Gaussian processes are widely employed as versatile modelling and predictive tools in spatial statistics, functional data analysis, computer modelling and diverse applications of machine learning. They have been widely studied over Euclidean spaces, where they are specified using covariance functions or covariograms for modelling complex dependencies. There is a growing literature on Gaussian processes over Riemannian manifolds in order to develop richer and more flexible inferential frameworks for non-Euclidean data. While numerical approximations through graph representations have been well studied for the Matern covariogram and heat kernel, the behaviour of asymptotic inference on the parameters of the covariogram has received relatively scant attention. We focus on asymptotic behaviour for Gaussian processes constructed over compact Riemannian manifolds. Building upon a recently introduced Matern covariogram on a compact Riemannian manifold, we employ formal notions and conditions for the equivalence of two Matern Gaussian random measures on compact manifolds to derive the parameter that is identifiable, also known as the microergodic parameter, and formally establish the consistency of the maximum likelihood estimate and the asymptotic optimality of the best linear unbiased predictor. The circle is studied as a specific example of compact Riemannian manifolds with numerical experiments to illustrate and corroborate the theory.
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Deep Learning/Theory
Inspired by fast algorithms in natural language processing, we study low rank approximation in the entrywise transformed setting where we want to find a good rank $k$ approximation to $f(U \cdot V)$, where $U, V^\top \in \mathbb{R}^{n \times r}$ are given, $r = O(\log(n))$, and $f(x)$ is a general scalar function. Previous work in sublinear low rank approximation has shown that if both (1) $U = V^\top$ and (2) $f(x)$ is a PSD kernel function, then there is an $O(nk^{\omega-1})$ time constant relative error approximation algorithm, where $\omega \approx 2.376$ is the exponent of matrix multiplication. We give the first conditional time hardness results for this problem, demonstrating that both conditions (1) and (2) are in fact necessary for getting better than $n^{2-o(1)}$ time for a relative error low rank approximation for a wide class of functions. We give novel reductions from the Strong Exponential Time Hypothesis (SETH) that rely on lower bounding the leverage scores of flat sparse vectors and hold even when the rank of the transformed matrix $f(UV)$ and the target rank are $n^{o(1)}$, and when $U = V^\top$. Furthermore, even when $f(x) = x^p$ is a simple polynomial, we give runtime lower bounds in the case when $U \neq V^\top$ of the form $\Omega(\min(n^{2-o(1)}, \Omega(2^p)))$. Lastly, we demonstrate that our lower bounds are tight by giving an $O(n \cdot \text{poly}(k, 2^p, 1/\epsilon))$ time relative error approximation algorithm and a fast $O(n \cdot \text{poly}(k, p, 1/\epsilon))$ additive error approximation using fast tensor-based sketching. Additionally, since our low rank algorithms rely on matrix-vector product subroutines, our lower bounds extend to show that computing $f(UV)W$, for even a small matrix $W$, requires $\Omega(n^{2-o(1)})$ time.
Theory/Active Learning and Interactive Learning
We consider the problem of Imitation Learning (IL) by actively querying noisy expert for feedback. While imitation learning has been empirically successful, much of prior work assumes access to noiseless expert feedback which is not practical in many applications. In fact, when one only has access to noisy expert feedback, algorithms that rely on purely offline data (non-interactive IL) can be shown to need a prohibitively large number of samples to be successful. In contrast, in this work, we provide an interactive algorithm for IL that uses selective sampling to actively query the noisy expert for feedback. Our contributions are twofold: First, we provide a new selective sampling algorithm that works with general function classes and multiple actions, and obtains the best-known bounds for the regret and the number of queries. Next, we extend this analysis to the problem of IL with noisy expert feedback and provide a new IL algorithm that makes limited queries. Our algorithm for selective sampling leverages function approximation, and relies on an online regression oracle w.r.t.~the given model class to predict actions, and to decide whether to query the expert for its label. On the theoretical side, the regret bound of our algorithm is upper bounded by the regret of the online regression oracle, while the query complexity additionally depends on the eluder dimension of the model class. We complement this with a lower bound that demonstrates that our results are tight. We extend our selective sampling algorithm for IL with general function approximation and provide bounds on both the regret and the number of queries made to the noisy expert. A key novelty here is that our regret and query complexity bounds only depend on the number of times the optimal policy (and not the noisy expert, or the learner) go to states that have a small margin.
Denoising diffusion models have proven to be a flexible and effective paradigm for generative modelling.Their recent extension to infinite dimensional Euclidean spaces has allowed for the modelling of stochastic processes.However, many problems in the natural sciences incorporate symmetries and involve data living in non-Euclidean spaces.In this work, we extend the framework of diffusion models to incorporate a series of geometric priors in infinite-dimension modelling.We do so by a) constructing a noising process which admits, as limiting distribution, a geometric Gaussian process that transforms under the symmetry group of interest, and b) approximating the score with a neural network that is equivariant w.r.t. this group.We show that with these conditions, the generative functional model admits the same symmetry.We demonstrate scalability and capacity of the model, using a novel Langevin-based conditional sampler, to fit complex scalar and vector fields, with Euclidean and spherical codomain, on synthetic and real-world weather data.
Theory/Everything Else
By universal formulas we understand parameterized analytic expressions that have a fixed complexity, but nevertheless can approximate any continuous function on a compact set. There exist various examples of such formulas, including some in the form of neural networks. In this paper we analyze the essential structural elements of these highly expressive models. We introduce a hierarchy of expressiveness classes connecting the global approximability property to the weaker property of infinite VC dimension, and prove a series of classification results for several increasingly complex functional families. In particular, we introduce a general family of polynomially-exponentially-algebraic functions that, as we prove, is subject to polynomial constraints. As a consequence, we show that fixed-size neural networks with not more than one layer of neurons having transcendental activations (e.g., sine or standard sigmoid) cannot in general approximate functions on arbitrary finite sets. On the other hand, we give examples of functional families, including two-hidden-layer neural networks, that approximate functions on arbitrary finite sets, but fail to do that on the whole domain of definition.
Theory/Probabilistic Methods
We prove concentration inequalities and associated PAC bounds for both continuous- and discrete-time additive functionals for possibly unbounded functions of multivariate, nonreversible diffusion processes. Our analysis relies on an approach via the Poisson equation allowing us to consider a very broad class of subexponentially ergodic, multivariate diffusion processes. These results add to existing concentration inequalities for additive functionals of diffusion processes which have so far been only available for either bounded functions or for unbounded functions of processes from a significantly smaller class. We demonstrate the power of these exponential inequalities by two examples of very different areas. Considering a possibly high-dimensional, parametric, nonlinear drift model under sparsity constraints we apply the continuous-time concentration results to validate the restricted eigenvalue condition for Lasso estimation, which is fundamental for the derivation of oracle inequalities. The results for discrete additive functionals are applied for an investigation of the unadjusted Langevin MCMC algorithm for sampling of moderately heavy tailed densities $\pi$. In particular, we provide PAC bounds for the sample Monte Carlo estimator of integrals $\pi(f)$ for polynomially growing functions $f$ that quantify sufficient sample and step sizes for approximation within a prescribed margin with high probability.
Optimization/Zero-order and Black-box Optimization
Finite-difference methods are a class of algorithms designed to solve black-box optimization problems by approximating a gradient of the target function on a set of directions. In black-box optimization, the non-smooth setting is particularly relevant since, in practice, differentiability and smoothness assumptions cannot be verified. To cope with nonsmoothness, several authors use a smooth approximation of the target function and show that finite difference methods approximate its gradient. Recently, it has been proved that imposing a structure in the directions allows improving performance. However, only the smooth setting was considered. To close this gap, we introduce and analyze O-ZD, the first structured finite-difference algorithm for non-smooth black-box optimization. Our method exploits a smooth approximation of the target function and we prove that it approximates its gradient on a subset of random {\em orthogonal} directions. We analyze the convergence of O-ZD under different assumptions. For non-smooth convex functions, we obtain the optimal complexity. In the non-smooth non-convex setting, we characterize the number of iterations needed to bound the expected norm of the smoothed gradient. For smooth functions, our analysis recovers existing results for structured zeroth-order methods for the convex case and extends them to the non-convex setting. We conclude with numerical simulations where assumptions are satisfied, observing that our algorithm has very good practical performances.
Optimization/Discrete and Combinatorial Optimization
Maximizing submodular functions has been increasingly used in many applications of machine learning, such as data summarization, recommendation systems, and feature selection. Moreover, there has been a growing interest in both submodular maximization and dynamic algorithms. In 2020, Monemizadeh and Lattanzi, Mitrovic, Norouzi-Fard, Tarnawski, and Zadimoghaddam initiated developing dynamic algorithms for the monotone submodular maximization problem under the cardinality constraint $k$. In 2022, Chen and Peng studied the complexity of this problem and raised an important open question: "\emph{Can we extend [fully dynamic] results (algorithm or hardness) to non-monotone submodular maximization?}". We affirmatively answer their question by demonstrating a reduction from maximizing a non-monotone submodular function under the cardinality constraint $k$ to maximizing a monotone submodular function under the same constraint. Through this reduction, we obtain the first dynamic algorithms to solve the non-monotone submodular maximization problem under the cardinality constraint $k$. Our algorithms maintain an $(8+\epsilon)$-approximate of the solution and use expected amortized $O(\epsilon^{-3}k^3\log^3(n)\log(k))$ or $O(\epsilon^{-1}k^2\log^3(k))$ oracle queries per update, respectively. Furthermore, we showcase the benefits of our dynamic algorithm for video summarization and max-cut problems on several real-world data sets.
Social Aspects/Accountability, Transparency and Interpretability
In real applications, interaction between machine learning models and domain experts is critical; however, the classical machine learning paradigm that usually produces only a single model does not facilitate such interaction. Approximating and exploring the Rashomon set, i.e., the set of all near-optimal models, addresses this practical challenge by providing the user with a searchable space containing a diverse set of models from which domain experts can choose. We present algorithms to efficiently and accurately approximate the Rashomon set of sparse, generalized additive models with ellipsoids for fixed support sets and use these ellipsoids to approximate Rashomon sets for many different support sets. The approximated Rashomon set serves as a cornerstone to solve practical challenges such as (1) studying the variable importance for the model class; (2) finding models under user-specified constraints (monotonicity, direct editing); and (3) investigating sudden changes in the shape functions. Experiments demonstrate the fidelity of the approximated Rashomon set and its effectiveness in solving practical challenges.
Reinforcement Learning/Planning
What are the functionals of the reward that can be computed and optimized exactly in Markov Decision Processes?In the finite-horizon, undiscounted setting, Dynamic Programming (DP) can only handle these operations efficiently for certain classes of statistics. We summarize the characterization of these classes for policy evaluation, and give a new answer for the planning problem. Interestingly, we prove that only generalized means can be optimized exactly, even in the more general framework of Distributional Reinforcement Learning (DistRL).DistRL permits, however, to evaluate other functionals approximately. We provide error bounds on the resulting estimators, and discuss the potential of this approach as well as its limitations.These results contribute to advancing the theory of Markov Decision Processes by examining overall characteristics of the return, and particularly risk-conscious strategies.
Reinforcement Learning/Deep RL
Deep reinforcement learning (RL) has achieved remarkable success in solving complex tasks through its integration with deep neural networks (DNNs) as function approximators. However, the reliance on DNNs has introduced a new challenge called primacy bias, whereby these function approximators tend to prioritize early experiences, leading to overfitting. To alleviate this bias, a reset method has been proposed, which involves periodic resets of a portion or the entirety of a deep RL agent while preserving the replay buffer. However, the use of this method can result in performance collapses after executing the reset, raising concerns from the perspective of safe RL and regret minimization. In this paper, we propose a novel reset-based method that leverages deep ensemble learning to address the limitations of the vanilla reset method and enhance sample efficiency. The effectiveness of the proposed method is validated through various experiments including those in the domain of safe RL. Numerical results demonstrate its potential for real-world applications requiring high sample efficiency and safety considerations.
Optimization/Bilevel optimization
We analyse a general class of bilevel problems, in which the upper-level problem consists in the minimization of a smooth objective function and the lower-level problem is to find the fixed point of a smooth contraction map. This type of problems include instances of meta-learning, equilibrium models, hyperparameter optimization and data poisoning adversarial attacks. Several recent works have proposed algorithms which warm-start the lower-level problem, i.e. they use the previous lower-level approximate solution as a staring point for the lower-level solver. This warm-start procedure allows one to improve the sample complexity in both the stochastic and deterministic settings, achieving in some cases the order-wise optimal sample complexity. However, there are situations, e.g., meta learning and equilibrium models, in which the warm-start procedure is not well-suited or ineffective. In this work we show that without warm-start, it is still possible to achieve order-wise (near) optimal sample complexity. In particular, we propose a simple method which uses (stochastic) fixed point iterations at the lower-level and projected inexact gradient descent at the upper-level, that reaches an $\epsilon$-stationary point using $O(\epsilon^{-2})$ and $\tilde{O}(\epsilon^{-1})$ samples for the stochastic and the deterministic setting, respectively. Finally, compared to methods using warm-start, our approach yields a simpler analysis that does not need to study the coupled interactions between the upper-level and lower-level iterates.
Deep Learning/Generative Models and Autoencoders
Manifold learning flows are a class of generative modelling techniques that assume a low-dimensional manifold description of the data. The embedding of such a manifold into the high-dimensional space of the data is achieved via learnable invertible transformations. Therefore, once the manifold is properly aligned via a reconstruction loss, the probability density is tractable on the manifold and maximum likelihood can be used to optimize the network parameters. Naturally, the lower-dimensional representation of the data requires an injective-mapping. Recent approaches were able to enforce that the density aligns with the modelled manifold, while efficiently calculating the density volume-change term when embedding to the higher-dimensional space. However, unless the injective-mapping is analytically predefined, the learned manifold is not necessarily an \emph{efficient representation} of the data. Namely, the latent dimensions of such models frequently learn an entangled intrinsic basis, with degenerate information being stored in each dimension. Alternatively, if a locally orthogonal and/or sparse basis is to be learned, here coined canonical intrinsic basis, it can serve in learning a more compact latent space representation. Toward this end, we propose a canonical manifold learning flow method, where a novel optimization objective enforces the transformation matrix to have few prominent and non-degenerate basis functions. We demonstrate that by minimizing the off-diagonal manifold metric elements $\ell_1$-norm, we can achieve such a basis, which is simultaneously sparse and/or orthogonal. Canonical manifold flow yields a more efficient use of the latent space, automatically generating fewer prominent and distinct dimensions to represent data, and consequently a better approximation of target distributions than other manifold flow methods in most experiments we conducted, resulting in lower FID scores.
Reinforcement Learning
Many reinforcement learning approaches rely on temporal-difference (TD) learning to learn a critic.However, TD-learning updates can be high variance.Here, we introduce a model-based RL framework, Taylor TD, which reduces this variance in continuous state-action settings. Taylor TD uses a first-order Taylor series expansion of TD updates.This expansion allows Taylor TD to analytically integrate over stochasticity in the action-choice, and some stochasticity in the state distribution for the initial state and action of each TD update.We include theoretical and empirical evidence that Taylor TD updates are indeed lower variance than standard TD updates. Additionally, we show Taylor TD has the same stable learning guarantees as standard TD-learning with linear function approximation under a reasonable assumption.Next, we combine Taylor TD with the TD3 algorithm, forming TaTD3.We show TaTD3 performs as well, if not better, than several state-of-the art model-free and model-based baseline algorithms on a set of standard benchmark tasks.
Optimization/Convex
We propose a family of recursive cutting-plane algorithms to solve feasibility problems with constrained memory, which can also be used for first-order convex optimization. Precisely, in order to find a point within a ball of radius $\epsilon$ with a separation oracle in dimension $d$---or to minimize $1$-Lipschitz convex functions to accuracy $\epsilon$ over the unit ball---our algorithms use $\mathcal O(\frac{d^2}{p}\ln \frac{1}{\epsilon})$ bits of memory, and make $\mathcal O((C\frac{d}{p}\ln \frac{1}{\epsilon})^p)$ oracle calls. The family is parametrized by $p\in[d]$ and provides an oracle-complexity/memory trade-off in the sub-polynomial regime $\ln\frac{1}{\epsilon}\gg\ln d$. While several works gave lower-bound trade-offs (impossibility results)---we explicit here their dependence with $\ln\frac{1}{\epsilon}$, showing that these also hold in any sub-polynomial regime---to the best of our knowledge this is the first class of algorithms that provides a positive trade-off between gradient descent and cutting-plane methods in any regime with $\epsilon\leq 1/\sqrt d$. The algorithms divide the $d$ variables into $p$ blocks and optimize over blocks sequentially, with approximate separation vectors constructed using a variant of Vaidya's method. In the regime $\epsilon \leq d^{-\Omega(d)}$, our algorithm with $p=d$ achieves the information-theoretic optimal memory usage and improves the oracle-complexity of gradient descent.
REPRODUCIBILITY SUMMARYScope of ReproducibilityIn this work, we study the reproducibility of the paper: Pre-Train Your Loss: Easy Bayesian Transfer Learning with Informative Priors. The paper proposes a three-step pipeline for replacing standard transfer learning with a pre-trained prior. The first step is training a prior, the second is re-scaling of a prior, and the third is inference.The authors claim that increasing the rank and the scaling factor improves performance on the downstream task. They also argue that using Bayesian learning with informative prior leads to a more data-efficient and improved performance compared to standard SGD transfer learning or using non-informative prior. We reproduce the main claims on one of the four data sets in the paper.MethodologyWe used a combination of the authors' and our code. The authors provided a training pipeline for the user but not the code to fully reproduce the paper. We modified the training pipeline to suit our needs and created a testing pipeline to evaluate the models. We reproduced the results for the Oxford-102-Flowers data set on an Nvidia RTX 3070 GPU using approximately 310 GPU hours for the main results.ResultsOur results confirm most of the claims tested, although we could not achieve the exact same accuracy due to missing hyper-parameters. We reproduced the trend in how scaling the prior impacts the performance and how a learned prior outperforms a non-learned prior.On contrary, we could not reproduce the effect of rank in low-rank covariance approximation on model performance, as well as the beneficial boost in performance of Bayesian learning compared to the standard SGD.What was easyThe authors' implementation provides various training and logging parameters. It is also helpful that the authors provided both the learned priors and scripts for the download, split and pre-processing of the data sets used in the study.What was difficultSetting the environment for the used packages to work correctly was difficult. Although many parameters are available for running the pipeline, their descriptions are misguiding, therefore a lot of time went into clarifying the parameter function and debugging different settings. The training also took a while, especially when training 5 models per data point. Communication with original authorsWe contacted the authors via e-mail about their pipeline and their use of hyper-parameters but did not hear back.
Optimization/Optimization and Learning under Uncertainty
Bayesian Optimization (BO) is a sample-efficient optimization algorithm widely employed across various applications. In some challenging BO tasks, input uncertainty arises due to the inevitable randomness in the optimization process, such as machining errors, execution noise, or contextual variability. This uncertainty deviates the input from the intended value before evaluation, resulting in significant performance fluctuations in the final result. In this paper, we introduce a novel robust Bayesian Optimization algorithm, AIRBO, which can effectively identify a robust optimum that performs consistently well under arbitrary input uncertainty. Our method directly models the uncertain inputs of arbitrary distributions by empowering the Gaussian Process with the Maximum Mean Discrepancy (MMD) and further accelerates the posterior inference via Nystrom approximation. Rigorous theoretical regret bound is established under MMD estimation error and extensive experiments on synthetic functions and real problems demonstrate that our approach can handle various input uncertainties and achieve a state-of-the-art performance.
Deep Learning/Algorithms
We introduce a probability distribution, combined with an efficient sampling algorithm, for weights and biases of fully-connected neural networks. In a supervised learning context, no iterative optimization or gradient computations of internal network parameters are needed to obtain a trained network. The sampling is based on the idea of random feature models. However, instead of a data-agnostic distribution, e.g., a normal distribution, we use both the input and the output training data to sample shallow and deep networks. We prove that sampled networks are universal approximators. For Barron functions, we show that the $L^2$-approximation error of sampled shallow networks decreases with the square root of the number of neurons. Our sampling scheme is invariant to rigid body transformations and scaling of the input data, which implies many popular pre-processing techniques are not required. In numerical experiments, we demonstrate that sampled networks achieve accuracy comparable to iteratively trained ones, but can be constructed orders of magnitude faster. Our test cases involve a classification benchmark from OpenML, sampling of neural operators to represent maps in function spaces, and transfer learning using well-known architectures.
Applications/Neuroscience, Cognitive Science
Inverse optimal control can be used to characterize behavior in sequential decision-making tasks. Most existing work, however, is limited to fully observable or linear systems, or requires the action signals to be known. Here, we introduce a probabilistic approach to inverse optimal control for partially observable stochastic non-linear systems with unobserved action signals, which unifies previous approaches to inverse optimal control with maximum causal entropy formulations. Using an explicit model of the noise characteristics of the sensory and motor systems of the agent in conjunction with local linearization techniques, we derive an approximate likelihood function for the model parameters, which can be computed within a single forward pass. We present quantitative evaluations on stochastic and partially observable versions of two classic control tasks and two human behavioral tasks. Importantly, we show that our method can disentangle perceptual factors and behavioral costs despite the fact that epistemic and pragmatic actions are intertwined in sequential decision-making under uncertainty, such as in active sensing and active learning. The proposed method has broad applicability, ranging from imitation learning to sensorimotor neuroscience.
Deep Learning/Generative Models and Autoencoders
We consider Generative Adversarial Networks (GANs) and address the underlying functional optimization problem ab initio within a variational setting. Strictly speaking, the optimization of the generator and discriminator functions must be carried out in accordance with the Euler-Lagrange conditions, which become particularly relevant in scenarios where the optimization cost involves regularizers comprising the derivatives of these functions. Considering Wasserstein GANs (WGANs) with a gradient-norm penalty, we show that the optimal discriminator is the solution to a Poisson differential equation. In principle, the optimal discriminator can be obtained in closed form without having to train a neural network. We illustrate this by employing a Fourier-series approximation to solve the Poisson differential equation. Experimental results based on synthesized Gaussian data demonstrate superior convergence behavior of the proposed approach in comparison with the baseline WGAN variants that employ weight-clipping, gradient or Lipschitz penalties on the discriminator on low-dimensional data. We also analyze the truncation error of the Fourier-series approximation and the estimation error of the Fourier coefficients in a high-dimensional setting. We demonstrate applications to real-world images considering latent-space prior matching in Wasserstein autoencoders and present performance comparisons on benchmark datasets such as MNIST, SVHN, CelebA, CIFAR-10, and Ukiyo-E. We demonstrate that the proposed approach achieves comparable reconstruction error and Frechet inception distance with faster convergence and up to two-fold improvement in image sharpness.