Bogdan Mazoure

Andrew Szot, Max Schwarzer, Harsh Agrawal et al. · apple-ml, gatech

We show that large language models (LLMs) can be adapted to be generalizable policies for embodied visual tasks. Our approach, called Large LAnguage model Reinforcement Learning Policy (LLaRP), adapts a pre-trained frozen LLM to take as input text instructions and visual egocentric observations and output actions directly in the environment. Using reinforcement learning, we train LLaRP to see and act solely through environmental interactions. We show that LLaRP is robust to complex paraphrasings of task instructions and can generalize to new tasks that require novel optimal behavior. In particular, on 1,000 unseen tasks it achieves 42% success rate, 1.7x the success rate of other common learned baselines or zero-shot applications of LLMs. Finally, to aid the community in studying language conditioned, massively multi-task, embodied AI problems we release a novel benchmark, Language Rearrangement, consisting of 150,000 training and 1,000 testing tasks for language-conditioned rearrangement. Video examples of LLaRP in unseen Language Rearrangement instructions are at https://llm-rl.github.io.

Bogdan Mazoure, Walter Talbott, Miguel Angel Bautista et al. · apple-ml

A fairly reliable trend in deep reinforcement learning is that the performance scales with the number of parameters, provided a complimentary scaling in amount of training data. As the appetite for large models increases, it is imperative to address, sooner than later, the potential problem of running out of high-quality demonstrations. In this case, instead of collecting only new data via costly human demonstrations or risking a simulation-to-real transfer with uncertain effects, it would be beneficial to leverage vast amounts of readily-available low-quality data. Since classical control algorithms such as behavior cloning or temporal difference learning cannot be used on reward-free or action-free data out-of-the-box, this solution warrants novel training paradigms for continuous control. We propose a simple algorithm called Diffused Value Function (DVF), which learns a joint multi-step model of the environment-robot interaction dynamics using a diffusion model. This model can be efficiently learned from state sequences (i.e., without access to reward functions nor actions), and subsequently used to estimate the value of each action out-of-the-box. We show how DVF can be used to efficiently capture the state visitation measure for multiple controllers, and show promising qualitative and quantitative results on challenging robotics benchmarks.

Bogdan Mazoure, Jake Bruce, Doina Precup et al.

Sequential decision-making agents struggle with long horizon tasks, since solving them requires multi-step reasoning. Most reinforcement learning (RL) algorithms address this challenge by improved credit assignment, introducing memory capability, altering the agent's intrinsic motivation (i.e. exploration) or its worldview (i.e. knowledge representation). Many of these components could be learned from offline data. In this work, we follow the hypothesis that exploration and representation learning can be improved by separately learning two different models from a single offline dataset. We show that learning a state representation using noise-contrastive estimation and a model of auxiliary reward separately from a single collection of human demonstrations can significantly improve the sample efficiency on the challenging NetHack benchmark. We also ablate various components of our experimental setting and highlight crucial insights.

R Devon Hjelm, Bogdan Mazoure, Florian Golemo et al. · mila

A broad challenge of research on generalization for sequential decision-making tasks in interactive environments is designing benchmarks that clearly landmark progress. While there has been notable headway, current benchmarks either do not provide suitable exposure nor intuitive control of the underlying factors, are not easy-to-implement, customizable, or extensible, or are computationally expensive to run. We built the Sandbox Environment for Generalizable Agent Research (SEGAR) with all of these things in mind. SEGAR improves the ease and accountability of generalization research in RL, as generalization objectives can be easy designed by specifying task distributions, which in turns allows the researcher to measure the nature of the generalization objective. We present an overview of SEGAR and how it contributes to these goals, as well as experiments that demonstrate a few types of research questions SEGAR can help answer.

Bogdan Mazoure, Benjamin Eysenbach, Ofir Nachum et al.

Model-based reinforcement learning (RL) methods are appealing in the offline setting because they allow an agent to reason about the consequences of actions without interacting with the environment. Prior methods learn a 1-step dynamics model, which predicts the next state given the current state and action. These models do not immediately tell the agent which actions to take, but must be integrated into a larger RL framework. Can we model the environment dynamics in a different way, such that the learned model does directly indicate the value of each action? In this paper, we propose Contrastive Value Learning (CVL), which learns an implicit, multi-step model of the environment dynamics. This model can be learned without access to reward functions, but nonetheless can be used to directly estimate the value of each action, without requiring any TD learning. Because this model represents the multi-step transitions implicitly, it avoids having to predict high-dimensional observations and thus scales to high-dimensional tasks. Our experiments demonstrate that CVL outperforms prior offline RL methods on complex continuous control benchmarks.

Tudor Cristea-Platon, Bogdan Mazoure, Josh Susskind et al.

Reinforcement learning practitioners often avoid hierarchical policies, especially in image-based observation spaces. Typically, the single-task performance improvement over flat-policy counterparts does not justify the additional complexity associated with implementing a hierarchy. However, by introducing multiple decision-making levels, hierarchical policies can compose lower-level policies to more effectively generalize between tasks, highlighting the need for multi-task evaluations. We analyze the benefits of hierarchy through simulated multi-task robotic control experiments from pixels. Our results show that hierarchical policies trained with task conditioning can (1) increase performance on training tasks, (2) lead to improved reward and state-space generalizations in similar tasks, and (3) decrease the complexity of fine tuning required to solve novel tasks. Thus, we believe that hierarchical policies should be considered when building reinforcement learning architectures capable of generalizing between tasks.

Andrew Szot, Bogdan Mazoure, Omar Attia et al. · apple-ml, gatech

We examine the capability of Multimodal Large Language Models (MLLMs) to tackle diverse domains that extend beyond the traditional language and vision tasks these models are typically trained on. Specifically, our focus lies in areas such as Embodied AI, Games, UI Control, and Planning. To this end, we introduce a process of adapting an MLLM to a Generalist Embodied Agent (GEA). GEA is a single unified model capable of grounding itself across these varied domains through a multi-embodiment action tokenizer. GEA is trained with supervised learning on a large dataset of embodied experiences and with online RL in interactive simulators. We explore the data and algorithmic choices necessary to develop such a model. Our findings reveal the importance of training with cross-domain data and online RL for building generalist agents. The final GEA model achieves strong generalization performance to unseen tasks across diverse benchmarks compared to other generalist models and benchmark-specific approaches.

Jason Ramapuram, Eeshan Gunesh Dhekane, Amitis Shidani et al.

Speech-only spoken language models (SLMs) lag behind text and text-speech models in performance, with recent discrete autoregressive (AR) SLMs indicating significant computational and data demands to match text models. Since discretizing continuous speech for AR creates bottlenecks, we explore whether continuous diffusion (CD) SLM is more viable. To quantify the SLMs linguistic quality, we introduce the phoneme Jensen-Shannon divergence (pJSD) metric. Our analysis reveals CD SLMs, mirroring AR behavior, exhibit scaling laws for validation loss and pJSD, and show optimal token-to-parameter ratios decreasing as compute scales. However, for the latter, loss becomes insensitive to choice of data and model sizes, showing potential for fast inference. Scaling CD SLMs to 16B parameters with tens of millions of hours of conversational data enables generation of emotive, prosodic, multi-speaker, multilingual speech, though achieving long-form coherence remains a significant challenge.

Ram Ramrakhya, Andrew Szot, Omar Attia et al. · apple-ml, gatech

Post-Training Multimodal Large Language Models (MLLMs) to build interactive agents holds promise across domains such as computer-use, web navigation, and robotics. A key challenge in scaling such post-training is lack of high-quality downstream agentic task datasets with tasks that are diverse, feasible, and verifiable. Existing approaches for task generation rely heavily on human annotation or prompting MLLM with limited downstream environment information, which is either costly or poorly scalable as it yield tasks with limited coverage. To remedy this, we present AutoPlay, a scalable pipeline for task generation that explicitly explores interactive environments to discover possible interactions and current state information to synthesize environment-grounded tasks. AutoPlay operates in two stages: (i) an exploration phase, where an MLLM explorer agent systematically uncovers novel environment states and functionalities, and (ii) a task generation phase, where a task generator leverages exploration trajectories and a set of task guideline prompts as context to synthesize diverse, executable, and verifiable tasks. We show AutoPlay generates 20k tasks across 20 Android applications and 10k tasks across 13 applications Ubuntu applications to train mobile-use and computer-use agents. AutoPlay generated tasks enable large-scale task demonstration synthesis without human annotation by employing an MLLM task executor and verifier. This data enables training MLLM-based UI agents that improve success rates up to $20.0\%$ on mobile-use and $10.9\%$ on computer-use scenarios. In addition, AutoPlay generated tasks combined with MLLM verifier-based rewards enable scaling reinforcement learning training of UI agents, leading to an additional $5.7\%$ gain. coverage. These results establish AutoPlay as a scalable approach for post-training capable MLLM agents reducing reliance on human annotation.

Silvia Sapora, Devon Hjelm, Alexander Toshev et al.

Inverse Reinforcement Learning aims to recover reward models from expert demonstrations, but traditional methods yield "black-box" models that are difficult to interpret and debug. In this work, we introduce GRACE (Generating Rewards As CodE), a method for using Large Language Models within an evolutionary search to reverse-engineer an interpretable, code-based reward function directly from expert trajectories. The resulting reward function is executable code that can be inspected and verified. We empirically validate GRACE on the BabyAI and AndroidWorld benchmarks, where it efficiently learns highly accurate rewards, even in complex, multi-task settings. Further, we demonstrate that the resulting reward leads to strong policies, compared to both competitive Imitation Learning and online RL approaches with ground-truth rewards. Finally, we show that GRACE is able to build complex reward APIs in multi-task setups.

Mohammadreza Bakhtyari, Bogdan Mazoure, Renato Cordeiro de Amorim et al.

We introduce ClustRecNet - a novel deep learning (DL)-based recommendation framework for determining the most suitable clustering algorithms for a given dataset, addressing the long-standing challenge of clustering algorithm selection in unsupervised learning. To enable supervised learning in this context, we construct a comprehensive data repository comprising 34,000 synthetic datasets with diverse structural properties. Each of them was processed using 10 popular clustering algorithms. The resulting clusterings were assessed via the Adjusted Rand Index (ARI) to establish ground truth labels, used for training and evaluation of our DL model. The proposed network architecture integrates convolutional, residual, and attention mechanisms to capture both local and global structural patterns from the input data. This design supports end-to-end training to learn compact representations of datasets and enables direct recommendation of the most suitable clustering algorithm, reducing reliance on handcrafted meta-features and traditional Cluster Validity Indices (CVIs). Comprehensive experiments across synthetic and real-world benchmarks demonstrate that our DL model consistently outperforms conventional CVIs (e.g. Silhouette, Calinski-Harabasz, Davies-Bouldin, and Dunn) as well as state-of-the-art AutoML clustering recommendation approaches (e.g. ML2DAC, AutoCluster, and AutoML4Clust). Notably, the proposed model achieves a 0.497 ARI improvement over the Calinski-Harabasz index on synthetic data and a 15.3% ARI gain over the best-performing AutoML approach on real-world data.

Andrew Szot, Bogdan Mazoure, Harsh Agrawal et al.

Multimodal Large Language Models (MLLMs) have demonstrated a wide range of capabilities across many domains, including Embodied AI. In this work, we study how to best ground a MLLM into different embodiments and their associated action spaces, with the goal of leveraging the multimodal world knowledge of the MLLM. We first generalize a number of methods through a unified architecture and the lens of action space adaptors. For continuous actions, we show that a learned tokenization allows for sufficient modeling precision, yielding the best performance on downstream tasks. For discrete actions, we demonstrate that semantically aligning these actions with the native output token space of the MLLM leads to the strongest performance. We arrive at these lessons via a thorough study of seven action space adapters on five different environments, encompassing over 114 embodied tasks.

Tianyu Li, Bogdan Mazoure, Guillaume Rabusseau

Weighted finite automata (WFAs) have been widely applied in many fields. One of the classic problems for WFAs is probability distribution estimation over sequences of discrete symbols. Although WFAs have been extended to deal with continuous input data, namely continuous WFAs (CWFAs), it is still unclear how to approximate density functions over sequences of continuous random variables using WFA-based models, due to the limitation on the expressiveness of the model as well as the tractability of approximating density functions via CWFAs. In this paper, we propose a nonlinear extension to the CWFA model to first improve its expressiveness, we refer to it as the nonlinear continuous WFAs (NCWFAs). Then we leverage the so-called RNADE method, which is a well-known density estimator based on neural networks, and propose the RNADE-NCWFA model. The RNADE-NCWFA model computes a density function by design. We show that this model is strictly more expressive than the Gaussian HMM model, which CWFA cannot approximate. Empirically, we conduct a synthetic experiment using Gaussian HMM generated data. We focus on evaluating the model's ability to estimate densities for sequences of varying lengths (longer length than the training data). We observe that our model performs the best among the compared baseline methods.

Bogdan Mazoure, Ilya Kostrikov, Ofir Nachum et al.

Reinforcement learning (RL) agents are widely used for solving complex sequential decision making tasks, but still exhibit difficulty in generalizing to scenarios not seen during training. While prior online approaches demonstrated that using additional signals beyond the reward function can lead to better generalization capabilities in RL agents, i.e. using self-supervised learning (SSL), they struggle in the offline RL setting, i.e. learning from a static dataset. We show that performance of online algorithms for generalization in RL can be hindered in the offline setting due to poor estimation of similarity between observations. We propose a new theoretically-motivated framework called Generalized Similarity Functions (GSF), which uses contrastive learning to train an offline RL agent to aggregate observations based on the similarity of their expected future behavior, where we quantify this similarity using \emph{generalized value functions}. We show that GSF is general enough to recover existing SSL objectives while also improving zero-shot generalization performance on a complex offline RL benchmark, offline Procgen.

Bogdan Mazoure, Ahmed M. Ahmed, Patrick MacAlpine et al.

A highly desirable property of a reinforcement learning (RL) agent -- and a major difficulty for deep RL approaches -- is the ability to generalize policies learned on a few tasks over a high-dimensional observation space to similar tasks not seen during training. Many promising approaches to this challenge consider RL as a process of training two functions simultaneously: a complex nonlinear encoder that maps high-dimensional observations to a latent representation space, and a simple linear policy over this space. We posit that a superior encoder for zero-shot generalization in RL can be trained by using solely an auxiliary SSL objective if the training process encourages the encoder to map behaviorally similar observations to similar representations, as reward-based signal can cause overfitting in the encoder (Raileanu et al., 2021). We propose Cross-Trajectory Representation Learning (CTRL), a method that runs within an RL agent and conditions its encoder to recognize behavioral similarity in observations by applying a novel SSL objective to pairs of trajectories from the agent's policies. CTRL can be viewed as having the same effect as inducing a pseudo-bisimulation metric but, crucially, avoids the use of rewards and associated overfitting risks. Our experiments ablate various components of CTRL and demonstrate that in combination with PPO it achieves better generalization performance on the challenging Procgen benchmark suite (Cobbe et al., 2020).

Bogdan Mazoure, Paul Mineiro, Pavithra Srinath et al.

We study session-based recommendation scenarios where we want to recommend items to users during sequential interactions to improve their long-term utility. Optimizing a long-term metric is challenging because the learning signal (whether the recommendations achieved their desired goals) is delayed and confounded by other user interactions with the system. Targeting immediately measurable proxies such as clicks can lead to suboptimal recommendations due to misalignment with the long-term metric. We develop a new reinforcement learning algorithm called Short Horizon Policy Improvement (SHPI) that approximates policy-induced drift in user behavior across sessions. SHPI is a straightforward modification of episodic RL algorithms for session-based recommendation, that additionally gives an appropriate termination bonus in each session. Empirical results on four recommendation tasks show that SHPI can outperform state-of-the-art recommendation techniques like matrix factorization with offline proxy signals, bandits with myopic online proxies, and RL baselines with limited amounts of user interaction.

Thang Doan, Mehdi Bennani, Bogdan Mazoure et al.

Continual learning (CL) is a setting in which an agent has to learn from an incoming stream of data during its entire lifetime. Although major advances have been made in the field, one recurring problem which remains unsolved is that of Catastrophic Forgetting (CF). While the issue has been extensively studied empirically, little attention has been paid from a theoretical angle. In this paper, we show that the impact of CF increases as two tasks increasingly align. We introduce a measure of task similarity called the NTK overlap matrix which is at the core of CF. We analyze common projected gradient algorithms and demonstrate how they mitigate forgetting. Then, we propose a variant of Orthogonal Gradient Descent (OGD) which leverages structure of the data through Principal Component Analysis (PCA). Experiments support our theoretical findings and show how our method can help reduce CF on classical CL datasets.

Bogdan Mazoure, Remi Tachet des Combes, Thang Doan et al.

We begin with the hypothesis that a model-free agent whose representations are predictive of properties of future states (beyond expected rewards) will be more capable of solving and adapting to new RL problems. To test that hypothesis, we introduce an objective based on Deep InfoMax (DIM) which trains the agent to predict the future by maximizing the mutual information between its internal representation of successive timesteps. We test our approach in several synthetic settings, where it successfully learns representations that are predictive of the future. Finally, we augment C51, a strong RL baseline, with our temporal DIM objective and demonstrate improved performance on a continual learning task and on the recently introduced Procgen environment.

Bogdan Mazoure, Thang Doan, Tianyu Li et al.

We propose a general framework for policy representation for reinforcement learning tasks. This framework involves finding a low-dimensional embedding of the policy on a reproducing kernel Hilbert space (RKHS). The usage of RKHS based methods allows us to derive strong theoretical guarantees on the expected return of the reconstructed policy. Such guarantees are typically lacking in black-box models, but are very desirable in tasks requiring stability. We conduct several experiments on classic RL domains. The results confirm that the policies can be robustly embedded in a low-dimensional space while the embedded policy incurs almost no decrease in return.

Tianyu Li, Bogdan Mazoure, Doina Precup et al.

Learning and planning in partially-observable domains is one of the most difficult problems in reinforcement learning. Traditional methods consider these two problems as independent, resulting in a classical two-stage paradigm: first learn the environment dynamics and then plan accordingly. This approach, however, disconnects the two problems and can consequently lead to algorithms that are sample inefficient and time consuming. In this paper, we propose a novel algorithm that combines learning and planning together. Our algorithm is closely related to the spectral learning algorithm for predicitive state representations and offers appealing theoretical guarantees and time complexity. We empirically show on two domains that our approach is more sample and time efficient compared to classical methods.

Thang Doan, Bogdan Mazoure, Moloud Abdar et al.

Continuous control tasks in reinforcement learning are important because they provide an important framework for learning in high-dimensional state spaces with deceptive rewards, where the agent can easily become trapped into suboptimal solutions. One way to avoid local optima is to use a population of agents to ensure coverage of the policy space, yet learning a population with the "best" coverage is still an open problem. In this work, we present a novel approach to population-based RL in continuous control that leverages properties of normalizing flows to perform attractive and repulsive operations between current members of the population and previously observed policies. Empirical results on the MuJoCo suite demonstrate a high performance gain for our algorithm compared to prior work, including Soft-Actor Critic (SAC).

Cody Mazza-Anthony, Bogdan Mazoure, Mark Coates

We address the task of identifying densely connected subsets of multivariate Gaussian random variables within a graphical model framework. We propose two novel estimators based on the Ordered Weighted $\ell_1$ (OWL) norm: 1) The Graphical OWL (GOWL) is a penalized likelihood method that applies the OWL norm to the lower triangle components of the precision matrix. 2) The column-by-column Graphical OWL (ccGOWL) estimates the precision matrix by performing OWL regularized linear regressions. Both methods can simultaneously identify highly correlated groups of variables and control the sparsity in the resulting precision matrix. We formulate GOWL such that it solves a composite optimization problem and establish that the estimator has a unique global solution. In addition, we prove sufficient grouping conditions for each column of the ccGOWL precision matrix estimate. We propose proximal descent algorithms to find the optimum for both estimators. For synthetic data where group structure is present, the ccGOWL estimator requires significantly reduced computation and achieves similar or greater accuracy than state-of-the-art estimators. Timing comparisons are presented and demonstrates the superior computational efficiency of the ccGOWL. We illustrate the grouping performance of the ccGOWL method on a cancer gene expression data set and an equities data set.

Bogdan Mazoure, Thang Doan, Audrey Durand et al.

The ability to discover approximately optimal policies in domains with sparse rewards is crucial to applying reinforcement learning (RL) in many real-world scenarios. Approaches such as neural density models and continuous exploration (e.g., Go-Explore) have been proposed to maintain the high exploration rate necessary to find high performing and generalizable policies. Soft actor-critic(SAC) is another method for improving exploration that aims to combine efficient learning via off-policy updates while maximizing the policy entropy. In this work, we extend SAC to a richer class of probability distributions (e.g., multimodal) through normalizing flows (NF) and show that this significantly improves performance by accelerating the discovery of good policies while using much smaller policy representations. Our approach, which we call SAC-NF, is a simple, efficient,easy-to-implement modification and improvement to SAC on continuous control baselines such as MuJoCo and PyBullet Roboschool domains. Finally, SAC-NF does this while being significantly parameter efficient, using as few as 5.5% the parameters for an equivalent SAC model.

Atta Norouzian, Bogdan Mazoure, Dermot Connolly et al.

Voice controlled virtual assistants (VAs) are now available in smartphones, cars, and standalone devices in homes. In most cases, the user needs to first "wake-up" the VA by saying a particular word/phrase every time he or she wants the VA to do something. Eliminating the need for saying the wake-up word for every interaction could improve the user experience. This would require the VA to have the capability to detect the speech that is being directed at it and respond accordingly. In other words, the challenge is to distinguish between system-directed and non-system-directed speech utterances. In this paper, we present a number of neural network architectures for tackling this classification problem based on using only acoustic features. These architectures are based on using convolutional, recurrent and feed-forward layers. In addition, we investigate the use of an attention mechanism applied to the output of the convolutional and the recurrent layers. It is shown that incorporating the proposed attention mechanism into the models always leads to significant improvement in classification accuracy. The best model achieved equal error rates of 16.25 and 15.62 percents on two distinct realistic datasets.

Thang Doan, Joao Monteiro, Isabela Albuquerque et al.

Generative Adversarial Networks (GANs) can successfully approximate a probability distribution and produce realistic samples. However, open questions such as sufficient convergence conditions and mode collapse still persist. In this paper, we build on existing work in the area by proposing a novel framework for training the generator against an ensemble of discriminator networks, which can be seen as a one-student/multiple-teachers setting. We formalize this problem within the full-information adversarial bandit framework, where we evaluate the capability of an algorithm to select mixtures of discriminators for providing the generator with feedback during learning. To this end, we propose a reward function which reflects the progress made by the generator and dynamically update the mixture weights allocated to each discriminator. We also draw connections between our algorithm and stochastic optimization methods and then show that existing approaches using multiple discriminators in literature can be recovered from our framework. We argue that less expressive discriminators are smoother and have a general coarse grained view of the modes map, which enforces the generator to cover a wide portion of the data distribution support. On the other hand, highly expressive discriminators ensure samples quality. Finally, experimental results show that our approach improves samples quality and diversity over existing baselines by effectively learning a curriculum. These results also support the claim that weaker discriminators have higher entropy improving modes coverage. Keywords: multiple discriminators, curriculum learning, multiple resolutions discriminators, multi-armed bandits, generative adversarial networks, smooth discriminators, multi-discriminator gan training, multiple experts.

Thang Doan, Bogdan Mazoure, Clare Lyle

Distributional reinforcement learning (distributional RL) has seen empirical success in complex Markov Decision Processes (MDPs) in the setting of nonlinear function approximation. However, there are many different ways in which one can leverage the distributional approach to reinforcement learning. In this paper, we propose GAN Q-learning, a novel distributional RL method based on generative adversarial networks (GANs) and analyze its performance in simple tabular environments, as well as OpenAI Gym. We empirically show that our algorithm leverages the flexibility and blackbox approach of deep learning models while providing a viable alternative to traditional methods.

Bogdan Mazoure, Riashat Islam

We investigate the use of alternative divergences to Kullback-Leibler (KL) in variational inference(VI), based on the Variational Dropout \cite{kingma2015}. Stochastic gradient variational Bayes (SGVB) \cite{aevb} is a general framework for estimating the evidence lower bound (ELBO) in Variational Bayes. In this work, we extend the SGVB estimator with using Alpha-Divergences, which are alternative to divergences to VI' KL objective. The Gaussian dropout can be seen as a local reparametrization trick of the SGVB objective. We extend the Variational Dropout to use alpha divergences for variational inference. Our results compare $α$-divergence variational dropout with standard variational dropout with correlated and uncorrelated weight noise. We show that the $α$-divergence with $α\rightarrow 1$ (or KL divergence) is still a good measure for use in variational inference, in spite of the efficient use of Alpha-divergences for Dropout VI \cite{Li17}. $α\rightarrow 1$ can yield the lowest training error, and optimizes a good lower bound for the evidence lower bound (ELBO) among all values of the parameter $α\in [0,\infty)$.