ICLR 2025 论文列表

The training of large models is memory-constrained, one direction to relieve this is training using local loss, like GIM, LoCo, and Forward-Forward algorithms. However, the local loss methods often face the issue of slow or non-convergence. In this paper, we propose a novel BP-modified local loss method that uses the true Backward Propagation (BP) gradient to modify the local loss gradient to improve the performance of local loss training. We use the stochastic modified equation to analyze our method and show that modified offset decreases the bias between the BP gradient and local loss gradient, but introduces additional variance, which results in a bias-variance balance. Numerical experiments on full-tuning and LoKr tuning on the ResNet-50 model and LoRA tuning on the ViT-b16 model on CIFAR-100 datasets show 20.5\% test top-1 accuracy improvement for the Forward-Forward algorithm, 18.6\% improvement for LoCo algorithm and achieve only on average 7.7\% of test accuracy loss compared to the BP algorithm, with up to 75\% memory savings.

We propose a novel asynchronous bundle method to solve distributed learning problems. Compared to existing asynchronous methods, our algorithm computes the next iterate based on a more accurate approximation of the objective function and does not require any prior information about the maximal information delay in the system. This makes the proposed method fast and easy to tune. We prove that the algorithm converges in both deterministic and stochastic (mini-batch) settings, and quantify how the convergence times depend on the level of asynchrony. The practical advantages of our method are illustrated through numerical experiments on classification problems of varying complexities and scales.

We initiate the study of one-pass streaming algorithms for underdetermined $\ell_p$ linear regression problems of the form $$ \min_{\mathbf A\mathbf x = \mathbf b} \lVert\mathbf x\rVert_p \,, \qquad \text{where } \mathbf A \in \mathbb R^{n \times d} \text{ with } n \ll d \,, $$ which generalizes basis pursuit ($p = 1$) and least squares solutions to underdetermined linear systems ($p = 2$). We study the column-arrival streaming model, in which the columns of $\mathbf A$ are presented one by one in a stream. When $\mathbf A$ is the incidence matrix of a graph, this corresponds to an edge insertion graph stream, and the regression problem captures $\ell_p$ flows which includes transshipment ($p = 1$), electrical flows ($p = 2$), and max flow ($p = \infty$) on undirected graphs as special cases. Our goal is to design algorithms which use space much less than the entire stream, which has a length of $d$. For the task of estimating the cost of the $\ell_p$ regression problem for $p\in[2,\infty]$, we show a streaming algorithm which constructs a sparse instance supported on $\tilde O(\varepsilon^{-2}n)$ columns of $\mathbf A$ which approximates the cost up to a $(1\pm\varepsilon)$ factor, which corresponds to $\tilde O(\varepsilon^{-2}n^2)$ bits of space in general and an $\tilde O(\varepsilon^{-2}n)$ space semi-streaming algorithm for constructing $\ell_p$ flow sparsifiers on graphs. This extends to $p\in(1, 2)$ with $\tilde O(\varepsilon^{2}n^{q/2})$ columns, where $q$ is the H\"older conjugate exponent of $p$. For $p = 2$, we show that $\Omega(n^2)$ bits of space are required in general even for outputting a constant factor solution. For $p = 1$, we show that the cost cannot be estimated even to an $o(\sqrt n)$ factor in $\mathrm{poly}(n)$ space. On the other hand, if we are interested in outputting a solution $\mathbf x$, then we show that $(1+\varepsilon)$-approximations require $\Omega(d)$ space for $p > 1$, and in general, $\kappa$-approximations require $\tilde\Omega(d/\kappa^{2q})$ space for $p > 1$. We complement these lower bounds with the first sublinear space upper bounds for this problem, showing that we can output a $\kappa$-approximation using space only $\mathrm{poly}(n) \cdot \tilde O(d/\kappa^q)$ for $p > 1$, as well as a $\sqrt n$-approximation using $\mathrm{poly}(n, \log d)$ space for $p = 1$.

We introduce a novel class of regularization functions, called Cauchy–Schwarz (CS) regularizers, which can be designed to induce a wide range of properties in solution vectors of optimization problems. To demonstrate the versatility of CS regularizers, we derive regularization functions that promote discrete-valued vectors, eigenvectors of a given matrix, and orthogonal matrices. The resulting CS regularizers are simple, differentiable, and can be free of spurious stationary points, making them suitable for gradient-based solvers and large-scale optimization problems. In addition, CS regularizers automatically adapt to the appropriate scale, which is, for example, beneficial when discretizing the weights of neural networks. To demonstrate the efficacy of CS regularizers, we provide results for solving underdetermined systems of linear equations and weight quantization in neural networks. Furthermore, we discuss specializations, variations, and generalizations, which lead to an even broader class of new and possibly more powerful regularizers.

In this paper, we systematically study quantum algorithms for finding an $\epsilon$-approximate second-order stationary point ($\epsilon$-SOSP) of a $d$-dimensional nonconvex function, a fundamental problem in nonconvex optimization, with noisy zeroth- or first-order oracles as inputs. We first prove that, up to noise of $O(\epsilon^{10}/d^5)$, perturbed accelerated gradient descent equipped with quantum gradient estimation takes $O(\log d/\epsilon^{1.75})$ quantum queries to find an $\epsilon$-SOSP. We then prove that standard perturbed gradient descent is robust to the noise of $O(\epsilon^6/d^4)$ and $O(\epsilon/d^{0.5+\zeta})$ for any $\zeta>0$ on the zeroth- and first-order oracles, respectively, which provides a quantum algorithm with poly-logarithmic query complexity. Furthermore, we propose a stochastic gradient descent algorithm using quantum mean estimation on the Gaussian smoothing of noisy oracles, which is robust to $O(\epsilon^{1.5}/d)$ and $O(\epsilon/\sqrt{d})$ noise on the zeroth- and first-order oracles, respectively. The quantum algorithm takes $O(d^{2.5}/\epsilon^{3.5})$ and $O(d^2/\epsilon^3)$ queries to the two oracles, giving a polynomial speedup over the classical counterparts. As a complement, we characterize the domains where quantum algorithms can find an $\epsilon$-SOSP with poly-logarithmic, polynomial, or exponential number of queries in $d$, or the problem is information-theoretically unsolvable even with an infinite number of queries. In addition, we prove an $\Omega(\epsilon^{-12/7})$ lower bound on $\epsilon$ for any randomized classical and quantum algorithm to find an $\epsilon$-SOSP using either noisy zeroth- or first-order oracles.

This paper presents a payoff perturbation technique, introducing a strong convexity to players' payoff functions in games. This technique is specifically designed for first-order methods to achieve last-iterate convergence in games where the gradient of the payoff functions is monotone in the strategy profile space, potentially containing additive noise. Although perturbation is known to facilitate the convergence of learning algorithms, the magnitude of perturbation requires careful adjustment to ensure last-iterate convergence. Previous studies have proposed a scheme in which the magnitude is determined by the distance from a periodically re-initialized anchoring or reference strategy. Building upon this, we propose Gradient Ascent with Boosting Payoff Perturbation, which incorporates a novel perturbation into the underlying payoff function, maintaining the periodically re-initializing anchoring strategy scheme. This innovation empowers us to provide faster last-iterate convergence rates against the existing payoff perturbed algorithms, even in the presence of additive noise.

We study the classical and parameterized complexity of computing the positive non-clashing teaching dimension of a set of concepts, that is, the smallest number of examples per concept required to successfully teach an intelligent learner under the considered, previously established model. For any class of concepts, it is known that this problem can be effortlessly transferred to the setting of balls in a graph $G$. We establish (1) the NP-hardness of the problem even when restricted to instances with positive non-clashing teaching dimension $k=2$ and where all balls in the graph are present, (2) near-tight running time upper and lower bounds for the problem on general graphs, (3) fixed-parameter tractability when parameterized by the vertex integrity of $G$, and (4) a lower bound excluding fixed-parameter tractability when parameterized by the feedback vertex number and pathwidth of $G$, even when combined with $k$. Our results provide a nearly complete understanding of the complexity landscape of computing the positive non-clashing teaching dimension and answer open questions from the literature.

Distributionally robust optimization (DRO) is a powerful technique to train robust machine learning models that perform well under distribution shifts. Compared with empirical risk minimization (ERM), DRO optimizes the expected loss under the worst-case distribution in an uncertainty set of distributions. This paper revisits the important problem of DRO with non-convex smooth loss functions. For this problem, Jin et al. (2021) showed that its dual problem is generalized $(L_0, L_1)$-smooth condition and gradient noise satisfies the affine variance condition, designed an algorithm of mini-batch normalized gradient descent with momentum, and proved its convergence and complexity. In this paper, we show that the dual problem and the gradient noise satisfy simpler yet more precise partially generalized smoothness condition and partially affine variance condition by studying the optimization variable and dual variable separately, which further yields much simpler algorithm design and convergence analysis. We develop a double stochastic gradient descent with clipping (D-SGD-C) algorithm that converges to an $\epsilon$-stationary point with $\mathcal O(\epsilon^{-4})$ gradient complexity, which matches with results in Jin et al. (2021). Our algorithm does not need to use momentum, and the proof is much simpler, thanks to the more precise characterization of partially generalized smoothness and partially affine variance noise. We further design a variance-reduced method that achieves a lower gradient complexity of $\mathcal O(\epsilon^{-3})$. Our theoretical results and insights are further verified numerically on a number of tasks, and our algorithms outperform the existing DRO method (Jin et al., 2021).

The regularity or implicit bias in neural network optimization has been typically studied via the parameter norms or the landscape curvature, often overlooking the trajectory leading to these parameters. However, properties of the trajectory --- particularly its directionality --- capture critical aspects of how gradient descent navigates the landscape to converge to a solution. In this work, we introduce the notion of a Trajectory Map and derive natural complexity measures that highlight the directional characteristics of optimization trajectories. Our comprehensive analysis across vision and language modeling tasks reveals that (a) the trajectory's directionality at the macro-level saturates by the initial phase of training, wherein weight decay and momentum play a crucial but understated role; and (b) in subsequent training, trajectory directionality manifests in micro-level behaviors, such as oscillations, for which we also provide a theoretical analysis. This implies that neural optimization trajectories have, overall, a more linear form than zig-zaggy, as evident by high directional similarity, especially towards the end. To further hone this point, we show that when the trajectory direction gathers such an inertia, optimization proceeds largely unaltered even if the network is severely decapacitated (by freezing >99% of the parameters), --- thereby demonstrating the potential for significant computational and resource savings without compromising performance.

We investigate the problem of designing differentially private (DP), revenue-maximizing single item auction. Specifically, we consider broadly applicable settings in mechanism design where agents' valuation distributions are **independent**, **non-identical**, and can be either **bounded** or **unbounded**. Our goal is to design such auctions with **pure**, i.e., $(\epsilon,0)$ privacy in polynomial time. In this paper, we propose two computationally efficient auction learning framework that achieves **pure** privacy under bounded and unbounded distribution settings. These frameworks reduces the problem of privately releasing a revenue-maximizing auction to the private estimation of pre-specified quantiles. Our solutions increase the running time by polylog factors compared to the non-private version. As an application, we show how to extend our results to the multi-round online auction setting with non-myopic bidders. To our best knowledge, this paper is the first to efficiently deliver a Myerson auction with **pure** privacy and near-optimal revenue, and the first to provide such auctions for **unbounded** distributions.

Post-Training low-bit Quantization (PTQ) is useful to accelerate DNNs due to its high efficiency, the current SOTAs of which mostly adopt feature reconstruction with self-distillation finetuning. However, when bitwidth goes to be extremely low, we find the current reconstruction optimization space is not optimal. Considering all possible parameters and the ignored fact that integer weight can be obtained early before actual inference, we thoroughly explore different optimization space by quant-step decoupling, where a wider PTQ optimization space, which consistently makes a better optimum, is found out. Based on these, we propose an Adaptive Quantization Transformation (AdaQTransform) for PTQ reconstruction, which makes the quantized output feature better fit the FP32 counterpart with adaptive per-channel transformation, thus achieves lower feature reconstruction error. In addition, it incurs negligible extra finetuning cost and no extra inference cost. Based on AdaQTransform, for the first time, we build a general quantization setting paradigm subsuming current PTQs, QATs and other potential forms. Experiments demonstrate AdaQTransform expands the optimization space for PTQ and helps current PTQs find a better optimum over CNNs, ViTs, LLMs and image super-resolution networks, e.g., it improves NWQ by 5.7% on ImageNet for W2A2-MobileNet-v2.

Despite being the cornerstone of deep learning, backpropagation is criticized for its inherent sequentiality, which can limit the scalability of very deep models. Such models faced convergence issues due to vanishing gradient, later resolved using residual connections. Variants of these are now widely used in modern architectures. However, the computational cost of backpropagation remains a major burden, accounting for most of the training time. Taking advantage of residual-like architectural designs, we introduce Highway backpropagation, a parallelizable iterative algorithm that approximates backpropagation, by alternatively i) accumulating the gradient estimates along the residual path, and ii) backpropagating them through every layer in parallel. This algorithm is naturally derived from a decomposition of the gradient as the sum of gradients flowing through all paths, and is adaptable to a diverse set of common architectures, ranging from ResNets and Transformers to recurrent neural networks. Through an extensive empirical study on a large selection of tasks and models, we evaluate Highway-BP and show that major speedups can be achieved with minimal performance degradation.

Deep neural networks trained using gradient descent with a fixed learning rate $\eta$ often operate in the regime of ``edge of stability'' (EOS), where the largest eigenvalue of the Hessian equilibrates about the stability threshold $2/\eta$. In this work, we present a fine-grained analysis of the learning dynamics of (deep) linear networks (DLNs) within the deep matrix factorization loss beyond EOS. For DLNs, loss oscillations beyond EOS follow a period-doubling route to chaos. We theoretically analyze the regime of the 2-period orbit and show that the loss oscillations occur within a small subspace, with the dimension of the subspace precisely characterized by the learning rate. The crux of our analysis lies in showing that the symmetry-induced conservation law for gradient flow, defined as the balancing gap among the singular values across layers, breaks at EOS and decays monotonically to zero. Overall, our results contribute to explaining two key phenomena in deep networks: (i) shallow models and simple tasks do not always exhibit EOS; and (ii) oscillations occur within top features}. We present experiments to support our theory, along with examples demonstrating how these phenomena occur in nonlinear networks and how they differ from those which have benign landscape such as in DLNs.

We examine the implicit bias of mirror flow in least squares error regression with wide and shallow neural networks. For a broad class of potential functions, we show that mirror flow exhibits lazy training and has the same implicit bias as ordinary gradient flow when the network width tends to infinity. For univariate ReLU networks, we characterize this bias through a variational problem in function space. Our analysis includes prior results for ordinary gradient flow as a special case and lifts limitations which required either an intractable adjustment of the training data or networks with skip connections. We further introduce \emph{scaled potentials} and show that for these, mirror flow still exhibits lazy training but is not in the kernel regime. For univariate networks with absolute value activations, we show that mirror flow with scaled potentials induces a rich class of biases, which generally cannot be captured by an RKHS norm. A takeaway is that whereas the parameter initialization determines how strongly the curvature of the learned function is penalized at different locations of the input space, the scaled potential determines how the different magnitudes of the curvature are penalized.

There is growing evidence of the effectiveness of Shampoo, a higher-order preconditioning method, over Adam in deep learning optimization tasks. However, Shampoo's drawbacks include additional hyperparameters and computational overhead when compared to Adam, which only updates running averages of first- and second-moment quantities. This work establishes a formal connection between Shampoo (implemented with the 1/2 power) and Adafactor --- a memory-efficient approximation of Adam --- showing that Shampoo is equivalent to running Adafactor in the eigenbasis of Shampoo's preconditioner. This insight leads to the design of a simpler and computationally efficient algorithm: **S**hampo**O** with **A**dam in the **P**reconditioner's eigenbasis (SOAP). With regards to improving Shampoo's computational efficiency, the most straightforward approach would be to simply compute Shampoo's eigendecomposition less frequently. Unfortunately, as our empirical results show, this leads to performance degradation that worsens with this frequency. SOAP mitigates this degradation by continually updating the running average of the second moment, just as Adam does, but in the current (slowly changing) coordinate basis. Furthermore, since SOAP is equivalent to running Adam in a rotated space, it introduces only one additional hyperparameter (the preconditioning frequency) compared to Adam. We empirically evaluate SOAP on language model pre-training with 360m and 660m sized models. In the large batch regime, SOAP reduces the number of iterations by over 40\% and wall clock time by over 35\% compared to AdamW, with approximately 20\% improvements in both metrics compared to Shampoo. An implementation of SOAP is available at https://github.com/nikhilvyas/SOAP.

Machine unlearning, the process of selectively removing data from trained models, is increasingly crucial for addressing privacy concerns and knowledge gaps post-deployment. Despite this importance, existing approaches are often heuristic and lack formal guarantees. In this paper, we analyze the fundamental utility, time, and space complexity trade-offs of approximate unlearning, providing rigorous certification analogous to differential privacy. For in-distribution forget data—data similar to the retain set—we show that a surprisingly simple and general procedure, empirical risk minimization with output perturbation, achieves tight unlearning-utility-complexity trade-offs, addressing a previous theoretical gap on the separation from unlearning ``for free" via differential privacy, which inherently facilitates the removal of such data. However, such techniques fail with out-of-distribution forget data—data significantly different from the retain set—where unlearning time complexity can exceed that of retraining, even for a single sample. To address this, we propose a new robust and noisy gradient descent variant that provably amortizes unlearning time complexity without compromising utility.

Trajectory inference seeks to recover the temporal dynamics of a population from snapshots of its (uncoupled) temporal marginals, i.e. where observed particles are \emph{not} tracked over time. Prior works addressed this challenging problem under a stochastic differential equation (SDE) model with a gradient-driven drift in the observed space, introducing a minimum entropy estimator relative to the Wiener measure and a practical grid-free mean-field Langevin (MFL) algorithm using Schr\"odinger bridges. Motivated by the success of observable state space models in the traditional paired trajectory inference problem (e.g. target tracking), we extend the above framework to a class of latent SDEs in the form of \emph{observable state space models}. In this setting, we use partial observations to infer trajectories in the latent space under a specified dynamics model (e.g. the constant velocity/acceleration models from target tracking). We introduce the PO-MFL algorithm to solve this latent trajectory inference problem and provide theoretical guarantees to the partially observed setting. Experiments validate the robustness of our method and the exponential convergence of the MFL dynamics, and demonstrate significant outperformance over the latent-free baseline in key scenarios.

Scaling Transformer to a large scale without using some technical tricks such as learning rate warump and an obviously lower learning rate, is an extremely challenging task, and is increasingly gaining more attention. In this paper, we provide a theoretical analysis for the process of training Transformer and reveal a key problem behind model crash phenomenon in the training process, termed *spectral energy concentration* of ${W_q}^{\top} W_k$, which is the reason for a malignant entropy collapse, where ${W_q}$ and $W_k$ are the projection matrices for the query and the key in Transformer, respectively. To remedy this problem, motivated by *Weyl's Inequality*, we present a novel optimization strategy, \ie, making the weight updating in successive steps steady---if the ratio $\frac{\sigma_{1}(\nabla W_t)}{\sigma_{1}(W_{t-1})}$ is larger than a threshold, we will automatically bound the learning rate to a weighted multiple of $\frac{\sigma_{1}(W_{t-1})}{\sigma_{1}(\nabla W_t)}$, where $\nabla W_t$ is the updating quantity in step $t$. Such an optimization strategy can prevent spectral energy concentration to only a few directions, and thus can avoid malignant entropy collapse which will trigger the model crash. We conduct extensive experiments using ViT, Swin-Transformer and GPT, showing that our optimization strategy can effectively and stably train these (Transformer) models without using learning rate warmup.

We study gradient methods for optimizing $(L_0, L_1)$-smooth functions, a class that generalizes Lipschitz-smooth functions and has gained attention for its relevance in machine learning. We provide new insights into the structure of this function class and develop a principled framework for analyzing optimization methods in this setting. While our convergence rate estimates recover existing results for minimizing the gradient norm in nonconvex problems, our approach significantly improves the best-known complexity bounds for convex objectives. Moreover, we show that the gradient method with Polyak stepsizes and the normalized gradient method achieve nearly the same complexity guarantees as methods that rely on explicit knowledge of $(L_0, L_1)$. Finally, we demonstrate that a carefully designed accelerated gradient method can be applied to $(L_0, L_1)$-smooth functions, further improving all previous results.