NeurIPS 2024 论文列表

This paper studies a risk minimization problem with decision dependent data distribution. The problem pertains to the performative prediction setting in which a trained model can affect the outcome estimated by the model. Such dependency creates a feedback loop that influences the stability of optimization algorithms such as stochastic gradient descent (SGD). We present the first study on performative prediction with smooth but possibly non-convex loss. We analyze a greedy deployment scheme with SGD (SGD-GD). Note that in the literature, SGD-GD is often studied with strongly convex loss. We first propose the definition of stationary performative stable (SPS) solutions through relaxing the popular performative stable condition. We then prove that SGD-GD converges to a biased SPS solution in expectation. We consider two conditions of sensitivity on the distribution shifts: (i) the sensitivity is characterized by Wasserstein-1 distance and the loss is Lipschitz w.r.t.~data samples, or (ii) the sensitivity is characterized by total variation (TV) divergence and the loss is bounded. In both conditions, the bias levels are proportional to the stochastic gradient's variance and sensitivity level. Our analysis is extended to a lazy deployment scheme where models are deployed once per several SGD updates, and we show that it converges to an SPS solution with reduced bias. Numerical experiments corroborate our theories.

Penalizing the nuclear norm of a function's Jacobian encourages it to locally behave like a low-rank linear map. Such functions vary locally along only a handful of directions, making the Jacobian nuclear norm a natural regularizer for machine learning problems. However, this regularizer is intractable for high-dimensional problems, as it requires computing a large Jacobian matrix and taking its SVD. We show how to efficiently penalize the Jacobian nuclear norm using techniques tailor-made for deep learning. We prove that for functions parametrized as compositions $f = g \circ h$, one may equivalently penalize the average squared Frobenius norm of $Jg$ and $Jh$. We then propose a denoising-style approximation that avoids the Jacobian computations altogether. Our method is simple, efficient, and accurate, enabling Jacobian nuclear norm regularization to scale to high-dimensional deep learning problems. We complement our theory with an empirical study of our regularizer's performance and investigate applications to denoising and representation learning.

Algorithms for bilevel optimization often encounter Hessian computations, which are prohibitive in high dimensions. While recent works offer first-order methods for unconstrained bilevel problems, the constrained setting remains relatively underexplored. We present first-order linearly constrained optimization methods with finite-time hypergradient stationarity guarantees. For linear equality constraints, we attain $\epsilon$-stationarity in $\widetilde{O}(\epsilon^{-2})$ gradient oracle calls, which is nearly-optimal. For linear inequality constraints, we attain $(\delta,\epsilon)$-Goldstein stationarity in $\widetilde{O}(d{\delta^{-1} \epsilon^{-3}})$ gradient oracle calls, where $d$ is the upper-level dimension. Finally, we obtain for the linear inequality setting dimension-free rates of $\widetilde{O}({\delta^{-1} \epsilon^{-4}})$ oracle complexity under the additional assumption of oracle access to the optimal dual variable. Along the way, we develop new nonsmooth nonconvex optimization methods with inexact oracles. Our numerical experiments verify these guarantees.

Bayesian optimization is a methodology to optimize black-box functions. Traditionally, it focuses on the setting where you can arbitrarily query the search space. However, many real-life problems do not offer this flexibility; in particular, the search space of the next query may depend on previous ones. Example challenges arise in the physical sciences in the form of local movement constraints, required monotonicity in certain variables, and transitions influencing the accuracy of measurements. Altogether, such *transition constraints* necessitate a form of planning. This work extends classical Bayesian optimization via the framework of Markov Decision Processes. We iteratively solve a tractable linearization of our utility function using reinforcement learning to obtain a policy that plans ahead for the entire horizon. This is a parallel to the optimization of an *acquisition function in policy space*. The resulting policy is potentially history-dependent and non-Markovian. We showcase applications in chemical reactor optimization, informative path planning, machine calibration, and other synthetic examples.

Algorithms with predictions is a recent framework for decision-making under uncertainty that leverages the power of machine-learned predictions without making any assumption about their quality. The goal in this framework is for algorithms to achieve an improved performance when the predictions are accurate while maintaining acceptable guarantees when the predictions are erroneous. A serious concern with algorithms that use predictions is that these predictions can be biased and, as a result, cause the algorithm to make decisions that are deemed unfair. We show that this concern manifests itself in the classical secretary problem in the learning-augmented setting---the state-of-the-art algorithm can have zero probability of accepting the best candidate, which we deem unfair, despite promising to accept a candidate whose expected value is at least $\max\{\Omega (1) , 1 - O(\varepsilon)\}$ times the optimal value, where $\varepsilon$ is the prediction error. We show how to preserve this promise while also guaranteeing to accept the best candidate with probability $\Omega(1)$. Our algorithm and analysis are based on a new ``pegging'' idea that diverges from existing works and simplifies/unifies some of their results. Finally, we extend to the $k$-secretary problem and complement our theoretical analysis with experiments.

The manifold hypothesis presumes that high-dimensional data lies on or near a low-dimensional manifold. While the utility of encoding geometric structure has been demonstrated empirically, rigorous analysis of its impact on the learnability of neural networks is largely missing. Several recent results have established hardness results for learning feedforward and equivariant neural networks under i.i.d. Gaussian or uniform Boolean data distributions. In this paper, we investigate the hardness of learning under the manifold hypothesis. We ask, which minimal assumptions on the curvature and regularity of the manifold, if any, render the learning problem efficiently learnable. We prove that learning is hard under input manifolds of bounded curvature by extending proofs of hardness in the SQ and cryptographic settings for boolean data inputs to the geometric setting. On the other hand, we show that additional assumptions on the volume of the data manifold alleviate these fundamental limitations and guarantee learnability via a simple interpolation argument. Notable instances of this regime are manifolds which can be reliably reconstructed via manifold learning. Looking forward, we comment on and empirically explore intermediate regimes of manifolds, which have heterogeneous features commonly found in real world data.

This paper introduces the notion of upper-linearizable/quadratizable functions, a class that extends concavity and DR-submodularity in various settings, including monotone and non-monotone cases over different types of convex sets. A general meta-algorithm is devised to convert algorithms for linear/quadratic maximization into ones that optimize upper-linearizable/quadratizable functions, offering a unified approach to tackling concave and DR-submodular optimization problems. The paper extends these results to multiple feedback settings, facilitating conversions between semi-bandit/first-order feedback and bandit/zeroth-order feedback, as well as between first/zeroth-order feedback and semi-bandit/bandit feedback. Leveraging this framework, new algorithms are derived using existing results as base algorithms for convex optimization, improving upon state-of-the-art results in various cases. Dynamic and adaptive regret guarantees are obtained for DR-submodular maximization, marking the first algorithms to achieve such guarantees in these settings. Notably, the paper achieves these advancements with fewer assumptions compared to existing state-of-the-art results, underscoring its broad applicability and theoretical contributions to non-convex optimization.

In this work we consider the problem of discretization of neural operators in a general setting. Using category theory, we give a no-go theorem that shows that diffeomorphisms between Hilbert spaces may not admit any continuous approximations by diffeomorphisms on finite spaces, even if the discretization is non-linear. This shows how infinite-dimensional Hilbert spaces and finite-dimensional vector spaces fundamentally differ. A key take-away is that to obtain discretization invariance, considerable effort is needed to ensure that finite-dimensional approximations of neural operator converge not only as sequences of functions, but that their representations converge in a suitable sense as well. With this perspective, we give several positive results. We first show that strongly monotone diffeomorphism operators always admit finite-dimensional strongly monotone diffeomorphisms. Next we show that bilipschitz neural operators may always be written via the repeated alternating composition of strongly monotone neural operators and invertible linear maps. We also show that such operators may be inverted locally via iteration provided that such inverse exists. Finally, we conclude by showing how our framework may be used `out of the box' to prove quantitative approximation results for discretization of neural operators.

Oja's algorithm for Streaming Principal Component Analysis (PCA) for $n$ data-points in a $d$ dimensional space achieves the same sin-squared error $O(r_{\mathsf{eff}}/n)$ as the offline algorithm in $O(d)$ space and $O(nd)$ time and a single pass through the datapoints. Here $r_{\mathsf{eff}}$ is the effective rank (ratio of the trace and the principal eigenvalue of the population covariance matrix $\Sigma$). Under this computational budget, we consider the problem of sparse PCA, where the principal eigenvector of $\Sigma$ is $s$-sparse, and $r_{\mathsf{eff}}$ can be large. In this setting, to our knowledge, *there are no known single-pass algorithms* that achieve the minimax error bound in $O(d)$ space and $O(nd)$ time without either requiring strong initialization conditions or assuming further structure (e.g., spiked) of the covariance matrix. We show that a simple single-pass procedure that thresholds the output of Oja's algorithm (the Oja vector) can achieve the minimax error bound under some regularity conditions in $O(d)$ space and $O(nd)$ time. We present a nontrivial and novel analysis of the entries of the unnormalized Oja vector, which involves the projection of a product of independent random matrices on a random initial vector. This is completely different from previous analyses of Oja's algorithm and matrix products, which have been done when the $r_{\mathsf{eff}}$ is bounded.

The typical training of neural networks using large stepsize gradient descent (GD) under the logistic loss often involves two distinct phases, where the empirical risk oscillates in the first phase but decreases monotonically in the second phase. We investigate this phenomenon in two-layer networks that satisfy a near-homogeneity condition. We show that the second phase begins once the empirical risk falls below a certain threshold, dependent on the stepsize. Additionally, we show that the normalized margin grows nearly monotonically in the second phase, demonstrating an implicit bias of GD in training non-homogeneous predictors. If the dataset is linearly separable and the derivative of the activation function is bounded away from zero, we show that the average empirical risk decreases, implying that the first phase must stop in finite steps. Finally, we demonstrate that by choosing a suitably large stepsize, GD that undergoes this phase transition is more efficient than GD that monotonically decreases the risk. Our analysis applies to networks of any width, beyond the well-known neural tangent kernel and mean-field regimes.

For constrained, not necessarily monotone submodular maximization, all known approximation algorithms with ratio greater than $1/e$ require continuous ideas, such as queries to the multilinear extension of a submodular function and its gradient, which are typically expensive to simulate with the original set function. For combinatorial algorithms, the best known approximation ratios for both size and matroid constraint are obtained by a simple randomized greedy algorithm of Buchbinder et al. [9]: $1/e \approx 0.367$ for size constraint and $0.281$ for the matroid constraint in $\mathcal O (kn)$ queries, where $k$ is the rank of the matroid. In this work, we develop the first combinatorial algorithms to break the $1/e$ barrier: we obtain approximation ratio of $0.385$ in $\mathcal O (kn)$ queries to the submodular set function for size constraint, and $0.305$ for a general matroid constraint. These are achieved by guiding the randomized greedy algorithm with a fast local search algorithm. Further, we develop deterministic versions of these algorithms, maintaining the same ratio and asymptotic time complexity. Finally, we develop a deterministic, nearly linear time algorithm with ratio $0.377$.

Multiple Instance Learning (MIL) has been increasingly adopted to mitigate the high costs and complexity associated with labeling individual instances, learning instead from bags of instances labeled at the bag level and enabling instance-level labeling. While existing research has primarily focused on the learnability of MIL at the bag level, there is an absence of theoretical exploration to check if a given MIL algorithm is learnable at the instance level. This paper proposes a theoretical framework based on probably approximately correct (PAC) learning theory to assess the instance-level learnability of deep multiple instance learning (Deep MIL) algorithms. Our analysis exposes significant gaps between current Deep MIL algorithms, highlighting the theoretical conditions that must be satisfied by MIL algorithms to ensure instance-level learnability. With these conditions, we interpret the learnability of the representative Deep MIL algorithms and validate them through empirical studies.

Despite their simple intuition, convolutions are more tedious to analyze than dense layers, which complicates the transfer of theoretical and algorithmic ideas to convolutions. We simplify convolutions by viewing them as tensor networks (TNs) that allow reasoning about the underlying tensor multiplications by drawing diagrams, manipulating them to perform function transformations like differentiation, and efficiently evaluating them with `einsum`. To demonstrate their simplicity and expressiveness, we derive diagrams of various autodiff operations and popular curvature approximations with full hyper-parameter support, batching, channel groups, and generalization to any convolution dimension. Further, we provide convolution-specific transformations based on the connectivity pattern which allow to simplify diagrams before evaluation. Finally, we probe performance. Our TN implementation accelerates a recently-proposed KFAC variant up to 4.5 x while removing the standard implementation's memory overhead, and enables new hardware-efficient tensor dropout for approximate backpropagation.

In federated learning (FL), accommodating clients' varied computational capacities poses a challenge, often limiting the participation of those with constrained resources in global model training. To address this issue, the concept of model heterogeneity through submodel extraction has emerged, offering a tailored solution that aligns the model's complexity with each client's computational capacity. In this work, we propose Federated Importance-Aware Submodel Extraction (FIARSE), a novel approach that dynamically adjusts submodels based on the importance of model parameters, thereby overcoming the limitations of previous static and dynamic submodel extraction methods. Compared to existing works, the proposed method offers a theoretical foundation for the submodel extraction and eliminates the need for additional information beyond the model parameters themselves to determine parameter importance, significantly reducing the overhead on clients. Extensive experiments are conducted on various datasets to showcase the superior performance of the proposed FIARSE.

Offline optimization has recently emerged as an increasingly popular approach to mitigate the prohibitively expensive cost of online experimentation. The key idea is to learn a surrogate of the black-box function that underlines the target experiment using a static (offline) dataset of its previous input-output queries. Such an approach is, however, fraught with an out-of-distribution issue where the learned surrogate becomes inaccurate outside the offline data regimes. To mitigate this, existing offline optimizers have proposed numerous conditioning techniques to prevent the learned surrogate from being too erratic. Nonetheless, such conditioning strategies are often specific to particular surrogate or search models, which might not generalize to a different model choice. This motivates us to develop a model-agnostic approach instead, which incorporates a notion of model sharpness into the training loss of the surrogate as a regularizer. Our approach is supported by a new theoretical analysis demonstrating that reducing surrogate sharpness on the offline dataset provably reduces its generalized sharpness on unseen data. Our analysis extends existing theories from bounding generalized prediction loss (on unseen data) with loss sharpness to bounding the worst-case generalized surrogate sharpness with its empirical estimate on training data, providing a new perspective on sharpness regularization. Our extensive experimentation on a diverse range of optimization tasks also shows that reducing surrogate sharpness often leads to significant improvement, marking (up to) a noticeable 9.6% performance boost. Our code is publicly available at https://github.com/cuong-dm/IGNITE.

Given a source and a target probability measure, the Monge problem studies efficient ways to map the former onto the latter. This efficiency is quantified by defining a *cost* function between source and target data. Such a cost is often set by default in the machine learning literature to the squared-Euclidean distance, $\ell^2\_2(\mathbf{x},\mathbf{y}):=\tfrac12\|\mathbf{x}-\mathbf{y}\|\_2^2$. The benefits of using *elastic* costs, defined using a regularizer $\tau$ as $c(\mathbf{x},\mathbf{y}):=\ell^2_2(\mathbf{x},\mathbf{y})+\tau(\mathbf{x}-\mathbf{y})$, was recently highlighted in (Cuturi et al. 2023). Such costs shape the *displacements* of Monge maps $T$, namely the difference between a source point and its image $T(\mathbf{x})-\mathbf{x}$, by giving them a structure that matches that of the proximal operator of $\tau$. In this work, we make two important contributions to the study of elastic costs:*(i)* For any elastic cost, we propose a numerical method to compute Monge maps that are provably optimal. This provides a much-needed routine to create synthetic problems where the ground-truth OT map is known, by analogy to the Brenier theorem, which states that the gradient of any convex potential is always a valid Monge map for the $\ell_2^2$ cost; *(ii)* We propose a loss to *learn* the parameter $\theta$ of a parameterized regularizer $\tau_\theta$, and apply it in the case where $\tau_{A}({\bf z}):=\|A^\perp {\bf z}\|^2_2$. This regularizer promotes displacements that lie on a low-dimensional subspace of $\mathbb{R}^d$, spanned by the $p$ rows of $A\in\mathbb{R}^{p\times d}$. We illustrate the soundness of our procedure on synthetic data, generated using our first contribution, in which we show near-perfect recovery of $A$'s subspace using only samples. We demonstrate the applicability of this method by showing predictive improvements on single-cell data tasks.

We consider the solvable neural scaling model with three parameters: data complexity, target complexity, and model-parameter-count. We use this neural scaling model to derive new predictions about the compute-limited, infinite-data scaling law regime. To train the neural scaling model, we run one-pass stochastic gradient descent on a mean-squared loss. We derive a representation of the loss curves which holds over all iteration counts and improves in accuracy as the model parameter count grows. We then analyze the compute-optimal model-parameter-count, and identify 4 phases (+3 subphases) in the data-complexity/target-complexity phase-plane. The phase boundaries are determined by the relative importance of model capacity, optimizer noise, and embedding of the features. We furthermore derive, with mathematical proof and extensive numerical evidence, the scaling-law exponents in all of these phases, in particular computing the optimal model-parameter-count as a function of floating point operation budget. We include a colab notebook https://tinyurl.com/2saj6bkj, nanoChinchilla, that reproduces some key results of the paper.

High-dimensional linear bandits with low-dimensional structure have received considerable attention in recent studies due to their practical significance. The most common structure in the literature is sparsity. However, it may not be available in practice. Symmetry, where the reward is invariant under certain groups of transformations on the set of arms, is another important inductive bias in the high-dimensional case that covers many standard structures, including sparsity. In this work, we study high-dimensional symmetric linear bandits where the symmetry is hidden from the learner, and the correct symmetry needs to be learned in an online setting. We examine the structure of a collection of hidden symmetry and provide a method based on model selection within the collection of low-dimensional subspaces. Our algorithm achieves a regret bound of $ O(d_0^{2/3} T^{2/3} \log(d))$, where $d$ is the ambient dimension which is potentially very large, and $d_0$ is the dimension of the true low-dimensional subspace such that $d_0 \ll d$. With an extra assumption on well-separated models, we can further improve the regret to $ O(d_0 \sqrt{T\log(d)} )$.

In this paper, we focus on simple bilevel optimization problems, where we minimize a convex smooth objective function over the optimal solution set of another convex smooth constrained optimization problem. We present a novel bilevel optimization method that locally approximates the solution set of the lower-level problem using a cutting plane approach and employs an accelerated gradient-based update to reduce the upper-level objective function over the approximated solution set. We measure the performance of our method in terms of suboptimality and infeasibility errors and provide non-asymptotic convergence guarantees for both error criteria. Specifically, when the feasible set is compact, we show that our method requires at most $\mathcal{O}(\max\\{1/\sqrt{\epsilon_{f}}, 1/\epsilon_g\\})$ iterations to find a solution that is $\epsilon_f$-suboptimal and $\epsilon_g$-infeasible. Moreover, under the additional assumption that the lower-level objective satisfies the $r$-th Hölderian error bound, we show that our method achieves an iteration complexity of $\mathcal{O}(\max\\{\epsilon_{f}^{-\frac{2r-1}{2r}},\epsilon_{g}^{-\frac{2r-1}{2r}}\\})$, which matches the optimal complexity of single-level convex constrained optimization when $r=1$.