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Deep Learning · Generative Models and Autoencoders

Giannis Daras, Alexandros Dimakis, Constantinos Daskalakis

Ambient diffusion is a recently proposed framework for training diffusion models using corrupted data. Both Ambient Diffusion and alternative SURE-based approaches for learning diffusion models from corrupted data resort to approximations which deteriorate performance. We present the first framework for training diffusion models that provably sample from the uncorrupted distribution given only noisy training data, solving an open problem in Ambient diffusion. Our key technical contribution is a method that uses a double application of Tweedie's formula and a consistency loss function that allows us to extend sampling at noise levels below the observed data noise. We also provide further evidence that diffusion models memorize from their training sets by identifying extremely corrupted images that are almost perfectly reconstructed, raising copyright and privacy concerns. Our method for training using corrupted samples can be used to mitigate this problem. We demonstrate this by fine-tuning Stable Diffusion XL to generate samples from a distribution using only noisy samples. Our framework reduces the amount of memorization of the fine-tuning dataset, while maintaining competitive performance.

Probabilistic Methods

Louis Sharrock, Jack Simons, Song Liu, Mark Beaumont

We introduce Sequential Neural Posterior Score Estimation (SNPSE), a score-based method for Bayesian inference in simulator-based models. Our method, inspired by the remarkable success of score-based methods in generative modelling, leverages conditional score-based diffusion models to generate samples from the posterior distribution of interest. The model is trained using an objective function which directly estimates the score of the posterior. We embed the model into a sequential training procedure, which guides simulations using the current approximation of the posterior at the observation of interest, thereby reducing the simulation cost. We also introduce several alternative sequential approaches, and discuss their relative merits. We then validate our method, as well as its amortised, non-sequential, variant on several numerical examples, demonstrating comparable or superior performance to existing state-of-the-art methods such as Sequential Neural Posterior Estimation (SNPE).

Theory · Probabilistic Methods

Song Liu, Jiahao Yu, Jack Simons, Mingxuan Yi, Mark Beaumont

Many machine learning problems can be seen as approximating a *target* distribution using a *particle* distribution by minimizing their statistical discrepancy. Wasserstein Gradient Flow can move particles along a path that minimizes the $f$-divergence between the target and particle distributions. To move particles, we need to calculate the corresponding velocity fields derived from a density ratio function between these two distributions. Previous works estimated such density ratio functions and then differentiated the estimated ratios. These approaches may suffer from overfitting, leading to a less accurate estimate of the velocity fields. Inspired by non-parametric curve fitting, we directly estimate these velocity fields using interpolation techniques. We prove that our estimators are consistent under mild conditions. We validate their effectiveness using novel applications on domain adaptation and missing data imputation. The code for reproducing our results can be found at https://github.com/anewgithubname/gradest2.

Simon Buchholz, Bernhard Schölkopf

We study the problem of unsupervised representation learning in slightly misspecified settings, and thus formalize the study of robustness of nonlinear representation learning. We focus on the case where the mixing is close to a local isometry in a suitable distance and show based on existing rigidity results that the mixing can be identified up to linear transformations and small errors. In a second step, we investigate Independent Component Analysis (ICA) with observations generated according to $x=f(s)=As+h(s)$ where $A$ is an invertible mixing matrix and $h$ a small perturbation. We show that we can approximately recover the matrix $A$ and the independent components. Together, these two results show approximate identifiability of nonlinear ICA with almost isometric mixing functions. Those results are a step towards identifiability results for unsupervised representation learning for real-world data that do not follow restrictive model classes.

Reinforcement Learning · Batch/Offline

Shengchao Hu, Ziqing Fan, Chaoqin Huang, Li Shen, Ya Zhang, Yanfeng Wang, Dacheng Tao

Recent advancements in offline reinforcement learning (RL) have underscored the capabilities of Conditional Sequence Modeling (CSM), a paradigm that learns the action distribution based on history trajectory and target returns for each state. However, these methods often struggle with stitching together optimal trajectories from sub-optimal ones due to the inconsistency between the sampled returns within individual trajectories and the optimal returns across multiple trajectories. Fortunately, Dynamic Programming (DP) methods offer a solution by leveraging a value function to approximate optimal future returns for each state, while these techniques are prone to unstable learning behaviors, particularly in long-horizon and sparse-reward scenarios. Building upon these insights, we propose the Q-value regularized Transformer (QT), which combines the trajectory modeling ability of the Transformer with the predictability of optimal future returns from DP methods. QT learns an action-value function and integrates a term maximizing action-values into the training loss of CSM, which aims to seek optimal actions that align closely with the behavior policy. Empirical evaluations on D4RL benchmark datasets demonstrate the superiority of QT over traditional DP and CSM methods, highlighting the potential of QT to enhance the state-of-the-art in offline RL.

Probabilistic Methods

JIAN XU, Delu Zeng, John Paisley

Deep Gaussian processes (DGPs) provide a robust paradigm in Bayesian deep learning. In DGPs, a set of sparse integration locations called inducing points are selected to approximate the posterior distribution of the model. This is done to reduce computational complexity and improve model efficiency. However, inferring the posterior distribution of inducing points is not straightforward. Traditional variational inference techniques methods to approximate the posterior often leads to significant bias. To address this issue, we propose an alternative named Denoising Diffusion Variational Inference (DDVI) that utilizes a denoising diffusion stochastic differential equation (SDE) for generating posterior samples of inducing variables. We refer to the score matching method in the denoising diffusion model to approximate challenging score functions using a neural network. Furthermore, by combining classical mathematical theory of SDE with the minimization of KL divergence between the approximate and true processes, we propose a novel explicit variational lower bound for the marginal likelihood function of DGP. Through extensive experiments on various datasets and comparisons with baseline methods, we empirically demonstrate the effectiveness of the DDVI method in posterior inference of inducing points for DGP models.

Reinforcement Learning · Batch/Offline

Mao Hong, Zhengling Qi, Yanxun Xu

We propose a model-based offline reinforcement learning (RL) algorithm for confounded partially observable Markov decision processes (POMDPs) under general function approximations and show it is provably efficient under some technical conditions such as the partial coverage imposed on the offline data distribution. Specifically, we first establish a novel model-based identification result for learning the effect of any action on the reward and future transitions in the confounded POMDP. Using this identification result, we then design a nonparametric two-stage estimation procedure to construct an estimator for off-policy evaluation (OPE), which permits general function approximations. Finally, we learn the optimal policy by performing a conservative policy optimization within the confidence regions based on the proposed estimation procedure for OPE. Under some mild conditions, we establish a finite-sample upper bound on the suboptimality of the learned policy in finding the optimal one, which depends on the sample size and the length of horizons polynomially.

Optimization · Discrete and Combinatorial Optimization

PAUL DUETTING, Federico Fusco, Silvio Lattanzi, Ashkan Norouzi-Fard, Morteza Zadimoghaddam

Maximizing monotone submodular functions under cardinality constraints is a classic optimization task with several applications in data mining and machine learning. In this paper, we study this problem in a dynamic environment with consistency constraints: elements arrive in a streaming fashion, and the goal is maintaining a constant approximation to the optimal solution while having a stable solution (i.e., the number of changes between two consecutive solutions is bounded). In this setting, we provide algorithms with different trade-offs between consistency and approximation quality. We also complement our theoretical results with an experimental analysis showing the effectiveness of our algorithms in real-world instances.

Optimization · Optimization and Learning under Uncertainty

Yucen Li, Tim G. J. Rudner, Andrew Gordon Wilson

Bayesian optimization is a highly efficient approach to optimizing objective functions which are expensive to query. These objectives are typically represented by Gaussian process (GP) surrogate models which are easy to optimize and support exact inference. While standard GP surrogates have been well-established in Bayesian optimization, Bayesian neural networks (BNNs) have recently become practical function approximators, with many benefits over standard GPs such as the ability to naturally handle non-stationarity and learn representations for high-dimensional data. In this paper, we study BNNs as alternatives to standard GP surrogates for optimization. We consider a variety of approximate inference procedures for finite-width BNNs, including high-quality Hamiltonian Monte Carlo, low-cost stochastic MCMC, and heuristics such as deep ensembles. We also consider infinite-width BNNs, linearized Laplace approximations, and partially stochastic models such as deep kernel learning. We evaluate this collection of surrogate models on diverse problems with varying dimensionality, number of objectives, non-stationarity, and discrete and continuous inputs. We find: (i) the ranking of methods is highly problem dependent, suggesting the need for tailored inductive biases; (ii) HMC is the most successful approximate inference procedure for fully stochastic BNNs; (iii) full stochasticity may be unnecessary as deep kernel learning is relatively competitive; (iv) deep ensembles perform relatively poorly; (v) infinite-width BNNs are particularly promising, especially in high dimensions.

Reinforcement Learning

Mao Hong, Zhengling Qi, Yanxun Xu

In this paper, we propose a policy gradient method for confounded partially observable Markov decision processes (POMDPs) with continuous state and observation spaces in the offline setting. We first establish a novel identification result to non-parametrically estimate any history-dependent policy gradient under POMDPs using the offline data. The identification enables us to solve a sequence of conditional moment restrictions and adopt the min-max learning procedure with general function approximation for estimating the policy gradient. We then provide a finite-sample non-asymptotic bound for estimating the gradient uniformly over a pre-specified policy class in terms of the sample size, length of horizon, concentratability coefficient and the measure of ill-posedness in solving the conditional moment restrictions. Lastly, by deploying the proposed gradient estimation in the gradient ascent algorithm, we show the global convergence of the proposed algorithm in finding the history-dependent optimal policy under some technical conditions. To the best of our knowledge, this is the first work studying the policy gradient method for POMDPs under the offline setting.

Reinforcement Learning · Batch/Offline

Chenjie Mao, Qiaosheng Zhang, Zhen Wang, Xuelong Li

We study offline reinforcement learning (RL) with general function approximation. General function approximation is a powerful tool for algorithm design and analysis, but its adaptation to offline RL encounters several challenges due to varying approximation targets and assumptions that blur the real meanings of function assumptions. In this paper, we try to formulate and clarify the treatment of general function approximation in offline RL in two aspects: (1) analyzing different types of assumptions and their practical usage, and (2) understanding its role as a restriction on underlying MDPs from information-theoretic perspectives. Additionally, we introduce a new insight for lower bound establishing: one can exploit model-realizability to establish general-purpose lower bounds that can be generalized into other functions. Building upon this insight, we propose two generic lower bounds that contribute to a better understanding of offline RL with general function approximation.

Deep Learning

Renbo Tu, Colin White, Jean Kossaifi, Boris Bonev, Gennady Pekhimenko, Kamyar Azizzadenesheli, anima anandkumar

Neural operators, such as Fourier Neural Operators (FNO), form a principled approach for learning solution operators for partial differential equations (PDE) and other mappings between function spaces. However, many real-world problems require high-resolution training data, and the training time and limited GPU memory pose big barriers. One solution is to train neural operators in mixed precision to reduce the memory requirement and increase training speed. However, existing mixed-precision training techniques are designed for standard neural networks, and we find that their direct application to FNO leads to numerical overflow and poor memory efficiency. Further, at first glance, it may appear that mixed precision in FNO will lead to drastic accuracy degradation since reducing the precision of the Fourier transform yields poor results in classical numerical solvers. We show that this is not the case; in fact, we prove that reducing the precision in FNO still guarantees a good approximation bound, when done in a targeted manner. Specifically, we build on the intuition that neural operator learning inherently induces an approximation error, arising from discretizing the infinite-dimensional ground-truth input function, implying that training in full precision is not needed. We formalize this intuition by rigorously characterizing the approximation and precision errors of FNO and bounding these errors for general input functions. We prove that the precision error is asymptotically comparable to the approximation error. Based on this, we design a simple method to optimize the memory-intensive half-precision tensor contractions by greedily finding the optimal contraction order. Through extensive experiments on different state-of-the-art neural operators, datasets, and GPUs, we demonstrate that our approach reduces GPU memory usage by up to 50% and improves throughput by 58% with little or no reduction in accuracy.

Ian Gemp, Luke Marris, Georgios Piliouras

We propose the first loss function for approximate Nash equilibria of normal-form games that is amenable to unbiased Monte Carlo estimation. This construction allows us to deploy standard non-convex stochastic optimization techniques for approximating Nash equilibria, resulting in novel algorithms with provable guarantees. We complement our theoretical analysis with experiments demonstrating that stochastic gradient descent can outperform previous state-of-the-art approaches.

Deep Learning · Theory

Rebekka Burkholz

Normalization techniques, for which Batch Normalization (BN) is a popular choice, is an integral part of many deep learning architectures and contributes significantly to the learning success. We provide a partial explanation for this phenomenon by proving that training normalization parameters alone is already sufficient for universal function approximation if the number of available, potentially random features matches or exceeds the weight parameters of the target networks that can be expressed. Our bound on the number of required features does not only improve on a recent result for fully-connected feed-forward architectures but also applies to CNNs with and without residual connections and almost arbitrary activation functions (which include ReLUs). Our explicit construction of a given target network solves a depth-width trade-off that is driven by architectural constraints and can explain why switching off entire neurons can have representational benefits, as has been observed empirically. To validate our theory, we explicitly match target networks that outperform experimentally obtained networks with trained BN parameters by utilizing a sufficient number of random features.

Optimization · Global Optimization

Carl Hvarfner, Frank Hutter, Luigi Nardi

The optimization of expensive-to-evaluate black-box functions is prevalent in various scientific disciplines. Bayesian optimization is an automatic, general and sample-efficient method to solve these problems with minimal knowledge of the the underlying function dynamics. However, the ability of Bayesian optimization to incorporate prior knowledge or beliefs about the function at hand in order to accelerate the optimization is limited, which reduces its appeal for knowledgeable practitioners with tight budgets. To allow domain experts to customize the optimization routine, we propose ColaBO, the first Bayesian-principled framework for incorporating prior beliefs beyond the typical kernel structure, such as the likely location of the optimizer or the optimal value. The generality of ColaBO makes it applicable across different Monte Carlo acquisition functions and types of user beliefs. We empirically demonstrate ColaBO's ability to substantially accelerate optimization when the prior information is accurate, and to retain approximately default performance when it is misleading.

Optimization · Optimization and Learning under Uncertainty

Rares Cristian, Georgia Perakis

We study contextual stochastic optimization problems. Optimization problems have uncertain parameters stemming from unknown, context-dependent, distributions. Due to the inherent uncertainty in these problems, one is often interested not only in minimizing expected cost, but also to be robust and protect against worst case scenarios. We propose a novel method that combines the learning stage with knowledge of the downstream optimization task. The method prescribes decisions which aim to maximize the likelihood that the cost is below a (user-controlled) threshold. The key idea is (1) to discretize the feasible region into subsets so that the uncertain objective function can be well approximated deterministically within each subset, and (2) devise a secondary optimization problem to prescribe decisions by integrating the individual approximations determined in step (1). We provide theoretical guarantees bounding the underlying regret of decisions proposed by our method. In addition, experimental results demonstrate that our approach is competitive in terms of average regret and yields more robust solutions than other methods proposed in the literature, including up to 20 times lower worst-case cost on a real-world electricity generation problem.

Reinforcement Learning · Function Approximation

Nuoya Xiong, Zhihan Liu, Zhaoran Wang, Zhuoran Yang

We study multi-agent reinforcement learning (MARL) for the general-sum Markov Games (MGs) under general function approximation. In order to find the minimum assumption for sample-efficient learning, we introduce a novel complexity measure called the Multi-Agent Decoupling Coefficient (MADC) for general-sum MGs. Using this measure, we propose the first unified algorithmic framework that ensures sample efficiency in learning Nash Equilibrium, Coarse Correlated Equilibrium, and Correlated Equilibrium for both model-based and model-free MARL problems with low MADC. We also show that our algorithm provides comparable sublinear regret to the existing works. Moreover, our algorithm combines an equilibrium-solving oracle with a single objective optimization subprocedure that solves for the regularized payoff of each deterministic joint policy, which avoids solving constrained optimization problems within data-dependent constraints (Jin et al. 2020; Wang et al. 2023) or executing sampling procedures with complex multi-objective optimization problems (Foster et al. 2023), thus being more amenable to empirical implementation.

Reinforcement Learning · Planning

Amin Rakhsha, Mete Kemertas, Mohammad Ghavamzadeh, Amir-massoud Farahmand

We propose and theoretically analyze an approach for planning with an approximate model in reinforcement learning that can reduce the adverse impact of model error. If the model is accurate enough, it accelerates the convergence to the true value function too. One of its key components is the MaxEnt Model Correction (MoCo) procedure that corrects the model’s next-state distributions based on a Maximum Entropy density estimation formulation. Based on MoCo, we introduce the Model Correcting Value Iteration (MoCoVI) algorithm, and its sampled-based variant MoCoDyna. We show that MoCoVI and MoCoDyna’s convergence can be much faster than the conventional model-free algorithms. Unlike traditional model-based algorithms, MoCoVI and MoCoDyna effectively utilize an approximate model and still converge to the correct value function.

Guoqiang Zhang, Kenta Niwa, W. Bastiaan Kleijn

A popular approach to sample a diffusion-based generative model is to solve an ordinary differential equation (ODE). In existing samplers, the coefficients of the ODE solvers are pre-determined by the ODE formulation, the reverse discrete timesteps, and the employed ODE methods. In this paper, we consider accelerating several popular ODE-based sampling processes (including EDM, DDIM, and DPM-Solver) by optimizing certain coefficients via improved integration approximation (IIA). We propose to minimize, for each time step, a mean squared error (MSE) function with respect to the selected coefficients. The MSE is constructed by applying the original ODE solver for a set of fine-grained timesteps, which in principle provides a more accurate integration approximation in predicting the next diffusion state. The proposed IIA technique does not require any change of a pre-trained model, and only introduces a very small computational overhead for solving a number of quadratic optimization problems. Extensive experiments show that considerably better FID scores can be achieved by using IIA-EDM, IIA-DDIM, and IIA-DPM-Solver than the original counterparts when the neural function evaluation (NFE) is small (i.e., less than 25).

Reinforcement Learning

Hanlin Zhu, Baihe Huang, Stuart Russell

We study the representation complexity of model-based and model-free reinforcement learning (RL) in the context of circuit complexity. We prove theoretically that there exists a broad class of MDPs such that their underlying transition and reward functions can be represented by constant depth circuits with polynomial size, while the optimal $Q$-function suffers an exponential circuit complexity in constant-depth circuits. By drawing attention to the approximation errors and building connections to complexity theory, our theory provides unique insights into why model-based algorithms usually enjoy better sample complexity than model-free algorithms from a novel representation complexity perspective: in some cases, the ground-truth rule (model) of the environment is simple to represent, while other quantities, such as $Q$-function, appear complex. We empirically corroborate our theory by comparing the approximation error of the transition kernel, reward function, and optimal $Q$-function in various Mujoco environments, which demonstrates that the approximation errors of the transition kernel and reward function are consistently lower than those of the optimal $Q$-function. To the best of our knowledge, this work is the first to study the circuit complexity of RL, which also provides a rigorous framework for future research.