Anthony L. Caterini

Dae Yon Hwang, Raunaq Suri, Valentin Villecroze et al.

LLM agents operate in two distinct regimes: open-weight agents amenable to reinforcement learning (RL) and black-box agents whose behaviour must be controlled purely at test time. Although black-box agents are often backed by state-of-the-art proprietary LLMs, API-only access precludes parameter-level optimization, rendering most RL methods inapplicable. To address this limitation, we turn to a known equivalence between RL and Bayesian inference. We propose Agentic Monte Carlo (AMC) to directly sample from the optimal policy of a black-box agent rather than training it through RL. The optimal policy is a posterior over trajectories whose prior we define as the fixed black-box LLM agent. We employ Sequential Monte Carlo to sample from this posterior by learning a value function to steer the agent while leaving the underlying black-box model unchanged. We validate AMC on three diverse environments from the AgentGym benchmark, demonstrating significant improvements over prompting baselines and even outperforming Group Relative Policy Optimization (GRPO) as we scale the test-time compute of our method. AMC demonstrates the feasibility of performing principled RL-style optimization of black-box LLM agents. Code is available at https://github.com/layer6ai-labs/Agentic-Monte-Carlo

George Stein, Jesse C. Cresswell, Rasa Hosseinzadeh et al.

We systematically study a wide variety of generative models spanning semantically-diverse image datasets to understand and improve the feature extractors and metrics used to evaluate them. Using best practices in psychophysics, we measure human perception of image realism for generated samples by conducting the largest experiment evaluating generative models to date, and find that no existing metric strongly correlates with human evaluations. Comparing to 17 modern metrics for evaluating the overall performance, fidelity, diversity, rarity, and memorization of generative models, we find that the state-of-the-art perceptual realism of diffusion models as judged by humans is not reflected in commonly reported metrics such as FID. This discrepancy is not explained by diversity in generated samples, though one cause is over-reliance on Inception-V3. We address these flaws through a study of alternative self-supervised feature extractors, find that the semantic information encoded by individual networks strongly depends on their training procedure, and show that DINOv2-ViT-L/14 allows for much richer evaluation of generative models. Next, we investigate data memorization, and find that generative models do memorize training examples on simple, smaller datasets like CIFAR10, but not necessarily on more complex datasets like ImageNet. However, our experiments show that current metrics do not properly detect memorization: none in the literature is able to separate memorization from other phenomena such as underfitting or mode shrinkage. To facilitate further development of generative models and their evaluation we release all generated image datasets, human evaluation data, and a modular library to compute 17 common metrics for 9 different encoders at https://github.com/layer6ai-labs/dgm-eval.

Bradley C. A. Brown, Anthony L. Caterini, Brendan Leigh Ross et al.

Deep learning has had tremendous success at learning low-dimensional representations of high-dimensional data. This success would be impossible if there was no hidden low-dimensional structure in data of interest; this existence is posited by the manifold hypothesis, which states that the data lies on an unknown manifold of low intrinsic dimension. In this paper, we argue that this hypothesis does not properly capture the low-dimensional structure typically present in image data. Assuming that data lies on a single manifold implies intrinsic dimension is identical across the entire data space, and does not allow for subregions of this space to have a different number of factors of variation. To address this deficiency, we consider the union of manifolds hypothesis, which states that data lies on a disjoint union of manifolds of varying intrinsic dimensions. We empirically verify this hypothesis on commonly-used image datasets, finding that indeed, observed data lies on a disconnected set and that intrinsic dimension is not constant. We also provide insights into the implications of the union of manifolds hypothesis in deep learning, both supervised and unsupervised, showing that designing models with an inductive bias for this structure improves performance across classification and generative modelling tasks. Our code is available at https://github.com/layer6ai-labs/UoMH.

Gabriel Loaiza-Ganem, Brendan Leigh Ross, Luhuan Wu et al.

Likelihood-based deep generative models have recently been shown to exhibit pathological behaviour under the manifold hypothesis as a consequence of using high-dimensional densities to model data with low-dimensional structure. In this paper we propose two methodologies aimed at addressing this problem. Both are based on adding Gaussian noise to the data to remove the dimensionality mismatch during training, and both provide a denoising mechanism whose goal is to sample from the model as though no noise had been added to the data. Our first approach is based on Tweedie's formula, and the second on models which take the variance of added noise as a conditional input. We show that surprisingly, while well motivated, these approaches only sporadically improve performance over not adding noise, and that other methods of addressing the dimensionality mismatch are more empirically adequate.

Jesse C. Cresswell, Brendan Leigh Ross, Gabriel Loaiza-Ganem et al.

Precision measurements and new physics searches at the Large Hadron Collider require efficient simulations of particle propagation and interactions within the detectors. The most computationally expensive simulations involve calorimeter showers. Advances in deep generative modelling - particularly in the realm of high-dimensional data - have opened the possibility of generating realistic calorimeter showers orders of magnitude more quickly than physics-based simulation. However, the high-dimensional representation of showers belies the relative simplicity and structure of the underlying physical laws. This phenomenon is yet another example of the manifold hypothesis from machine learning, which states that high-dimensional data is supported on low-dimensional manifolds. We thus propose modelling calorimeter showers first by learning their manifold structure, and then estimating the density of data across this manifold. Learning manifold structure reduces the dimensionality of the data, which enables fast training and generation when compared with competing methods.

Gabriel Loaiza-Ganem, Brendan Leigh Ross, Jesse C. Cresswell et al.

Likelihood-based, or explicit, deep generative models use neural networks to construct flexible high-dimensional densities. This formulation directly contradicts the manifold hypothesis, which states that observed data lies on a low-dimensional manifold embedded in high-dimensional ambient space. In this paper we investigate the pathologies of maximum-likelihood training in the presence of this dimensionality mismatch. We formally prove that degenerate optima are achieved wherein the manifold itself is learned but not the distribution on it, a phenomenon we call manifold overfitting. We propose a class of two-step procedures consisting of a dimensionality reduction step followed by maximum-likelihood density estimation, and prove that they recover the data-generating distribution in the nonparametric regime, thus avoiding manifold overfitting. We also show that these procedures enable density estimation on the manifolds learned by implicit models, such as generative adversarial networks, hence addressing a major shortcoming of these models. Several recently proposed methods are instances of our two-step procedures; we thus unify, extend, and theoretically justify a large class of models.

Bradley C. A. Brown, Jordan Juravsky, Anthony L. Caterini et al.

Given a pair of models with similar training set performance, it is natural to assume that the model that possesses simpler internal representations would exhibit better generalization. In this work, we provide empirical evidence for this intuition through an analysis of the intrinsic dimension (ID) of model activations, which can be thought of as the minimal number of factors of variation in the model's representation of the data. First, we show that common regularization techniques uniformly decrease the last-layer ID (LLID) of validation set activations for image classification models and show how this strongly affects generalization performance. We also investigate how excessive regularization decreases a model's ability to extract features from data in earlier layers, leading to a negative effect on validation accuracy even while LLID continues to decrease and training accuracy remains near-perfect. Finally, we examine the LLID over the course of training of models that exhibit grokking. We observe that well after training accuracy saturates, when models ``grok'' and validation accuracy suddenly improves from random to perfect, there is a co-occurent sudden drop in LLID, thus providing more insight into the dynamics of sudden generalization.

Brendan Leigh Ross, Gabriel Loaiza-Ganem, Anthony L. Caterini et al.

Natural data observed in $\mathbb{R}^n$ is often constrained to an $m$-dimensional manifold $\mathcal{M}$, where $m < n$. This work focuses on the task of building theoretically principled generative models for such data. Current generative models learn $\mathcal{M}$ by mapping an $m$-dimensional latent variable through a neural network $f_θ: \mathbb{R}^m \to \mathbb{R}^n$. These procedures, which we call pushforward models, incur a straightforward limitation: manifolds cannot in general be represented with a single parameterization, meaning that attempts to do so will incur either computational instability or the inability to learn probability densities within the manifold. To remedy this problem, we propose to model $\mathcal{M}$ as a neural implicit manifold: the set of zeros of a neural network. We then learn the probability density within $\mathcal{M}$ with a constrained energy-based model, which employs a constrained variant of Langevin dynamics to train and sample from the learned manifold. In experiments on synthetic and natural data, we show that our model can learn manifold-supported distributions with complex topologies more accurately than pushforward models.

Hamidreza Kamkari, Brendan Leigh Ross, Jesse C. Cresswell et al.

Likelihood-based deep generative models (DGMs) commonly exhibit a puzzling behaviour: when trained on a relatively complex dataset, they assign higher likelihood values to out-of-distribution (OOD) data from simpler sources. Adding to the mystery, OOD samples are never generated by these DGMs despite having higher likelihoods. This two-pronged paradox has yet to be conclusively explained, making likelihood-based OOD detection unreliable. Our primary observation is that high-likelihood regions will not be generated if they contain minimal probability mass. We demonstrate how this seeming contradiction of large densities yet low probability mass can occur around data confined to low-dimensional manifolds. We also show that this scenario can be identified through local intrinsic dimension (LID) estimation, and propose a method for OOD detection which pairs the likelihoods and LID estimates obtained from a pre-trained DGM. Our method can be applied to normalizing flows and score-based diffusion models, and obtains results which match or surpass state-of-the-art OOD detection benchmarks using the same DGM backbones. Our code is available at https://github.com/layer6ai-labs/dgm_ood_detection.

Junwei Ma, Nour Shaheen, Alex Labach et al.

Deep tabular modelling increasingly relies on in-context learning where, during inference, a model receives a set of $(x,y)$ pairs as context and predicts labels for new inputs without weight updates. We challenge the prevailing view that broad generalization here requires pre-training on large synthetic corpora (e.g., TabPFN priors) or a large collection of real data (e.g., TabDPT training datasets), discovering that a relatively small amount of data suffices for generalization. We find that simple self-supervised pre-training on just a \emph{single} real table can produce surprisingly strong transfer across heterogeneous benchmarks. By systematically pre-training and evaluating on many diverse datasets, we analyze what aspects of the data are most important for building a Tabular Foundation Model (TFM) generalizing across domains. We then connect this to the pre-training procedure shared by most TFMs and show that the number and quality of \emph{tasks} one can construct from a dataset is key to downstream performance.

Junwei Ma, Valentin Thomas, Rasa Hosseinzadeh et al. · mila

Tabular data is one of the most ubiquitous sources of information worldwide, spanning a wide variety of domains. This inherent heterogeneity has slowed the development of Tabular Foundation Models (TFMs) capable of fast generalization to unseen datasets. In-Context Learning (ICL) has recently emerged as a promising solution for TFMs, enabling dynamic adaptation to new tasks without additional tuning. While many studies have attempted to re-purpose large language models for tabular ICL, they have had limited success, so recent works have focused on developing tabular-specific foundation models. In this work, we propose an approach to combine ICL-based retrieval with self supervised learning to train tabular foundation models. We also investigate the utility of real vs. synthetic data for model pre-training, and show that real data can contain useful signal not easily captured in synthetic training. Specifically, we show that incorporating real data during the pre-training phase can lead to significantly faster training and better downstream generalization to unseen data. Our resulting model, TabDPT, achieves top performance on both regression (CTR23) and classification (CC18) benchmarks. Importantly, we also demonstrate that with our pre-training procedure, scaling both model and data size leads to consistent performance improvements that follow power laws. This echoes scaling laws in LLMs and other foundation models, and suggests that Internet-scale TFMs can be achievable. We open-source our full pipeline: inference code including trained model weights can be found at github.com/layer6ai-labs/TabDPT-inference, and the training code to reproduce experiments can be found at github.com/layer6ai-labs/TabDPT-training.

Anthony L. Caterini, Gabriel Loaiza-Ganem, Geoff Pleiss et al.

Normalizing flows are invertible neural networks with tractable change-of-volume terms, which allow optimization of their parameters to be efficiently performed via maximum likelihood. However, data of interest are typically assumed to live in some (often unknown) low-dimensional manifold embedded in a high-dimensional ambient space. The result is a modelling mismatch since -- by construction -- the invertibility requirement implies high-dimensional support of the learned distribution. Injective flows, mappings from low- to high-dimensional spaces, aim to fix this discrepancy by learning distributions on manifolds, but the resulting volume-change term becomes more challenging to evaluate. Current approaches either avoid computing this term entirely using various heuristics, or assume the manifold is known beforehand and therefore are not widely applicable. Instead, we propose two methods to tractably calculate the gradient of this term with respect to the parameters of the model, relying on careful use of automatic differentiation and techniques from numerical linear algebra. Both approaches perform end-to-end nonlinear manifold learning and density estimation for data projected onto this manifold. We study the trade-offs between our proposed methods, empirically verify that we outperform approaches ignoring the volume-change term by more accurately learning manifolds and the corresponding distributions on them, and show promising results on out-of-distribution detection. Our code is available at https://github.com/layer6ai-labs/rectangular-flows.

Panteha Naderian, Gabriel Loaiza-Ganem, Harry J. Braviner et al.

Multi-goal reaching is an important problem in reinforcement learning needed to achieve algorithmic generalization. Despite recent advances in this field, current algorithms suffer from three major challenges: high sample complexity, learning only a single way of reaching the goals, and difficulties in solving complex motion planning tasks. In order to address these limitations, we introduce the concept of cumulative accessibility functions, which measure the reachability of a goal from a given state within a specified horizon. We show that these functions obey a recurrence relation, which enables learning from offline interactions. We also prove that optimal cumulative accessibility functions are monotonic in the planning horizon. Additionally, our method can trade off speed and reliability in goal-reaching by suggesting multiple paths to a single goal depending on the provided horizon. We evaluate our approach on a set of multi-goal discrete and continuous control tasks. We show that our method outperforms state-of-the-art goal-reaching algorithms in success rate, sample complexity, and path optimality. Our code is available at https://github.com/layer6ai-labs/CAE, and additional visualizations can be found at https://sites.google.com/view/learning-cae/.

Claudius Krause, Michele Faucci Giannelli, Gregor Kasieczka et al.

We present the results of the "Fast Calorimeter Simulation Challenge 2022" - the CaloChallenge. We study state-of-the-art generative models on four calorimeter shower datasets of increasing dimensionality, ranging from a few hundred voxels to a few tens of thousand voxels. The 31 individual submissions span a wide range of current popular generative architectures, including Variational AutoEncoders (VAEs), Generative Adversarial Networks (GANs), Normalizing Flows, Diffusion models, and models based on Conditional Flow Matching. We compare all submissions in terms of quality of generated calorimeter showers, as well as shower generation time and model size. To assess the quality we use a broad range of different metrics including differences in 1-dimensional histograms of observables, KPD/FPD scores, AUCs of binary classifiers, and the log-posterior of a multiclass classifier. The results of the CaloChallenge provide the most complete and comprehensive survey of cutting-edge approaches to calorimeter fast simulation to date. In addition, our work provides a uniquely detailed perspective on the important problem of how to evaluate generative models. As such, the results presented here should be applicable for other domains that use generative AI and require fast and faithful generation of samples in a large phase space.

Gabriel Loaiza-Ganem, Brendan Leigh Ross, Rasa Hosseinzadeh et al.

In recent years there has been increased interest in understanding the interplay between deep generative models (DGMs) and the manifold hypothesis. Research in this area focuses on understanding the reasons why commonly-used DGMs succeed or fail at learning distributions supported on unknown low-dimensional manifolds, as well as developing new models explicitly designed to account for manifold-supported data. This manifold lens provides both clarity as to why some DGMs (e.g. diffusion models and some generative adversarial networks) empirically surpass others (e.g. likelihood-based models such as variational autoencoders, normalizing flows, or energy-based models) at sample generation, and guidance for devising more performant DGMs. We carry out the first survey of DGMs viewed through this lens, making two novel contributions along the way. First, we formally establish that numerical instability of likelihoods in high ambient dimensions is unavoidable when modelling data with low intrinsic dimension. We then show that DGMs on learned representations of autoencoders can be interpreted as approximately minimizing Wasserstein distance: this result, which applies to latent diffusion models, helps justify their outstanding empirical results. The manifold lens provides a rich perspective from which to understand DGMs, and we aim to make this perspective more accessible and widespread.

Anthony L. Caterini, Gabriel Loaiza-Ganem

Deep generative models trained by maximum likelihood remain very popular methods for reasoning about data probabilistically. However, it has been observed that they can assign higher likelihoods to out-of-distribution (OOD) data than in-distribution data, thus calling into question the meaning of these likelihood values. In this work we provide a novel perspective on this phenomenon, decomposing the average likelihood into a KL divergence term and an entropy term. We argue that the latter can explain the curious OOD behaviour mentioned above, suppressing likelihood values on datasets with higher entropy. Although our idea is simple, we have not seen it explored yet in the literature. This analysis provides further explanation for the success of OOD detection methods based on likelihood ratios, as the problematic entropy term cancels out in expectation. Finally, we discuss how this observation relates to recent success in OOD detection with manifold-supported models, for which the above decomposition does not hold directly.

Rob Cornish, Anthony L. Caterini, George Deligiannidis et al.

We show that normalising flows become pathological when used to model targets whose supports have complicated topologies. In this scenario, we prove that a flow must become arbitrarily numerically noninvertible in order to approximate the target closely. This result has implications for all flow-based models, and especially Residual Flows (ResFlows), which explicitly control the Lipschitz constant of the bijection used. To address this, we propose Continuously Indexed Flows (CIFs), which replace the single bijection used by normalising flows with a continuously indexed family of bijections, and which can intuitively "clean up" mass that would otherwise be misplaced by a single bijection. We show theoretically that CIFs are not subject to the same topological limitations as normalising flows, and obtain better empirical performance on a variety of models and benchmarks.

Anthony L. Caterini, Arnaud Doucet, Dino Sejdinovic

Variational Auto-Encoders (VAEs) have become very popular techniques to perform inference and learning in latent variable models as they allow us to leverage the rich representational power of neural networks to obtain flexible approximations of the posterior of latent variables as well as tight evidence lower bounds (ELBOs). Combined with stochastic variational inference, this provides a methodology scaling to large datasets. However, for this methodology to be practically efficient, it is necessary to obtain low-variance unbiased estimators of the ELBO and its gradients with respect to the parameters of interest. While the use of Markov chain Monte Carlo (MCMC) techniques such as Hamiltonian Monte Carlo (HMC) has been previously suggested to achieve this [23, 26], the proposed methods require specifying reverse kernels which have a large impact on performance. Additionally, the resulting unbiased estimator of the ELBO for most MCMC kernels is typically not amenable to the reparameterization trick. We show here how to optimally select reverse kernels in this setting and, by building upon Hamiltonian Importance Sampling (HIS) [17], we obtain a scheme that provides low-variance unbiased estimators of the ELBO and its gradients using the reparameterization trick. This allows us to develop a Hamiltonian Variational Auto-Encoder (HVAE). This method can be reinterpreted as a target-informed normalizing flow [20] which, within our context, only requires a few evaluations of the gradient of the sampled likelihood and trivial Jacobian calculations at each iteration.

Anthony L. Caterini, Dong Eui Chang

In this paper, a geometric framework for neural networks is proposed. This framework uses the inner product space structure underlying the parameter set to perform gradient descent not in a component-based form, but in a coordinate-free manner. Convolutional neural networks are described in this framework in a compact form, with the gradients of standard --- and higher-order --- loss functions calculated for each layer of the network. This approach can be applied to other network structures and provides a basis on which to create new networks.