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Reinforcement Learning · Batch/Offline

Hyungkyu Kang, Min-hwan Oh

In this paper, we study offline preference-based reinforcement learning (PbRL), where learning is based on pre-collected preference feedback over pairs of trajectories. While offline PbRL has demonstrated remarkable empirical success, existing theoretical approaches face challenges in ensuring conservatism under uncertainty, requiring computationally intractable confidence set constructions. We address this limitation by proposing Adversarial Preference-based Policy Optimization (APPO), a computationally efficient algorithm for offline PbRL that guarantees sample complexity bounds without relying on explicit confidence sets. By framing PbRL as a two-player game between a policy and a model, our approach enforces conservatism in a tractable manner. Using standard assumptions on function approximation and bounded trajectory concentrability, we derive a sample complexity bound. To our knowledge, APPO is the first offline PbRL algorithm to offer both statistical efficiency and practical applicability. Experimental results on continuous control tasks demonstrate that APPO effectively learns from complex datasets, showing comparable performance with existing state-of-the-art methods.

Jiachen (Tianhao) Wang, Dawn Song, James Y Zou, Prateek Mittal, Ruoxi Jia

Traditional data influence estimation methods, like influence function, assume that learning algorithms are permutation-invariant with respect to training data. However, modern training paradigms—especially for foundation models using stochastic algorithms and non-convergent, multi-stage curricula—are sensitive to data ordering, thus violating this assumption. This mismatch renders influence functions inadequate for answering some critical questions in current machine learning: How can we differentiate the influence of the same data contributing at different stages of training? More generally, how can we capture the dependence of data influence on the optimization trajectory during training? To address this gap, we formalize the concept of \emph{trajectory-specific leave-one-out (LOO) influence}, which quantifies the impact of removing a data point from a specific iteration during training, accounting for the exact sequence of data encountered and the model's optimization trajectory. However, exactly evaluating the trajectory-specific LOO presents a significant computational challenge. To address this, we propose \emph{data value embedding}, a novel technique enabling efficient approximation of trajectory-specific LOO. Specifically, we compute a training data embedding that encapsulates the cumulative interactions between data and the evolving model parameters. The LOO can then be efficiently approximated through a simple dot-product between the data value embedding and the gradient of the given test data. As data value embedding captures training data ordering, it offers valuable insights into model training dynamics. In particular, we uncover distinct phases of data influence, revealing that data points in the early and late stages of training exert a greater impact on the final model. These insights translate into actionable strategies for managing the computational overhead of data selection by strategically timing the selection process, potentially opening new avenues in data curation research.

Reinforcement Learning · Function Approximation

Runzhe Wu, Ayush Sekhari, Akshay Krishnamurthy, Wen Sun

We study computationally and statistically efficient Reinforcement Learning algorithms for the *linear Bellman Complete* setting. This setting uses linear function approximation to capture value functions and unifies existing models like linear Markov Decision Processes (MDP) and Linear Quadratic Regulators (LQR). While it is known from the prior works that this setting is statistically tractable, it remained open whether a computationally efficient algorithm exists. Our work provides a computationally efficient algorithm for the linear Bellman complete setting that works for MDPs with large action spaces, random initial states, and random rewards but relies on the underlying dynamics to be deterministic. Our approach is based on randomization: we inject random noise into least squares regression problems to perform optimistic value iteration. Our key technical contribution is to carefully design the noise to only act in the null space of the training data to ensure optimism while circumventing a subtle error amplification issue.

Theory · Reinforcement Learning and Planning

Ally Du, Lin Yang, Ruosong Wang

The recent work by Dong and Yang (2023) showed for misspecified sparse linear bandits, one can obtain an $O(\epsilon)$-optimal policy using a polynomial number of samples when the sparsity is a constant, where $\epsilon$ is the misspecification error. This result is in sharp contrast to misspecified linear bandits without sparsity, which require an exponential number of samples to get the same guarantee. In order to study whether the analog result is possible in the reinforcement learning setting, we consider the following problem: assuming the optimal $Q$-function is a $d$-dimensional linear function with sparsity $k$ and misspecification error $\epsilon$, whether we can obtain an $O(\epsilon)$-optimal policy using number of samples polynomially in the feature dimension $d$. We first demonstrate why the standard approach based on Bellman backup or the existing optimistic value function elimination approach such as OLIVE (Jiang et al., 2017) achieves suboptimal guarantees for this problem. We then design a novel elimination-based algorithm to show one can obtain an $O(H\epsilon)$-optimal policy with sample complexity polynomially in the feature dimension $d$ and planning horizon $H$. Lastly, we complement our upper bound with an $\tilde \Omega(H\epsilon)$ suboptimality lower bound, giving a complete picture of this problem.

Jiachen (Tianhao) Wang, Dawn Song, James Y Zou, Prateek Mittal, Ruoxi Jia

Traditional data influence estimation methods, like influence function, assume that learning algorithms are permutation-invariant with respect to training data. However, modern training paradigms—especially for foundation models using stochastic algorithms and non-convergent, multi-stage curricula—are sensitive to data ordering, thus violating this assumption. This mismatch renders influence functions inadequate for answering some critical questions in current machine learning: How can we differentiate the influence of the same data contributing at different stages of training? More generally, how can we capture the dependence of data influence on the optimization trajectory during training? To address this gap, we formalize the concept of \emph{trajectory-specific leave-one-out (LOO) influence}, which quantifies the impact of removing a data point from a specific iteration during training, accounting for the exact sequence of data encountered and the model's optimization trajectory. However, exactly evaluating the trajectory-specific LOO presents a significant computational challenge. To address this, we propose \emph{data value embedding}, a novel technique enabling efficient approximation of trajectory-specific LOO. Specifically, we compute a training data embedding that encapsulates the cumulative interactions between data and the evolving model parameters. The LOO can then be efficiently approximated through a simple dot-product between the data value embedding and the gradient of the given test data. As data value embedding captures training data ordering, it offers valuable insights into model training dynamics. In particular, we uncover distinct phases of data influence, revealing that data points in the early and late stages of training exert a greater impact on the final model. These insights translate into actionable strategies for managing the computational overhead of data selection by strategically timing the selection process, potentially opening new avenues in data curation research.

Theory · Optimization

Cedar Site Bai, Brian Bullins

In this paper, we provide tight lower bounds for the oracle complexity of minimizing high-order Hölder smooth and uniformly convex functions. Specifically, for a function whose $p^{th}$-order derivatives are Hölder continuous with degree $\nu$ and parameter $H$, and that is uniformly convex with degree $q$ and parameter $\sigma$, we focus on two asymmetric cases: (1) $q > p + \nu$, and (2) $q < p+\nu$. Given up to $p^{th}$-order oracle access, we establish worst-case oracle complexities of $\Omega\left( \left( \frac{H}{\sigma}\right)^\frac{2}{3(p+\nu)-2}\left( \frac{\sigma}{\epsilon}\right)^\frac{2(q-p-\nu)}{q(3(p+\nu)-2)}\right)$ in the first case with an $\ell_\infty$-ball-truncated-Gaussian smoothed hard function and $\Omega\left(\left(\frac{H}{\sigma}\right)^\frac{2}{3(p+\nu)-2}+ \log\log\left(\left(\frac{\sigma^{p+\nu}}{H^q}\right)^\frac{1}{p+\nu-q}\frac{1}{\epsilon}\right)\right)$ in the second case, for reaching an $\epsilon$-approximate solution in terms of the optimality gap. Our analysis generalizes previous lower bounds for functions under first- and second-order smoothness as well as those for uniformly convex functions, and furthermore our results match the corresponding upper bounds in this general setting.

Deep Learning · Sequential Models, Time series

Woojin Cho, Minju Jo, Kookjin Lee, Noseong Park

As function approximators, deep neural networks have served as an effective tool to represent various signal types. Recent approaches utilize multi-layer perceptrons (MLPs) to learn a nonlinear mapping from a coordinate to its corresponding signal, facilitating the learning of continuous neural representations from discrete data points. Despite notable successes in learning diverse signal types, coordinate-based MLPs often face issues of overfitting and limited generalizability beyond the training region, resulting in subpar extrapolation performance. This study addresses scenarios where the underlying true signals exhibit periodic properties, either spatially or temporally. We propose a novel network architecture, which extracts periodic patterns from measurements and leverages this information to represent the signal, thereby enhancing generalization and improving extrapolation performance. We demonstrate the efficacy of the proposed method through comprehensive experiments, including the learning of the periodic solutions for differential equations, and time series imputation (interpolation) and forecasting (extrapolation) on real-world datasets.

Deep Learning · Generative Models and Autoencoders

Shivam Gupta, Linda Cai, Sitan Chen

Sampling algorithms play an important role in controlling the quality and runtime of diffusion model inference. In recent years, a number of works (Chen et al., 2023c;b; Benton et al., 2023; Lee et al., 2022) have analyzed algorithms for diffusion sampling with provable guarantees; these works show that for essentially any data distribution, one can approximately sample in polynomial time given a sufficiently accurate estimate of its score functions at different noise levels. In this work, we propose a new scheme inspired by Shen and Lee's randomized midpoint method for log-concave sampling (Shen & Lee, 2019). We prove that this approach achieves the best known dimension dependence for sampling from arbitrary smooth distributions in total variation distance ($\widetilde O(d^{5/12})$ compared to $\widetilde O(\sqrt{d})$ from prior work). We also show that our algorithm can be parallelized to run in only $\widetilde O(\log^2 d)$ parallel rounds, constituting the first provable guarantees for parallel sampling with diffusion models. As a byproduct of our methods, for the well-studied problem of log-concave sampling in total variation distance, we give an algorithm and simple analysis achieving dimension dependence $\widetilde O(d^{5/12})$ compared to $\widetilde O(\sqrt{d})$ from prior work.

Miscellaneous Aspects of Machine Learning · Online Learning, Active Learning and Bandits

Aldo Pacchiano

Many works have developed no-regret algorithms for contextual bandits with function approximation, where the mean rewards over context-action pairs belong to a function class $\mathcal{F}$. Although there are many approaches to this problem, algorithms based on the principle of optimism, such as optimistic least squares have gained in importance. It can be shown the regret of this algorithm scales as $\widetilde{\mathcal{O}}\left(\sqrt{d_{\mathrm{eluder}}(\mathcal{F}) \log(\mathcal{F}) T }\right)$ where $d_{\mathrm{eluder}}(\mathcal{F})$ is a statistical measure of the complexity of the function class $\mathcal{F}$ known as eluder dimension. Unfortunately, even if the variance of the measurement noise of the rewards at time $t$ equals $\sigma_t^2$ and these are close to zero, the optimistic least squares algorithm’s regret scales with $\sqrt{T}$. In this work we are the first to develop algorithms that satisfy regret bounds for contextual bandits with function approximation of the form $\widetilde{\mathcal{O}}\left( \sigma \sqrt{\log(\mathcal{F})d_{\mathrm{eluder}}(\mathcal{F}) T } + d_{\mathrm{eluder}}(\mathcal{F}) \cdot \log(|\mathcal{F}|)\right) $ when the variances are unknown and satisfy $\sigma_t^2 = \sigma$ for all $t$ and $\widetilde{\mathcal{O}}\left( d_{\mathrm{eluder}}(\mathcal{F})\sqrt{\log(\mathcal{F})\sum_{t=1}^T \sigma_t^2 } + d_{\mathrm{eluder}}(\mathcal{F}) \cdot \log(|\mathcal{F}|)\right) $ when the variances change every time-step. These bounds generalize existing techniques for deriving second order bounds in contextual linear problems.

Applications · Everything Else

Sujay Bhatt, Alec Koppel, Sumitra Ganesh, Sihan Zeng

Mechanism design in resource allocation studies dividing limited resources among self-interested agents whose satisfaction with the allocation depends on privately held utilities. We consider the problem in a payment-free setting, with the aim of maximizing social welfare while enforcing incentive compatibility (IC), i.e., agents cannot inflate allocations by misreporting their utilities. The well-known proportional fairness (PF) mechanism achieves the maximum possible social welfare but incurs an undesirably high exploitability (the maximum unilateral inflation in utility from misreport and a measure of deviation from IC). In fact, it is known that no mechanism can achieve the maximum social welfare and exact incentive compatibility (IC) simultaneously without the use of monetary incentives (Cole et al., 2013). Motivated by this fact, we propose learning an approximate mechanism that desirably trades off the competing objectives. Our main contribution is to design an innovative neural network architecture tailored to the resource allocation problem, which we name Regularized Proportional Fairness Network (RPF-Net). RPF-Net regularizes the output of the PF mechanism by a learned function approximator of the most exploitable allocation, with the aim of reducing the incentive for any agent to misreport. We derive generalization bounds that guarantee the mechanism performance when trained under finite and out-of-distribution samples and experimentally demonstrate the merits of the proposed mechanism compared to the state-of-the-art. The PF mechanism acts as an important benchmark for comparing the social welfare of any mechanism. However, there exists no established way of computing its exploitability. The challenge here is that we need to find the maximizer of an optimization problem for which the gradient is only implicitly defined. We for the first time provide a systematic method for finding such (sub)gradients, which enables the evaluation of the exploitability of the PF mechanism through iterative (sub)gradient ascent.

Miscellaneous Aspects of Machine Learning · General Machine Learning Techniques

Yixuan Wang, Jonathan Siegel, Ziming Liu, Thomas Hou

Kolmogorov-Arnold Networks (KAN) \cite{liu2024kan} were very recently proposed as a potential alternative to the prevalent architectural backbone of many deep learning models, the multi-layer perceptron (MLP). KANs have seen success in various tasks of AI for science, with their empirical efficiency and accuracy demonstrated in function regression, PDE solving, and many more scientific problems. In this article, we revisit the comparison of KANs and MLPs, with emphasis on a theoretical perspective. On the one hand, we compare the representation and approximation capabilities of KANs and MLPs. We establish that MLPs can be represented using KANs of a comparable size. This shows that the approximation and representation capabilities of KANs are at least as good as MLPs. Conversely, we show that KANs can be represented using MLPs, but that in this representation the number of parameters increases by a factor of the KAN grid size. This suggests that KANs with a large grid size may be more efficient than MLPs at approximating certain functions. On the other hand, from the perspective of learning and optimization, we study the spectral bias of KANs compared with MLPs. We demonstrate that KANs are less biased toward low frequencies than MLPs. We highlight that the multi-level learning feature specific to KANs, i.e. grid extension of splines, improves the learning process for high-frequency components. Detailed comparisons with different choices of depth, width, and grid sizes of KANs are made, shedding some light on how to choose the hyperparameters in practice.

Michael Sander, Gabriel Peyré

Causal Transformers are trained to predict the next token for a given context. While it is widely accepted that self-attention is crucial for encoding the causal structure of sequences, the precise underlying mechanism behind this in-context autoregressive learning ability remains unclear. In this paper, we take a step towards understanding this phenomenon by studying the approximation ability of Transformers for next-token prediction. Specifically, we explore the capacity of causal Transformers to predict the next token $x_{t+1}$ given an autoregressive sequence $(x_1, \dots, x_t)$ as a prompt, where $ x_{t+1} = f(x_t) $, and $ f $ is a context-dependent function that varies with each sequence.On the theoretical side, we focus on specific instances, namely when $ f $ is linear or when $ (x_t)$ is periodic. We explicitly construct a Transformer (with linear, exponential, or softmax attention) that learns the mapping $f$ in-context through a causal kernel descent method. The causal kernel descent method we propose provably estimates $x_{t+1} $ based solely on past and current observations $ (x_1, \dots, x_t) $, with connections to the Kaczmarz algorithm in Hilbert spaces. We present experimental results that validate our theoretical findings and suggest their applicability to more general mappings $f$.

Optimization · Large Scale, Parallel and Distributed

Matteo Pagliardini, Pierre Ablin, David Grangier

Momentum based optimizers are central to a wide range of machine learning applications. These typically rely on an Exponential Moving Average (EMA) of gradients, which decays exponentially the present contribution of older gradients. This accounts for gradients being local linear approximations which lose their relevance as the iterate moves along the loss landscape. This work questions the use of a single EMA to accumulate past gradients and empirically demonstrates how this choice can be sub-optimal: a single EMA cannot simultaneously give a high weight to the immediate past, and a non-negligible weight to older gradients. Building on this observation, we propose AdEMAMix, a simple modification of the Adam optimizer with a mixture of two EMAs to better take advantage of past gradients. Our experiments on language modeling and image classification show---quite surprisingly---that gradients can stay relevant for tens of thousands of steps. They help to converge faster, and often to lower minima: e.g., a $1.3$B parameter AdEMAMix LLM trained on $101$B tokens performs comparably to an AdamW model trained on $197$B tokens ($+95\%$). Moreover, our method significantly slows-down model forgetting during training. Our work motivates further exploration of different types of functions to leverage past gradients, beyond EMAs.

Nicole Feng, Keenan Crane

We introduce a method for approximating the signed distance function (SDF) of geometry corrupted by holes, noise, or self-intersections. The method implicitly defines a completed version of the shape, rather than explicitly repairing the given input. Our starting point is a modified version of the heat method for geodesic distance, which diffuses normal vectors rather than a scalar distribution. This formulation provides robustness akin to generalized winding numbers (GWN), but provides distance function rather than just an inside/outside classification. Our formulation also offers several features not common to classic distance algorithms, such as the ability to simultaneously fit multiple level sets, a notion of distance for geometry that does not topologically bound any region, and the ability to mix and match signed and unsigned distance. The method can be applied in any dimension and to any spatial discretization, including triangle meshes, tet meshes, point clouds, polygonal meshes, voxelized surfaces, and regular grids. We evaluate the method on several challenging examples, implementing normal offsets and other morphological operations directly on imperfect curve and surface data. In many cases we also obtain an inside/outside classification dramatically more robust than the one obtained provided by GWN.

Masaaki Miki, Toby Mitchell

In architecture, shapes of surfaces that can withstand gravity with no bending action are considered ideal for shell structures. Those shells have special geometries through which they can stream gravitational force toward the ground via stresses strictly tangent to the surface, making them highly efficient. The process of finding these special forms is called form-finding. Recently, [Miki and Mitchell 2022] presented a method to reliably produce mixed tension-compression continuum shells, a type of shells known to be especially difficult to form-find. The key to this method was to use the concept of the Airy stress function to derive a valid bending-free shell shape by iterating on both the shell shape and the Airy stress function; this turns a problem that is over-constrained in general into a problem with many solutions. In [Miki and Mitchell 2022], it was proposed that the method could also be used to design grid shells by tracing curves on a continuum shell such that the resulting grid has bars that are both bending-free and form flat panels, a property useful for construction of real grid shells made of glass and steel. However, this special type of grid is not guaranteed to exist in general on a mixed-tension compression shell, even when the shell is in bending-free equilibrium [Miki and Mitchell 2023]. Additional conditions must be imposed on the shell shape to guarantee the existence of simultaneously bending-free and conjugate grid directions. The current study resolves the existence issue by adding alignment conditions. We consider several practical curve alignment conditions: alignment with the lines of curvature of the shell, approximate alignment with a bidirectional set of user-prescribed guide curves, and exact alignment with a single direction of user-prescribed guide curves. We report that the variable projection method originally used to solve the form-finding problem in the work of [Miki and Mitchell 2022] can be successfully extended to solve the newly introduced alignment conditions, and conclude with results for several practical design examples. To our knowledge, this is the first method that can take a user-input grid and find a "nearby" grid that is both flat-panelled and in bending-free equilibrium for the general case of mixed tension-compression grid shells.

Qiujie Dong, Rui Xu 0016, Pengfei Wang, Shuang-Min Chen, Shiqing Xin, Xiaohong Jia 0001, Wenping Wang 0001, Changhe Tu

Despite recent advances in reconstructing an organic model with the neural signed distance function (SDF), the high-fidelity reconstruction of a CAD model directly from low-quality unoriented point clouds remains a significant challenge. In this paper, we address this challenge based on the prior observation that the surface of a CAD model is generally composed of piecewise surface patches, each approximately developable even around the feature line. Our approach, named NeurCADRecon, is self-supervised, and its loss includes a developability term to encourage the Gaussian curvature toward 0 while ensuring fidelity to the input points (see the teaser figure). Noticing that the Gaussian curvature is non-zero at tip points, we introduce a double-trough curve to tolerate the existence of these tip points. Furthermore, we develop a dynamic sampling strategy to deal with situations where the given points are incomplete or too sparse. Since our resulting neural SDFs can clearly manifest sharp feature points/lines, one can easily extract the feature-aligned triangle mesh from the SDF and then decompose it into smooth surface patches, greatly reducing the difficulty of recovering the parametric CAD design. A comprehensive comparison with existing state-of-the-art methods shows the significant advantage of our approach in reconstructing faithful CAD shapes.

Michael Fischer 0011, Tobias Ritschel 0001

Gradient-based optimization is now ubiquitous across graphics, but unfortunately can not be applied to problems with undefined or zero gradients. To circumvent this issue, the loss function can be manually replaced by a "surrogate" that has similar minima but is differentiable. Our proposed framework, ZeroGrads, automates this process by learning a neural approximation of the objective function, which in turn can be used to differentiate through arbitrary black-box graphics pipelines. We train the surrogate on an actively smoothed version of the objective and encourage locality, focusing the surrogate's capacity on what matters at the current training episode. The fitting is performed online, alongside the parameter optimization, and self-supervised, without pre-computed data or pre-trained models. As sampling the objective is expensive (it requires a full rendering or simulator run), we devise an efficient sampling scheme that allows for tractable run-times and competitive performance at little overhead. We demonstrate optimizing diverse non-convex, non-differentiable black-box problems in graphics, such as visibility in rendering, discrete parameter spaces in procedural modelling or optimal control in physics-driven animation. In contrast to other derivative-free algorithms, our approach scales well to higher dimensions, which we demonstrate on problems with up to 35k interlinked variables.

Ladislav Kavan, John Doublestein, Martin Prazák, Matthew Cioffi, Doug Roble

We present a new method to bake classical facial animation blendshapes into a fast linear blend skinning representation. Previous work explored skinning decomposition methods that approximate general animated meshes using a dense set of bone transformations; these optimizers typically alternate between optimizing for the bone transformations and the skinning weights. We depart from this alternating scheme and propose a new approach based on proximal algorithms, which effectively means adding a projection step to the popular Adam optimizer. This approach is very flexible and allows us to quickly experiment with various additional constraints and/or loss functions. Specifically, we depart from the classical skinning paradigms and restrict the transformation coefficients to contain only about 90% non-zeros, while achieving similar accuracy and visual quality as the state-of-the-art. The sparse storage enables our method to deliver significant savings in terms of both memory and run-time speed. We include a compact implementation of our new skinning decomposition method in PyTorch, which is easy to experiment with and modify to related problems.

Zilu Li, Guandao Yang, Qingqing Zhao, Xi Deng, Leonidas J. Guibas, Bharath Hariharan, Gordon Wetzstein

This paper presents a method to leverage arbitrary neural network architecture for control variates. Control variates are crucial in reducing the variance of Monte Carlo integration, but they hinge on finding a function that both correlates with the integrand and has a known analytical integral. Traditional approaches rely on heuristics to choose this function, which might not be expressive enough to correlate well with the integrand. Recent research alleviates this issue by modeling the integrands with a learnable parametric model, such as a neural network. However, the challenge remains in creating an expressive parametric model with a known analytical integral. This paper proposes a novel approach to construct learnable parametric control variates functions from arbitrary neural network architectures. Instead of using a network to approximate the integrand directly, we employ the network to approximate the anti-derivative of the integrand. This allows us to use automatic differentiation to create a function whose integration can be constructed by the antiderivative network. We apply our method to solve partial differential equations using the Walk-on-sphere algorithm [Sawhney and Crane 2020]. Our results indicate that this approach is unbiased using various network architectures and achieves lower variance than other control variate methods.

Jirí Minarcík, Sam Estep, Wode Ni, Keenan Crane

This paper describes an optimization-based framework for finding arrangements of 2D shapes subject to pairwise constraints. Such arrangements naturally arise in tasks such as vector illustration and diagram generation, but enforcing these criteria robustly is surprisingly challenging. We approach this problem through the minimization of novel energetic penalties, derived from the signed distance function of the Minkowski difference between interacting shapes. This formulation provides useful gradients even when initialized from a wildly infeasible state, and, unlike many common collision penalties, can handle open curves that do not have a well-defined inside and outside. Moreover, it supports rich features beyond the basic no-overlap condition, such as tangency, containment, and precise padding, which are especially valuable in the vector illustration context. We develop closed-form expressions and efficient approximations of our penalty for standard vector graphics primitives, yielding efficient evaluation and easy implementation within existing automatic differentiation pipelines. The method has already been “battle-tested” as a component of public-facing open source software; we demonstrate the utility of the framework via examples from illustration, data visualization, diagram generation, and geometry processing.