrate that DPaI significantly outperforms current state-of-the-art PaI methods on various architectures, such as Convolutional Neural Networks and Vision-Transformers. Code is available at https://github.com/QuanNguyen-Tri/DPaI.git

Social Aspects · Accountability, Transparency and Interpretability

Vinitra Swamy, Syrielle Montariol, Julian Blackwell, Jibril Frej, Martin Jaggi, Tanja Käser

In human-centric settings like education or healthcare, model accuracy and model explainability are key factors for user adoption. Towards these two goals, intrinsically interpretable deep learning models have gained popularity, focusing on accurate predictions alongside faithful explanations. However, there exists a gap in the human-centeredness of these approaches, which often produce nuanced and complex explanations that are not easily actionable for downstream users. We present InterpretCC (interpretable conditional computation), a family of intrinsically interpretable neural networks at a unique point in the design space that optimizes for ease of human understanding and explanation faithfulness, while maintaining comparable performance to state-of-the-art models. InterpretCC achieves this through adaptive sparse activation of features before prediction, allowing the model to use a different, minimal set of features for each instance. We extend this idea into an interpretable, global mixture-of-experts (MoE) model that allows users to specify topics of interest, discretely separates the feature space for each data point into topical subnetworks, and adaptively and sparsely activates these topical subnetworks for prediction. We apply InterpretCC for text, time series and tabular data across several real-world datasets, demonstrating comparable performance with non-interpretable baselines and outperforming intrinsically interpretable baselines. Through a user study involving 56 teachers, InterpretCC explanations are found to have higher actionability and usefulness over other intrinsically interpretable approaches.

Deep Learning · Generative Models and Autoencoders

Vinh Tong, Trung-Dung Hoang, Anji Liu, Guy Van den Broeck, Mathias Niepert

Diffusion Probabilistic Models (DPMs) are generative models showing competitive performance in various domains, including image synthesis and 3D point cloud generation. Sampling from pre-trained DPMs involves multiple neural function evaluations (NFEs) to transform Gaussian noise samples into images, resulting in higher computational costs compared to single-step generative models such as GANs or VAEs. Therefore, reducing the number of NFEs while preserving generation quality is crucial. To address this, we propose LD3, a lightweight framework designed to learn the optimal time discretization for sampling. LD3 can be combined with various samplers and consistently improves generation quality without having to retrain resource-intensive neural networks. We demonstrate analytically and empirically that LD3 improves sampling efficiency with much less computational overhead. We evaluate our method with extensive experiments on 7 pre-trained models, covering unconditional and conditional sampling in both pixel-space and latent-space DPMs. We achieve FIDs of 2.38 (10 NFE), and 2.27 (10 NFE) on unconditional CIFAR10 and AFHQv2 in 5-10 minutes of training. LD3 offers an efficient approach to sampling from pre-trained diffusion models. Code is available at https://github.com/vinhsuhi/LD3.

Deep Learning · Generative Models and Autoencoders

Vinh Tong, Trung-Dung Hoang, Anji Liu, Guy Van den Broeck, Mathias Niepert

Diffusion Probabilistic Models (DPMs) are generative models showing competitive performance in various domains, including image synthesis and 3D point cloud generation. Sampling from pre-trained DPMs involves multiple neural function evaluations (NFEs) to transform Gaussian noise samples into images, resulting in higher computational costs compared to single-step generative models such as GANs or VAEs. Therefore, reducing the number of NFEs while preserving generation quality is crucial. To address this, we propose LD3, a lightweight framework designed to learn the optimal time discretization for sampling. LD3 can be combined with various samplers and consistently improves generation quality without having to retrain resource-intensive neural networks. We demonstrate analytically and empirically that LD3 improves sampling efficiency with much less computational overhead. We evaluate our method with extensive experiments on 7 pre-trained models, covering unconditional and conditional sampling in both pixel-space and latent-space DPMs. We achieve FIDs of 2.38 (10 NFE), and 2.27 (10 NFE) on unconditional CIFAR10 and AFHQv2 in 5-10 minutes of training. LD3 offers an efficient approach to sampling from pre-trained diffusion models. Code is available at https://github.com/vinhsuhi/LD3.

Social Aspects · Trustworthy Machine Learning

Tobias Leemann, Periklis Petridis, Giuseppe Vietri, Dionysis Manousakas, Aaron Roth, Sergul Aydore

While retrieval-augmented generation (RAG) has been shown to enhance factuality of large language model (LLM) outputs, LLMs still suffer from hallucination, generating incorrect or irrelevant information. A common detection strategy involves prompting the LLM again to assess whether its response is grounded in the retrieved evidence, but this approach is costly. Alternatively, lightweight natural language inference (NLI) models for efficient grounding verification can be used at inference time. While existing pre-trained NLI models offer potential solutions, their performance remains subpar compared to larger models on realistic RAG inputs. RAG inputs are more complex than most datasets used for training NLI models and have characteristics specific to the underlying knowledge base, requiring adaptation of the NLI models to a specific target domain. Additionally, the lack of labeled instances in the target domain makes supervised domain adaptation, e.g., through fine-tuning, infeasible. To address these challenges, we introduce Automatic Generative Domain Adaptation (Auto-GDA). Our framework enables unsupervised domain adaptation through synthetic data generation.Unlike previous methods that rely on handcrafted filtering and augmentation strategies, Auto-GDA employs an iterative process to continuously improve the quality of generated samples using weak labels from less efficient teacher models and discrete optimization to select the most promising augmented samples. Experimental results demonstrate the effectiveness of our approach, with models fine-tuned on synthetic data using Auto-GDA often surpassing the performance of the teacher model and reaching the performance level of LLMs at 10% of their computational cost.

Reinforcement Learning · Multi-agent

Songyuan Zhang, Oswin So, Mitchell Black, Chuchu Fan

Control policies that can achieve high task performance and satisfy safety constraints are desirable for any system, including multi-agent systems (MAS). One promising technique for ensuring the safety of MAS is distributed control barrier functions (CBF). However, it is difficult to design distributed CBF-based policies for MAS that can tackle unknown discrete-time dynamics, partial observability, changing neighborhoods, and input constraints, especially when a distributed high-performance nominal policy that can achieve the task is unavailable. To tackle these challenges, we propose **DGPPO**, a new framework that *simultaneously* learns both a *discrete* graph CBF which handles neighborhood changes and input constraints, and a distributed high-performance safe policy for MAS with unknown discrete-time dynamics.We empirically validate our claims on a suite of multi-agent tasks spanning three different simulation engines. The results suggest that, compared with existing methods, our DGPPO framework obtains policies that achieve high task performance (matching baselines that ignore the safety constraints), and high safety rates (matching the most conservative baselines), with a *constant* set of hyperparameters across all environments.

Reinforcement Learning · Deep RL

Qifan Liang, Yixiang Shan, Haipeng Liu, Zhengbang Zhu, Ting Long, Weinan Zhang, Yuan Tian

An important challenge in multi-agent reinforcement learning is partial observability, where agents cannot access the global state of the environment during execution and can only receive observations within their field of view. To address this issue, previous works typically use the dimensional-wise state, which is obtained by applying MLP or dimensional-based attention on the global state, for decision-making during training and relying on a reconstructed dimensional-wise state during execution. However, dimensional-wise states tend to divert agent attention to specific features, neglecting potential dependencies between agents, making it difficult to make optimal decisions. Moreover, the inconsistency between the states used in training and execution further increases additional errors. To resolve these issues, we propose a method called Reconstruction-Guided Policy (RGP) to reconstruct the agent-wise state, which represents the information of inter-agent relationships, as input for decision-making during both training and execution. This not only preserves the potential dependencies between agents but also ensures consistency between the states used in training and execution. We conducted extensive experiments on both discrete and continuous action environments to evaluate RGP, and the results demonstrates its superior effectiveness. Our code is public in https://anonymous.4open.science/r/RGP-9F79

Optimization · Non-Convex

Kanan Gupta, Stephan Wojtowytsch

While momentum-based optimization algorithms are commonly used in the notoriously non-convex optimization problems of deep learning, their analysis has historically been restricted to the convex and strongly convex setting. In this article, we partially close this gap between theory and practice and demonstrate that virtually identical guarantees can be obtained in optimization problems with a 'benign' non-convexity. We show that these weaker geometric assumptions are well justified in overparametrized deep learning, at least locally. Variations of this result are obtained for a continuous time model of Nesterov's accelerated gradient descent algorithm (NAG), the classical discrete time version of NAG, and versions of NAG with stochastic gradient estimates with purely additive noise and with noise that exhibits both additive and multiplicative scaling.

Applications · Physics

Philipp Scholl, Katharina Bieker, Hillary Hauger, Gitta Kutyniok

The problem of symbolic regression (SR) arises in many different applications, such as identifying physical laws or deriving mathematical equations describing the behavior of financial markets from given data. Various methods exist to address the problem of SR, often based on genetic programming. However, these methods are usually complicated and involve various hyperparameters. In this paper, we present our new approach ParFam that utilizes parametric families of suitable symbolic functions to translate the discrete symbolic regression problem into a continuous one, resulting in a more straightforward setup compared to current state-of-the-art methods. In combination with a global optimizer, this approach results in a highly effective method to tackle the problem of SR. We theoretically analyze the expressivity of ParFam and demonstrate its performance with extensive numerical experiments based on the common SR benchmark suit SRBench, showing that we achieve state-of-the-art results. Moreover, we present an extension incorporating a pre-trained transformer network (DL-ParFam) to guide ParFam, accelerating the optimization process by up to two magnitudes. Our code and results can be found at https://anonymous.4open.science/r/parfam-D402.

Applications · Computer Vision

Sua Lee, Kyubum Shin, Jung Ho Park

Recent advances in pre-trained Vision Language Models (VLM) have shown promising potential for effectively adapting to downstream tasks through _prompt learning_, without the need for additional annotated paired datasets.To supplement the text information in VLM trained on correlations with vision data, new approaches leveraging Large Language Models (LLM) in prompts have been proposed, enhancing robustness to unseen and diverse data.Existing methods typically extract text-based responses (i.e., _descriptions_) from LLM to incorporate into prompts; however, this approach suffers from high variability and low reliability.In this work, we propose **De**scription-free **Mul**ti-prompt Learning(**DeMul**), a novel method that eliminates the process of extracting descriptions and instead directly distills knowledge from LLM into prompts.By adopting a description-free approach, prompts can encapsulate richer semantics while still being represented as continuous vectors for optimization, thereby eliminating the need for discrete pre-defined templates.Additionally, in a multi-prompt setting, we empirically demonstrate the potential of prompt weighting in reflecting the importance of different prompts during training.Experimental results show that our approach achieves superior performance across 11 recognition datasets.

Probabilistic Methods · Variational Inference

Sayan Banerjee, Krishna Balasubramanian, PROMIT GHOSAL

We provide finite-particle convergence rates for the Stein Variational Gradient Descent (SVGD) algorithm in the Kernelized Stein Discrepancy ($\KSD$) and Wasserstein-2 metrics. Our key insight is that the time derivative of the relative entropy between the joint density of $N$ particle locations and the $N$-fold product target measure, starting from a regular initial distribution, splits into a dominant 'negative part' proportional to $N$ times the expected $\KSD^2$ and a smaller 'positive part'. This observation leads to $\KSD$ rates of order $1/\sqrt{N}$, in both continuous and discrete time, providing a near optimal (in the sense of matching the corresponding i.i.d. rates) double exponential improvement over the recent result by~\cite{shi2024finite}. Under mild assumptions on the kernel and potential, these bounds also grow polynomially in the dimension $d$. By adding a bilinear component to the kernel, the above approach is used to further obtain Wasserstein-2 convergence in continuous time. For the case of `bilinear + Mat\'ern' kernels, we derive Wasserstein-2 rates that exhibit a curse-of-dimensionality similar to the i.i.d. setting. We also obtain marginal convergence and long-time propagation of chaos results for the time-averaged particle laws.

Optimization · Zero-order and Black-box Optimization

Yanbo Zhang, Benedikt Hartl, Hananel Hazan, Michael Levin

In a convergence of machine learning and biology, we reveal that diffusion models are evolutionary algorithms. By considering evolution as a denoising process and reversed evolution as diffusion, we mathematically demonstrate that diffusion models inherently perform evolutionary algorithms, naturally encompassing selection, mutation, and reproductive isolation. Building on this equivalence, we propose the Diffusion Evolution method: an evolutionary algorithm utilizing iterative denoising -- as originally introduced in the context of diffusion models -- to heuristically refine solutions in parameter spaces. Unlike traditional approaches, Diffusion Evolution efficiently identifies multiple optimal solutions and outperforms prominent mainstream evolutionary algorithms. Furthermore, leveraging advanced concepts from diffusion models, namely latent space diffusion and accelerated sampling, we introduce Latent Space Diffusion Evolution, which finds solutions for evolutionary tasks in high-dimensional complex parameter space while significantly reducing computational steps. This parallel between diffusion and evolution not only bridges two different fields but also opens new avenues for mutual enhancement, raising questions about open-ended evolution and potentially utilizing non-Gaussian or discrete diffusion models in the context of Diffusion Evolution.

Saptarshi Roy, Vansh Bansal, Purnamrita Sarkar, Alessandro Rinaldo

Diffusion models have emerged as a powerful tool for image generation and denoising. Typically, generative models learn a trajectory between the starting noise distribution and the target data distribution. Recently Liu et al. (2023b) designed a novel alternative generative model Rectified Flow(RF), which aims to learn straight flow trajectories from noise to data using a sequence of convex optimization problems with close ties to optimal transport. If the trajectory is curved, one must use many Euler discretization steps or novel strategies, such as exponential integrators, to achieve a satisfactory generation quality. In contrast, RF has been shown to theoretically straighten the trajectory through successive rectifications, reducing the number of function evaluations (NFEs) while sampling. It has also been shown empirically that RF may improve the straightness in two rectifications if one can solve the underlying optimization problem within a sufficiently small error. In this paper, we make two key theoretical contributions: 1) we provide the first theoretical analysis of theWasserstein distance between the sampling distribution of RF and the target distribution. Our error rate is characterized by the number of discretization steps and a new formulation of straightness stronger than that in the original work. 2) under a mild regularity assumption, we show that for a rectified flow from a Gaussian to any general target distribution with finite first moment (e.g. mixture of Gaussians), two rectifications are sufficient to achieve a straight flow, which is in line with the previous empirical findings. Additionally, we also present empirical results on both simulated and real datasets to validate our theoretical findings.

Deep Learning · Generative Models and Autoencoders

Hanyu Wang, Saksham Suri, Yixuan Ren, Hao Chen, Abhinav Shrivastava

We present LARP, a novel video tokenizer designed to overcome limitations in current video tokenization methods for autoregressive (AR) generative models. Unlike traditional patchwise tokenizers that directly encode local visual patches into discrete tokens, LARP introduces a holistic tokenization scheme that gathers information from the visual content using a set of learned holistic queries. This design allows LARP to capture more global and semantic representations, rather than being limited to local patch-level information. Furthermore, it offers flexibility by supporting an arbitrary number of discrete tokens, enabling adaptive and efficient tokenization based on the specific requirements of the task. To align the discrete token space with downstream AR generation tasks, LARP integrates a lightweight AR transformer as a training-time prior model that predicts the next token on its discrete latent space. By incorporating the prior model during training, LARP learns a latent space that is not only optimized for video reconstruction but is also structured in a way that is more conducive to autoregressive generation. Moreover, this process defines a sequential order for the discrete tokens, progressively pushing them toward an optimal configuration during training, ensuring smoother and more accurate AR generation at inference time. Comprehensive experiments demonstrate LARPs strong performance, achieving state-of-the-art FVD on the UCF101 class-conditional video generation benchmark. LARP enhances the compatibility of AR models with videos and opens up the potential to build unified high-fidelity multimodal large language models (MLLMs). Project page: https://hywang66.github.io/larp/

Deep Learning · Generative Models and Autoencoders

Hanyu Wang, Saksham Suri, Yixuan Ren, Hao Chen, Abhinav Shrivastava

We present LARP, a novel video tokenizer designed to overcome limitations in current video tokenization methods for autoregressive (AR) generative models. Unlike traditional patchwise tokenizers that directly encode local visual patches into discrete tokens, LARP introduces a holistic tokenization scheme that gathers information from the visual content using a set of learned holistic queries. This design allows LARP to capture more global and semantic representations, rather than being limited to local patch-level information. Furthermore, it offers flexibility by supporting an arbitrary number of discrete tokens, enabling adaptive and efficient tokenization based on the specific requirements of the task. To align the discrete token space with downstream AR generation tasks, LARP integrates a lightweight AR transformer as a training-time prior model that predicts the next token on its discrete latent space. By incorporating the prior model during training, LARP learns a latent space that is not only optimized for video reconstruction but is also structured in a way that is more conducive to autoregressive generation. Moreover, this process defines a sequential order for the discrete tokens, progressively pushing them toward an optimal configuration during training, ensuring smoother and more accurate AR generation at inference time. Comprehensive experiments demonstrate LARPs strong performance, achieving state-of-the-art FVD on the UCF101 class-conditional video generation benchmark. LARP enhances the compatibility of AR models with videos and opens up the potential to build unified high-fidelity multimodal large language models (MLLMs). Project page: https://hywang66.github.io/larp/

Probabilistic Methods · Monte Carlo and Sampling Methods

Byoungwoo Park, Hyungi Lee, Juho Lee

Many real-world datasets, such as healthcare, climate, and economics, are often collected as irregular time series, which poses challenges for accurate modeling. In this paper, we propose the Amortized Control of continuous State Space Model (ACSSM) for continuous dynamical modeling of time series for irregular and discrete observations. We first present a multi-marginal Doob's $h$-transform to construct a continuous dynamical system conditioned on these irregular observations. Following this, we introduce a variational inference algorithm with a tight evidence lower bound (ELBO), leveraging stochastic optimal control (SOC) theory to approximate the intractable Doob's $h$-transform and simulate the conditioned dynamics. To improve efficiency and scalability during both training and inference, ACSSM leverages auxiliary variable to flexibly parameterize the latent dynamics and amortized control. Additionally, it incorporates a simulation-free latent dynamics framework and a transformer-based data assimilation scheme, facilitating parallel inference of the latent states and ELBO computation. Through empirical evaluations across a variety of real-world datasets, ACSSM demonstrates superior performance in tasks such as classification, regression, interpolation, and extrapolation, while maintaining computational efficiency.

Probabilistic Methods · Monte Carlo and Sampling Methods

Byoungwoo Park, Hyungi Lee, Juho Lee

Many real-world datasets, such as healthcare, climate, and economics, are often collected as irregular time series, which poses challenges for accurate modeling. In this paper, we propose the Amortized Control of continuous State Space Model (ACSSM) for continuous dynamical modeling of time series for irregular and discrete observations. We first present a multi-marginal Doob's $h$-transform to construct a continuous dynamical system conditioned on these irregular observations. Following this, we introduce a variational inference algorithm with a tight evidence lower bound (ELBO), leveraging stochastic optimal control (SOC) theory to approximate the intractable Doob's $h$-transform and simulate the conditioned dynamics. To improve efficiency and scalability during both training and inference, ACSSM leverages auxiliary variable to flexibly parameterize the latent dynamics and amortized control. Additionally, it incorporates a simulation-free latent dynamics framework and a transformer-based data assimilation scheme, facilitating parallel inference of the latent states and ELBO computation. Through empirical evaluations across a variety of real-world datasets, ACSSM demonstrates superior performance in tasks such as classification, regression, interpolation, and extrapolation, while maintaining computational efficiency.

Deep Learning · Robustness

Bhaskar Mukhoty, Hilal AlQuabeh, Bin Gu

Rate-encoded spiking neural networks (SNNs) are known to offer superior adversarial robustness compared to direct-encoded SNNs but have relatively poor generalization on clean input. While the latter offers good generalization on clean input it suffers poor adversarial robustness under standard training. A key reason for this difference is the input noise introduced by the rate encoding, which encodes a pixel intensity with $T$ independent Bernoulli samples. To improve the generalization of rate-encoded SNNs, we propose the *signed rate encoding* (sRATE) that allows mean centering of the input and helps reduce the randomness introduced by the encoding, resulting in improved clean accuracy. In contrast to rate encoding, where input restricted to $[0,1]^d$ is encoded in $\\{0,1\\}^{d\times T}$, the signed rate encoding allows input in $[-1,1]^d$ to be encoded with spikes in $\\{-1,0,1\\}^{d\times T}$, where positive (negative) inputs are encoded with positive (negative) spikes. We further construct efficient \textit{Sparse Encoding Attack} (SEA) on standard and signed rate encoded input, which performs $l_0$-norm restricted adversarial attack in the discrete encoding space. We prove the theoretical optimality of the attack under the first-order approximation of the loss and compare it empirically with the existing attacks on the input space. Adversarial training performed with SEA, under signed rate encoding, offers superior adversarial robustness to the existing attacks and itself. Experiments conducted on standard datasets show the effectiveness of sign rate encoding in improving accuracy across all settings including adversarial robustness. The code is available at https://github.com/BhaskarMukhoty/SignedRateEncoding

Reinforcement Learning · Multi-agent

Qixin ZHANG, Zongqi Wan, Yu Yang, Li Shen, Dacheng Tao

Coordinating multiple agents to collaboratively maximize submodular functions in unpredictable environments is a critical task with numerous applications in machine learning, robot planning and control. The existing approaches, such as the OSG algorithm, are often hindered by their poor approximation guarantees and the rigid requirement for a fully connected communication graph. To address these challenges, we firstly present a $\textbf{MA-OSMA}$ algorithm, which employs the multi-linear extension to transfer the discrete submodular maximization problem into a continuous optimization, thereby allowing us to reduce the strict dependence on a complete graph through consensus techniques. Moreover, $\textbf{MA-OSMA}$ leverages a novel surrogate gradient to avoid sub-optimal stationary points. To eliminate the computationally intensive projection operations in $\textbf{MA-OSMA}$, we also introduce a projection-free $\textbf{MA-OSEA}$ algorithm, which effectively utilizes the KL divergence by mixing a uniform distribution. Theoretically, we confirm that both algorithms achieve a regret bound of $\widetilde{O}(\sqrt{\frac{C_{T}T}{1-\beta}})$ against a  $(\frac{1-e^{-c}}{c})$-approximation to the best comparator in hindsight, where $C_{T}$ is the deviation of maximizer sequence, $\beta$ is the spectral gap of the network and $c$ is the joint curvature of submodular objectives. This result significantly improves the $(\frac{1}{1+c})$-approximation provided by the state-of-the-art OSG algorithm. Finally, we demonstrate the effectiveness of our proposed algorithms through simulation-based multi-target tracking.

Reinforcement Learning · Deep RL

Ruohong Liu, Yuxin Pan, Linjie Xu, Lei Song, Pengcheng You, Yize Chen, Jiang Bian

Multi-objective reinforcement learning (MORL) excels at handling rapidly changing preferences in tasks that involve multiple criteria, even for unseen preferences. However, previous dominating MORL methods typically generate a fixed policy set or preference-conditioned policy through multiple training iterations exclusively for sampled preference vectors, and cannot ensure the efficient discovery of the Pareto front. Furthermore, integrating preferences into the input of policy or value functions presents scalability challenges, in particular as the dimension of the state and preference space grow, which can complicate the learning process and hinder the algorithm's performance on more complex tasks. To address these issues, we propose a two-stage Pareto front discovery algorithm called Constrained MORL (C-MORL), which serves as a seamless bridge between constrained policy optimization and MORL. Concretely, a set of policies is trained in parallel in the initialization stage, with each optimized towards its individual preference over the multiple objectives. Then, to fill the remaining vacancies in the Pareto front, the constrained optimization steps are employed to maximize one objective while constraining the other objectives to exceed a predefined threshold. Empirically, compared to recent advancements in MORL methods, our algorithm achieves more consistent and superior performances in terms of hypervolume, expected utility, and sparsity on both discrete and continuous control tasks, especially with numerous objectives (up to nine objectives in our experiments).