NeurIPS 2023 Papers

Conference on Neural Information Processing Systems

Decision breakdown

3,584papers

Counted from the official paper list; may differ slightly from the number the conference announced.

  • Oral
    772.1%
  • Spotlight
    40011.2%
  • Poster
    3,10786.7%

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Online Convex Optimization with Unbounded Memory

Raunak Kumar, Sarah Dean, Robert Kleinberg

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Online convex optimization (OCO) is a widely used framework in online learning. In each round, the learner chooses a decision in a convex set and an adversary chooses a convex loss function, and then the learner suffers the loss associated with their current decision. However, in many applications the learner's loss depends not only on the current decision but on the entire history of decisions until that point. The OCO framework and its existing generalizations do not capture this, and they can only be applied to many settings of interest after a long series of approximation arguments. They also leave open the question of whether the dependence on memory is tight because there are no non-trivial lower bounds. In this work we introduce a generalization of the OCO framework, ``Online Convex Optimization with Unbounded Memory'', that captures long-term dependence on past decisions. We introduce the notion of $p$-effective memory capacity, $H_p$, that quantifies the maximum influence of past decisions on present losses. We prove an $O(\sqrt{H_p T})$ upper bound on the policy regret and a matching (worst-case) lower bound. As a special case, we prove the first non-trivial lower bound for OCO with finite memory~\citep{anavaHM2015online}, which could be of independent interest, and also improve existing upper bounds. We demonstrate the broad applicability of our framework by using it to derive regret bounds, and to improve and simplify existing regret bound derivations, for a variety of online learning problems including online linear control and an online variant of performative prediction.

Poster
What can Large Language Models do in chemistry? A comprehensive benchmark on eight tasks

Taicheng Guo, Kehan Guo, Bozhao Nan, Zhenwen Liang, Zhichun Guo, Nitesh V Chawla, Olaf Wiest, Xiangliang Zhang

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Large Language Models (LLMs) with strong abilities in natural language processing tasks have emerged and have been applied in various kinds of areas such as science, finance and software engineering. However, the capability of LLMs to advance the field of chemistry remains unclear. In this paper, rather than pursuing state-of-the-art performance, we aim to evaluate capabilities of LLMs in a wide range of tasks across the chemistry domain. We identify three key chemistry-related capabilities including understanding, reasoning and explaining to explore in LLMs and establish a benchmark containing eight chemistry tasks. Our analysis draws on widely recognized datasets facilitating a broad exploration of the capacities of LLMs within the context of practical chemistry. Five LLMs (GPT-4,GPT-3.5, Davinci-003, Llama and Galactica) are evaluated for each chemistry task in zero-shot and few-shot in-context learning settings with carefully selected demonstration examples and specially crafted prompts. Our investigation found that GPT-4 outperformed other models and LLMs exhibit different competitive levels in eight chemistry tasks. In addition to the key findings from the comprehensive benchmark analysis, our work provides insights into the limitation of current LLMs and the impact of in-context learning settings on LLMs’ performance across various chemistry tasks. The code and datasets used in this study are available at https://github.com/ChemFoundationModels/ChemLLMBench.

Poster
Energy Guided Diffusion for Generating Neurally Exciting Images

Paweł A. Pierzchlewicz, Konstantin Friedrich Willeke, Arne Nix, Pavithra Elumalai, Kelli Restivo, Tori Shinn, Cate Nealley, Gabrielle Rodriguez, Saumil Patel, Katrin Franke, Andreas S. Tolias, Fabian H. Sinz

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In recent years, most exciting inputs (MEIs) synthesized from encoding models of neuronal activity have become an established method for studying tuning properties of biological and artificial visual systems. However, as we move up the visual hierarchy, the complexity of neuronal computations increases. Consequently, it becomes more challenging to model neuronal activity, requiring more complex models. In this study, we introduce a novel readout architecture inspired by the mechanism of visual attention. This new architecture, which we call attention readout, together with a data-driven convolutional core outperforms previous task-driven models in predicting the activity of neurons in macaque area V4. However, as our predictive network becomes deeper and more complex, synthesizing MEIs via straightforward gradient ascent (GA) can struggle to produce qualitatively good results and overfit to idiosyncrasies of a more complex model, potentially decreasing the MEI's model-to-brain transferability. To solve this problem, we propose a diffusion-based method for generating MEIs via Energy Guidance (EGG). We show that for models of macaque V4, EGG generates single neuron MEIs that generalize better across varying model architectures than the state-of-the-art GA, while at the same time reducing computational costs by a factor of 4.7x, facilitating experimentally challenging closed-loop experiments. Furthermore, EGG diffusion can be used to generate other neurally exciting images, like most exciting naturalistic images that are on par with a selection of highly activating natural images, or image reconstructions that generalize better across architectures. Finally, EGG is simple to implement, requires no retraining of the diffusion model, and can easily be generalized to provide other characterizations of the visual system, such as invariances. Thus, EGG provides a general and flexible framework to study the coding properties of the visual system in the context of natural images.

Poster
Adjustable Robust Reinforcement Learning for Online 3D Bin Packing

Yuxin Pan, Yize Chen, Fangzhen Lin

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Designing effective policies for the online 3D bin packing problem (3D-BPP) has been a long-standing challenge, primarily due to the unpredictable nature of incoming box sequences and stringent physical constraints. While current deep reinforcement learning (DRL) methods for online 3D-BPP have shown promising results in optimizing average performance over an underlying box sequence distribution, they often fail in real-world settings where some worst-case scenarios can materialize. Standard robust DRL algorithms tend to overly prioritize optimizing the worst-case performance at the expense of performance under normal problem instance distribution. To address these issues, we first introduce a permutation-based attacker to investigate the practical robustness of both DRL-based and heuristic methods proposed for solving online 3D-BPP. Then, we propose an adjustable robust reinforcement learning (AR2L) framework that allows efficient adjustment of robustness weights to achieve the desired balance of the policy's performance in average and worst-case environments. Specifically, we formulate the objective function as a weighted sum of expected and worst-case returns, and derive the lower performance bound by relating to the return under a mixture dynamics. To realize this lower bound, we adopt an iterative procedure that searches for the associated mixture dynamics and improves the corresponding policy. We integrate this procedure into two popular robust adversarial algorithms to develop the exact and approximate AR2L algorithms. Experiments demonstrate that AR2L is versatile in the sense that it improves policy robustness while maintaining an acceptable level of performance for the nominal case.

Poster
Exploring the Optimal Choice for Generative Processes in Diffusion Models: Ordinary vs Stochastic Differential Equations

Yu Cao, Jingrun Chen, Yixin Luo, Xiang Zhou

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The diffusion model has shown remarkable success in computer vision, but it remains unclear whether the ODE-based probability flow or the SDE-based diffusion model is more superior and under what circumstances. Comparing the two is challenging due to dependencies on data distributions, score training, and other numerical issues. In this paper, we study the problem mathematically for two limiting scenarios: the zero diffusion (ODE) case and the large diffusion case. We first introduce a pulse-shape error to perturb the score function and analyze error accumulation of sampling quality, followed by a thorough analysis for generalization to arbitrary error. Our findings indicate that when the perturbation occurs at the end of the generative process, the ODE model outperforms the SDE model with a large diffusion coefficient. However, when the perturbation occurs earlier, the SDE model outperforms the ODE model, and we demonstrate that the error of sample generation due to such a pulse-shape perturbation is exponentially suppressed as the diffusion term's magnitude increases to infinity. Numerical validation of this phenomenon is provided using Gaussian, Gaussian mixture, and Swiss roll distribution, as well as realistic datasets like MNIST and CIFAR-10.

Poster
Laplacian Canonization: A Minimalist Approach to Sign and Basis Invariant Spectral Embedding

George Ma, Yifei Wang, Yisen Wang

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Spectral embedding is a powerful graph embedding technique that has received a lot of attention recently due to its effectiveness on Graph Transformers. However, from a theoretical perspective, the universal expressive power of spectral embedding comes at the price of losing two important invariance properties of graphs, sign and basis invariance, which also limits its effectiveness on graph data. To remedy this issue, many previous methods developed costly approaches to learn new invariants and suffer from high computation complexity. In this work, we explore a minimal approach that resolves the ambiguity issues by directly finding canonical directions for the eigenvectors, named Laplacian Canonization (LC). As a pure pre-processing method, LC is light-weighted and can be applied to any existing GNNs. We provide a thorough investigation, from theory to algorithm, on this approach, and discover an efficient algorithm named Maximal Axis Projection (MAP) that works for both sign and basis invariance and successfully canonizes more than 90\% of all eigenvectors. Experiments on real-world benchmark datasets like ZINC, MOLTOX21, and MOLPCBA show that MAP consistently outperforms existing methods while bringing minimal computation overhead. Code is available at https://github.com/PKU-ML/LaplacianCanonization.

Poster
Exponentially Convergent Algorithms for Supervised Matrix Factorization

Joowon Lee, Hanbaek Lyu, Weixin Yao

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Supervised matrix factorization (SMF) is a classical machine learning method that simultaneously seeks feature extraction and classification tasks, which are not necessarily a priori aligned objectives. Our goal is to use SMF to learn low-rank latent factors that offer interpretable, data-reconstructive, and class-discriminative features, addressing challenges posed by high-dimensional data. Training SMF model involves solving a nonconvex and possibly constrained optimization with at least three blocks of parameters. Known algorithms are either heuristic or provide weak convergence guarantees for special cases. In this paper, we provide a novel framework that `lifts' SMF as a low-rank matrix estimation problem in a combined factor space and propose an efficient algorithm that provably converges exponentially fast to a global minimizer of the objective with arbitrary initialization under mild assumptions. Our framework applies to a wide range of SMF-type problems for multi-class classification with auxiliary features. To showcase an application, we demonstrate that our algorithm successfully identified well-known cancer-associated gene groups for various cancers.

Poster
Implicit Transfer Operator Learning: Multiple Time-Resolution Models for Molecular Dynamics

Mathias Schreiner, Ole Winther, Simon Olsson

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Computing properties of molecular systems rely on estimating expectations of the (unnormalized) Boltzmann distribution. Molecular dynamics (MD) is a broadly adopted technique to approximate such quantities. However, stable simulations rely on very small integration time-steps ($10^{-15}\,\mathrm{s}$), whereas convergence of some moments, e.g. binding free energy or rates, might rely on sampling processes on time-scales as long as $10^{-1}\, \mathrm{s}$, and these simulations must be repeated for every molecular system independently. Here, we present Implicit Transfer Operator (ITO) Learning, a framework to learn surrogates of the simulation process with multiple time-resolutions. We implement ITO with denoising diffusion probabilistic models with a new SE(3) equivariant architecture and show the resulting models can generate self-consistent stochastic dynamics across multiple time-scales, even when the system is only partially observed. Finally, we present a coarse-grained CG-SE3-ITO model which can quantitatively model all-atom molecular dynamics using only coarse molecular representations. As such, ITO provides an important step towards multiple time- and space-resolution acceleration of MD. Code is available at \href{https://github.com/olsson-group/ito}{https://github.com/olsson-group/ito}.

Poster
Revealing the unseen: Benchmarking video action recognition under occlusion

Shresth Grover, Vibhav Vineet, Yogesh S Rawat

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In this work, we study the effect of occlusion on video action recognition. To facilitate this study, we propose three benchmark datasets and experiment with seven different video action recognition models. These datasets include two synthetic benchmarks, UCF-101-O and K-400-O, which enabled understanding the effects of fundamental properties of occlusion via controlled experiments. We also propose a real-world occlusion dataset, UCF-101-Y-OCC, which helps in further validating the findings of this study. We find several interesting insights such as 1) transformers are more robust than CNN counterparts, 2) pretraining make models robust against occlusions, and 3) augmentation helps, but does not generalize well to real-world occlusions. In addition, we propose a simple transformer based compositional model, termed as CTx-Net, which generalizes well under this distribution shift. We observe that CTx-Net outperforms models which are trained using occlusions as augmentation, performing significantly better under natural occlusions. We believe this benchmark will open up interesting future research in robust video action recognition

Poster
Neural Sculpting: Uncovering hierarchically modular task structure in neural networks through pruning and network analysis

Shreyas Malakarjun Patil, Loizos Michael, Constantine Dovrolis

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Natural target functions and tasks typically exhibit hierarchical modularity -- they can be broken down into simpler sub-functions that are organized in a hierarchy. Such sub-functions have two important features: they have a distinct set of inputs (input-separability) and they are reused as inputs higher in the hierarchy (reusability). Previous studies have established that hierarchically modular neural networks, which are inherently sparse, offer benefits such as learning efficiency, generalization, multi-task learning, and transfer. However, identifying the underlying sub-functions and their hierarchical structure for a given task can be challenging. The high-level question in this work is: if we learn a task using a sufficiently deep neural network, how can we uncover the underlying hierarchy of sub-functions in that task? As a starting point, we examine the domain of Boolean functions, where it is easier to determine whether a task is hierarchically modular. We propose an approach based on iterative unit and edge pruning (during training), combined with network analysis for module detection and hierarchy inference. Finally, we demonstrate that this method can uncover the hierarchical modularity of a wide range of Boolean functions and two vision tasks based on the MNIST digits dataset.

Poster
Towards Efficient Image Compression Without Autoregressive Models

Muhammad Salman Ali, Yeongwoong Kim, Maryam Qamar, Sung-Chang Lim, Donghyun Kim, Chaoning Zhang, Sung-Ho Bae, Hui Yong Kim

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Recently, learned image compression (LIC) has garnered increasing interest with its rapidly improving performance surpassing conventional codecs. A key ingredient of LIC is a hyperprior-based entropy model, where the underlying joint probability of the latent image features is modeled as a product of Gaussian distributions from each latent element. Since latents from the actual images are not spatially independent, autoregressive (AR) context based entropy models were proposed to handle the discrepancy between the assumed distribution and the actual distribution. Though the AR-based models have proven effective, the computational complexity is significantly increased due to the inherent sequential nature of the algorithm. In this paper, we present a novel alternative to the AR-based approach that can provide a significantly better trade-off between performance and complexity. To minimize the discrepancy, we introduce a correlation loss that forces the latents to be spatially decorrelated and better fitted to the independent probability model. Our correlation loss is proved to act as a general plug-in for the hyperprior (HP) based learned image compression methods. The performance gain from our correlation loss is ‘free’ in terms of computation complexity for both inference time and decoding time. To our knowledge, our method gives the best trade-off between the complexity and performance: combined with the Checkerboard-CM, it attains **90%** and when combined with ChARM-CM, it attains **98%** of the AR-based BD-Rate gains yet is around **50 times** and **30 times** faster than AR-based methods respectively

Poster
Find What You Want: Learning Demand-conditioned Object Attribute Space for Demand-driven Navigation

Hongcheng Wang, Andy Guan Hong Chen, Xiaoqi Li, Mingdong Wu, Hao Dong

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The task of Visual Object Navigation (VON) involves an agent's ability to locate a particular object within a given scene. To successfully accomplish the VON task, two essential conditions must be fulfiled: 1) the user knows the name of the desired object; and 2) the user-specified object actually is present within the scene. To meet these conditions, a simulator can incorporate predefined object names and positions into the metadata of the scene. However, in real-world scenarios, it is often challenging to ensure that these conditions are always met. Humans in an unfamiliar environment may not know which objects are present in the scene, or they may mistakenly specify an object that is not actually present. Nevertheless, despite these challenges, humans may still have a demand for an object, which could potentially be fulfilled by other objects present within the scene in an equivalent manner. Hence, this paper proposes Demand-driven Navigation (DDN), which leverages the user's demand as the task instruction and prompts the agent to find an object which matches the specified demand. DDN aims to relax the stringent conditions of VON by focusing on fulfilling the user's demand rather than relying solely on specified object names. This paper proposes a method of acquiring textual attribute features of objects by extracting common sense knowledge from a large language model (LLM). These textual attribute features are subsequently aligned with visual attribute features using Contrastive Language-Image Pre-training (CLIP). Incorporating the visual attribute features as prior knowledge, enhances the navigation process. Experiments on AI2Thor with the ProcThor dataset demonstrate that the visual attribute features improve the agent's navigation performance and outperform the baseline methods commonly used in the VON and VLN task and methods with LLMs. The codes and demonstrations can be viewed at https://sites.google.com/view/demand-driven-navigation.

Poster
Momentum Provably Improves Error Feedback!

Ilyas Fatkhullin, Alexander Tyurin, Peter Richtárik

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Due to the high communication overhead when training machine learning models in a distributed environment, modern algorithms invariably rely on lossy communication compression. However, when untreated, the errors caused by compression propagate, and can lead to severely unstable behavior, including exponential divergence. Almost a decade ago, Seide et al. [2014] proposed an error feedback (EF) mechanism, which we refer to as EF14, as an immensely effective heuristic for mitigating this issue. However, despite steady algorithmic and theoretical advances in the EF field in the last decade, our understanding is far from complete. In this work we address one of the most pressing issues. In particular, in the canonical nonconvex setting, all known variants of EF rely on very large batch sizes to converge, which can be prohibitive in practice. We propose a surprisingly simple fix which removes this issue both theoretically, and in practice: the application of Polyak's momentum to the latest incarnation of EF due to Richtárik et al. [2021] known as EF21. Our algorithm, for which we coin the name EF21-SGDM, improves the communication and sample complexities of previous error feedback algorithms under standard smoothness and bounded variance assumptions, and does not require any further strong assumptions such as bounded gradient dissimilarity. Moreover, we propose a double momentum version of our method that improves the complexities even further. Our proof seems to be novel even when compression is removed form the method, and as such, our proof technique is of independent interest in the study of nonconvex stochastic optimization enriched with Polyak's momentum.

Poster
Reliable learning in challenging environments

Nina Balcan, Steve Hanneke, Rattana Pukdee, Dravyansh Sharma

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The problem of designing learners that provide guarantees that their predictions are provably correct is of increasing importance in machine learning. However, learning theoretic guarantees have only been considered in very specific settings. In this work, we consider the design and analysis of reliable learners in challenging test-time environments as encountered in modern machine learning problems: namely adversarial test-time attacks (in several variations) and natural distribution shifts. In this work, we provide a reliable learner with provably optimal guarantees in such settings. We discuss computationally feasible implementations of the learner and further show that our algorithm achieves strong positive performance guarantees on several natural examples: for example, linear separators under log-concave distributions or smooth boundary classifiers under smooth probability distributions.

Poster
Exact Verification of ReLU Neural Control Barrier Functions

Hongchao Zhang, Junlin Wu, Yevgeniy Vorobeychik, Andrew Clark

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Control Barrier Functions (CBFs) are a popular approach for safe control of nonlinear systems. In CBF-based control, the desired safety properties of the system are mapped to nonnegativity of a CBF, and the control input is chosen to ensure that the CBF remains nonnegative for all time. Recently, machine learning methods that represent CBFs as neural networks (neural control barrier functions, or NCBFs) have shown great promise due to the universal representability of neural networks. However, verifying that a learned CBF guarantees safety remains a challenging research problem. This paper presents novel exact conditions and algorithms for verifying safety of feedforward NCBFs with ReLU activation functions. The key challenge in doing so is that, due to the piecewise linearity of the ReLU function, the NCBF will be nondifferentiable at certain points, thus invalidating traditional safety verification methods that assume a smooth barrier function. We resolve this issue by leveraging a generalization of Nagumo's theorem for proving invariance of sets with nonsmooth boundaries to derive necessary and sufficient conditions for safety. Based on this condition, we propose an algorithm for safety verification of NCBFs that first decomposes the NCBF into piecewise linear segments and then solves a nonlinear program to verify safety of each segment as well as the intersections of the linear segments. We mitigate the complexity by only considering the boundary of the safe region and by pruning the segments with Interval Bound Propagation (IBP) and linear relaxation. We evaluate our approach through numerical studies with comparison to state-of-the-art SMT-based methods. Our code is available at https://github.com/HongchaoZhang-HZ/exactverif-reluncbf-nips23.

Poster
Learning Unseen Modality Interaction

Yunhua Zhang, Hazel Doughty, Cees G. M. Snoek

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Multimodal learning assumes all modality combinations of interest are available during training to learn cross-modal correspondences. In this paper, we challenge this modality-complete assumption for multimodal learning and instead strive for generalization to unseen modality combinations during inference. We pose the problem of unseen modality interaction and introduce a first solution. It exploits a module that projects the multidimensional features of different modalities into a common space with rich information preserved. This allows the information to be accumulated with a simple summation operation across available modalities. To reduce overfitting to less discriminative modality combinations during training, we further improve the model learning with pseudo-supervision indicating the reliability of a modality’s prediction. We demonstrate that our approach is effective for diverse tasks and modalities by evaluating it for multimodal video classification, robot state regression, and multimedia retrieval. Project website: https://xiaobai1217.github.io/Unseen-Modality-Interaction/.

Poster
A Theoretical Analysis of Optimistic Proximal Policy Optimization in Linear Markov Decision Processes

Han Zhong, Tong Zhang

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The proximal policy optimization (PPO) algorithm stands as one of the most prosperous methods in the field of reinforcement learning (RL). Despite its success, the theoretical understanding of PPO remains deficient. Specifically, it is unclear whether PPO or its optimistic variants can effectively solve linear Markov decision processes (MDPs), which are arguably the simplest models in RL with function approximation. To bridge this gap, we propose an optimistic variant of PPO for episodic adversarial linear MDPs with full-information feedback, and establish a $\tilde{\mathcal{O}}(d^{3/4}H^2K^{3/4})$ regret for it. Here $d$ is the ambient dimension of linear MDPs, $H$ is the length of each episode, and $K$ is the number of episodes. Compared with existing policy-based algorithms, we achieve the state-of-the-art regret bound in both stochastic linear MDPs and adversarial linear MDPs with full information. Additionally, our algorithm design features a novel multi-batched updating mechanism and the theoretical analysis utilizes a new covering number argument of value and policy classes, which might be of independent interest.

Poster
EPIC Fields: Marrying 3D Geometry and Video Understanding

Vadim Tschernezki, Ahmad Darkhalil, Zhifan Zhu, David Fouhey, Iro Laina, Diane Larlus, Dima Damen, Andrea Vedaldi

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Neural rendering is fuelling a unification of learning, 3D geometry and video understanding that has been waiting for more than two decades. Progress, however, is still hampered by a lack of suitable datasets and benchmarks. To address this gap, we introduce EPIC Fields, an augmentation of EPIC-KITCHENS with 3D camera information. Like other datasets for neural rendering, EPIC Fields removes the complex and expensive step of reconstructing cameras using photogrammetry, and allows researchers to focus on modelling problems. We illustrate the challenge of photogrammetry in egocentric videos of dynamic actions and propose innovations to address them. Compared to other neural rendering datasets, EPIC Fields is better tailored to video understanding because it is paired with labelled action segments and the recent VISOR segment annotations. To further motivate the community, we also evaluate two benchmark tasks in neural rendering and segmenting dynamic objects, with strong baselines that showcase what is not possible today. We also highlight the advantage of geometry in semi-supervised video object segmentations on the VISOR annotations. EPIC Fields reconstructs 96\% of videos in EPIC-KITCHENS, registering 19M frames in 99 hours recorded in 45 kitchens, and is available from: http://epic-kitchens.github.io/epic-fields

Poster
Incentivized Communication for Federated Bandits

Zhepei Wei, Chuanhao Li, Haifeng Xu, Hongning Wang

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Most existing works on federated bandits take it for granted that all clients are altruistic about sharing their data with the server for the collective good whenever needed. Despite their compelling theoretical guarantee on performance and communication efficiency, this assumption is overly idealistic and oftentimes violated in practice, especially when the algorithm is operated over self-interested clients, who are reluctant to share data without explicit benefits. Negligence of such self-interested behaviors can significantly affect the learning efficiency and even the practical operability of federated bandit learning. In light of this, we aim to spark new insights into this under-explored research area by formally introducing an incentivized communication problem for federated bandits, where the server shall motivate clients to share data by providing incentives. Without loss of generality, we instantiate this bandit problem with the contextual linear setting and propose the first incentivized communication protocol, namely, Inc-FedUCB, that achieves near-optimal regret with provable communication and incentive cost guarantees. Extensive empirical experiments on both synthetic and real-world datasets further validate the effectiveness of the proposed method across various environments.

Poster
Unified Lower Bounds for Interactive High-dimensional Estimation under Information Constraints

Jayadev Acharya, Clement Louis Canonne, Ziteng Sun, Himanshu Tyagi

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We consider distributed parameter estimation using interactive protocols subject to local information constraints such as bandwidth limitations, local differential privacy, and restricted measurements. We provide a unified framework enabling us to derive a variety of (tight) minimax lower bounds for different parametric families of distributions, both continuous and discrete, under any $\ell_p$ loss. Our lower bound framework is versatile and yields “plug-and-play” bounds that are widely applicable to a large range of estimation problems, and, for the prototypical case of the Gaussian family, circumvents limitations of previous techniques. In particular, our approach recovers bounds obtained using data processing inequalities and Cramér–Rao bounds, two other alternative approaches for proving lower bounds in our setting of interest. Further, for the families considered, we complement our lower bounds with matching upper bounds.

Poster