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Deep Learning · Theory

David Robin, Kevin Scaman, marc lelarge

We present a framework to define a large class of neural networks for which, by construction, training by gradient flow provably reaches arbitrarily low loss when the number of parameters grows. Distinct from the fixed-space global optimality of non-convex optimization, this new form of convergence, and the techniques introduced to prove such convergence, pave the way for a usable deep learning convergence theory in the near future, without overparameterization assumptions relating the number of parameters and training samples. We define these architectures from a simple computation graph and a mechanism to lift it, thus increasing the number of parameters, generalizing the idea of increasing the widths of multi-layer perceptrons. We show that architectures similar to most common deep learning models are present in this class, obtained by sparsifying the weight tensors of usual architectures at initialization. Leveraging tools of algebraic topology and random graph theory, we use the computation graph’s geometry to propagate properties guaranteeing convergence to any precision for these large sparse models.

Hong Liu, Zhiyuan Li, David Hall, Percy Liang, Tengyu Ma

Given the massive cost of language model pre-training, a non-trivial improvement of the optimization algorithm would lead to a material reduction on the time and cost of training. Adam and its variants have been state-of-the-art for years, and more sophisticated second-order (Hessian-based) optimizers often incur too much per-step overhead. In this paper, we propose Sophia, a simple scalable second-order optimizer that uses a light-weight estimate of the diagonal Hessian as the pre-conditioner. The update is the moving average of the gradients divided by the moving average of the estimated Hessian, followed by element-wise clipping. The clipping controls the worst-case update size and tames the negative impact of non-convexity and rapid change of Hessian along the trajectory. Sophia only estimates the diagonal Hessian every handful of iterations, which has negligible average per-step time and memory overhead. On language modeling with GPT models of sizes ranging from 125M to 1.5B, Sophia achieves a 2x speed-up compared to Adam in the number of steps, total compute, and wall-clock time, achieving the same perplexity with 50\% fewer steps, less total compute, and reduced wall-clock time.

Haozhao Wang, Haoran Xu, Yichen Li, Yuan Xu, Ruixuan Li, Tianwei Zhang

In Federated Learning (FL), model aggregation is pivotal. It involves a global server iteratively aggregating client local trained models in successive rounds without accessing private data. Traditional methods typically aggregate the local models from the current round alone. However, due to the statistical heterogeneity across clients, the local models from different clients may be greatly diverse, making the obtained global model incapable of maintaining the specific knowledge of each local model. In this paper, we introduce a novel method, FedCDA, which selectively aggregates cross-round local models, decreasing discrepancies between the global model and local models.The principle behind FedCDA is that due to the different global model parameters received in different rounds and the non-convexity of deep neural networks, the local models from each client may converge to different local optima across rounds. Therefore, for each client, we select a local model from its several recent local models obtained in multiple rounds, where the local model is selected by minimizing its divergence from the local models of other clients. This ensures the aggregated global model remains close to all selected local models to maintain their data knowledge. Extensive experiments conducted on various models and datasets reveal our approach outperforms state-of-the-art aggregation methods.

Qinzi Zhang, Hoang Tran, Ashok Cutkosky

We introduce a new zeroth-order algorithm for private stochastic optimization on nonconvex and nonsmooth objectives.Given a dataset of size $M$, our algorithm ensures $(\alpha,\alpha\rho^2/2)$-Renyi differential privacy and finds a $(\delta,\epsilon)$-stationary point so long as $M=\tilde\Omega(\frac{d}{\delta\epsilon^3} + \frac{d^{3/2}}{\rho\delta\epsilon^2})$.This matches the optimal complexity found in its non-private zeroth-order analog. Notably, although the objective is not smooth, we have privacy ``for free'' when $\rho \ge \sqrt{d}\epsilon$.

Jie Hao, Xiaochuan Gong, Mingrui Liu

Bilevel optimization is an important formulation for many machine learning problems, such as meta-learning and hyperparameter optimization. Current bilevel optimization algorithms assume that the gradient of the upper-level function is Lipschitz (i.e., the upper-level function has a bounded smoothness parameter). However, recent studies reveal that certain neural networks such as recurrent neural networks (RNNs) and long-short-term memory networks (LSTMs) exhibit potential unbounded smoothness, rendering conventional bilevel optimization algorithms unsuitable for these neural networks. In this paper, we design a new bilevel optimization algorithm, namely BO-REP, to address this challenge. This algorithm updates the upper-level variable using normalized momentum and incorporates two novel techniques for updating the lower-level variable: \textit{initialization refinement} and \textit{periodic updates}. Specifically, once the upper-level variable is initialized, a subroutine is invoked to obtain a refined estimate of the corresponding optimal lower-level variable, and the lower-level variable is updated only after every specific period instead of each iteration. When the upper-level problem is nonconvex and unbounded smooth, and the lower-level problem is strongly convex, we prove that our algorithm requires $\widetilde{O}(1/\epsilon^4)$ \footnote{Here $\widetilde{O}(\cdot)$ compresses logarithmic factors of $1/\epsilon$ and $1/\delta$, where $\delta\in(0,1)$ denotes the failure probability.} iterations to find an $\epsilon$-stationary point in the stochastic setting, where each iteration involves calling a stochastic gradient or Hessian-vector product oracle. Notably, this result matches the state-of-the-art complexity results under the bounded smoothness setting and without mean-squared smoothness of the stochastic gradient, up to logarithmic factors. Our proof relies on novel technical lemmas for the periodically updated lower-level variable, which are of independent interest. Our experiments on hyper-representation learning, hyperparameter optimization, and data hyper-cleaning for text classification tasks demonstrate the effectiveness of our proposed algorithm. The code is available at [https://github.com/MingruiLiu-ML-Lab/Bilevel-Optimization-under-Unbounded-Smoothness](https://github.com/MingruiLiu-ML-Lab/Bilevel-Optimization-under-Unbounded-Smoothness).

Optimization · Non-Convex

Zhuqing Liu, Xin Zhang, Jia Liu, Zhengyuan Zhu, Songtao Lu

Learning an accurate value function for a given policy is a critical step in solving reinforcement learning (RL) problems. So far, however, the convergence speed and sample complexity performances of most existing policy evaluation algorithms remain unsatisfactory, particularly with non-linear function approximation. This challenge motivates us to develop a new path-integrated primal-dual stochastic gradient (PILOT) method, that is able to achieve a fast convergence speed for RL policy evaluation with nonlinear function approximation. To further alleviate the periodic full gradient evaluation requirement, we further propose an enhanced method with an adaptive-batch adjustment called PILOT$^+$. The main advantages of our methods include: i) PILOT allows the use of {\em{constant}} step sizes and achieves the $\mathcal{O}(1/K)$ convergence rate to first-order stationary points of non-convex policy evaluation problems; ii) PILOT is a generic {\em{single}}-timescale algorithm that is also applicable for solving a large class of non-convex strongly-concave minimax optimization problems; iii) By adaptively adjusting the batch size via historical stochastic gradient information, PILOT$^+$ is more sample-efficient empirically without loss of theoretical convergence rate. Our extensive numerical experiments verify our theoretical findings and showcase the high efficiency of the proposed PILOT and PILOT$^+$ algorithms compared with the state-of-the-art methods.

Optimization · Bilevel optimization

Bo Zhou, Ruiwei Jiang, Siqian Shen

Bilevel programs (BPs) find a wide range of applications in fields such as energy, transportation, and machine learning. As compared to BPs with continuous (linear/convex) optimization problems in both levels, the BPs with discrete decision variables have received much less attention, largely due to the ensuing computational intractability and the incapability of gradient-based algorithms for handling discrete optimization formulations. In this paper, we develop deep learning techniques to address this challenge. Specifically, we consider a BP with binary tender, wherein the upper and lower levels are linked via binary variables. We train a neural network to approximate the optimal value of the lower-level problem, as a function of the binary tender. Then, we obtain a single-level reformulation of the BP through a mixed-integer representation of the value function. Furthermore, we conduct a comparative analysis between two types of neural networks: general neural networks and the novel input supermodular neural networks, studying their representational capacities. To solve high-dimensional BPs, we introduce an enhanced sampling method to generate higher-quality samples and implement an iterative process to refine solutions. We demonstrate the performance of these approaches through extensive numerical experiments, whose lower-level problems are linear and mixed-integer programs, respectively.

Deep Learning

Adam Block, Dylan Foster, Akshay Krishnamurthy, Max Simchowitz, Cyril Zhang

This work studies training instabilities of behavior cloning with deep neural networks. We observe that minibatch SGD updates to the policy network during training result in sharp oscillations in long-horizon rewards, despite negligibly affecting the behavior cloning loss. We empirically disentangle the statistical and computational causes of these oscillations, and find them to stem from the chaotic propagation of minibatch SGD noise through unstable closed-loop dynamics. While SGD noise is benign in the single-step action prediction objective, it results in catastrophic error accumulation over long horizons, an effect we term *gradient variance amplification* (GVA). We demonstrate that many standard mitigation techniques do not alleviate GVA, but that taking an exponential moving average (EMA) of iterates is surprisingly effective at doing so. Furthermore, we illustrate the generality of the phenomenon by showing both the existence of GVA and its amelioration by EMA in autoregressive language generation. Finally, we provide theoretical vignettes both exhibiting the benefits of EMA in alleviating GVA and illustrating the extent to which classical convex models help in understanding the benefits of iterate averaging in deep learning.

Jeongyeol Kwon, Dohyun Kwon, Stephen Wright, Robert Nowak

In this work, we study first-order algorithms for solving Bilevel Optimization (BO) where the objective functions are smooth but possibly nonconvex in both levels and the variables are restricted to closed convex sets. As a first step, we study the landscape of BO through the lens of penalty methods, in which the upper- and lower-level objectives are combined in a weighted sum with penalty parameter $\sigma > 0$. In particular, we establish a strong connection between the penalty function and the hyper-objective by explicitly characterizing the conditions under which the values and derivatives of the two must be $O(\sigma)$-close. A by-product of our analysis is the explicit formula for the gradient of hyper-objective when the lower-level problem has multiple solutions under minimal conditions, which could be of independent interest. Next, viewing the penalty formulation as $O(\sigma)$-approximation of the original BO, we propose first-order algorithms that find an $\epsilon$-stationary solution by optimizing the penalty formulation with $\sigma = O(\epsilon)$. When the perturbed lower-level problem uniformly satisfies the {\it small-error} proximal error-bound (EB) condition, we propose a first-order algorithm that converges to an $\epsilon$-stationary point of the penalty function using in total $O(\epsilon^{-7})$ accesses to first-order stochastic gradient oracles. Under an additional assumption on stochastic oracles, we show that the algorithm can be implemented in a fully {\it single-loop} manner, {\it i.e.,} with $O(1)$ samples per iteration, and achieves the improved oracle-complexity of $O(\epsilon^{-5})$.

Optimization · Large Scale, Parallel and Distributed

Hamidreza Almasi, Harsh Mishra, Balajee Vamanan, Sathya N. Ravi

Modern ML applications increasingly rely on complex deep learning models and large datasets. There has been an exponential growth in the amount of computation needed to train the largest models. Therefore, to scale computation and data, these models are inevitably trained in a distributed manner in clusters of nodes, and their updates are aggregated before being applied to the model. However, a distributed setup is prone to Byzantine failures of individual nodes, components, and software. With data augmentation added to these settings, there is a critical need for robust and efficient aggregation systems. We define the quality of workers as reconstruction ratios $\in (0,1]$, and formulate aggregation as a Maximum Likelihood Estimation procedure using Beta densities. We show that the Regularized form of log-likelihood wrt subspace can be approximately solved using iterative least squares solver, and provide convergence guarantees using recent Convex Optimization landscape results. Our empirical findings demonstrate that our approach significantly enhances the robustness of state-of-the-art Byzantine resilient aggregators. We evaluate our method in a distributed setup with a parameter server, and show simultaneous improvements in communication efficiency and accuracy across various tasks.

Deep Learning · Theory

Linara Adilova, Maksym Andriushchenko, Michael Kamp, Asja Fischer, Martin Jaggi

Averaging neural network parameters is an intuitive method for fusing the knowledge of two independent models. It is most prominently used in federated learning. If models are averaged at the end of training, this can only lead to a good performing model if the loss surface of interest is very particular, i.e., the loss in the midpoint between the two models needs to be sufficiently low. This is impossible to guarantee for the non-convex losses of state-of-the-art networks. For averaging models trained on vastly different datasets, it was proposed to average only the parameters of particular layers or combinations of layers, resulting in better performing models. To get a better understanding of the effect of layer-wise averaging, we analyse the performance of the models that result from averaging single layers, or groups of layers. Based on our empirical and theoretical investigation, we introduce a novel notion of the layer-wise linear connectivity, and show that deep networks do not have layer-wise barriers between them.

Optimization · Large Scale, Parallel and Distributed

Yuan Gao, Rustem Islamov, Sebastian Stich

Modern distributed training relies heavily on communication compression to reduce the communication overhead. In this work, we study algorithms employing a popular class of contractive compressors in order to reduce communication overhead. However, the naive implementation often leads to unstable convergence or even exponential divergence due to the compression bias. Error Compensation (EC) is an extremely popular mechanism to mitigate the aforementioned issues during the training of models enhanced by contractive compression operators. Compared to the effectiveness of EC in the data homogeneous regime, the understanding of the practicality and theoretical foundations of EC in the data heterogeneous regime is limited. Existing convergence analyses typically rely on strong assumptions such as bounded gradients, bounded data heterogeneity, or large batch accesses, which are often infeasible in modern Machine Learning Applications. We resolve the majority of current issues by proposing EControl, a novel mechanism that can regulate error compensation by controlling the strength of the feedback signal. We prove fast convergence for EControl in standard strongly convex, general convex, and nonconvex settings without any additional assumptions on the problem or data heterogeneity. We conduct extensive numerical evaluations to illustrate the efficacy of our method and support our theoretical findings.

Miscellaneous Aspects of Machine Learning · Transfer, Multitask and Meta-learning

Tian Yu Liu, Aditya Golatkar, Stefano Soatto

We introduce Tangent Attention Fine-Tuning (TAFT), a method for fine-tuning linearized transformers obtained by computing a First-order Taylor Expansion around a pre-trained initialization. We show that the Jacobian-Vector Product resulting from linearization can be computed efficiently in a single forward pass, reducing training and inference cost to the same order of magnitude as its original non-linear counterpart, while using the same number of parameters. Furthermore, we show that, when applied to various downstream visual classification tasks, the resulting Tangent Transformer fine-tuned with TAFT can perform comparably with fine-tuning the original non-linear network. Since Tangent Transformers are linear with respect to the new set of weights, and the resulting fine-tuning loss is convex, we show that TAFT enjoys several advantages compared to non-linear fine-tuning when it comes to model composition, parallel training, machine unlearning, and differential privacy. Our code is available at: https://github.com/tianyu139/tangent-model-composition

Optimization

Antoine Bambade, Fabian Schramm, Adrien Taylor, Justin Carpentier

Optimization layers within neural network architectures have become increasingly popular for their ability to solve a wide range of machine learning tasks and to model domain-specific knowledge. However, designing optimization layers requires careful consideration as the underlying optimization problems might be infeasible during training. Motivated by applications in learning, control and robotics, this work focuses on convex quadratic programming (QP) layers. The specific structure of this type of optimization layer can be efficiently exploited for faster computations while still allowing rich modeling capabilities. We leverage primal-dual augmented Lagrangian techniques for computing derivatives of both feasible and infeasible QP solutions. More precisely, we propose a unified approach which tackles the differentiability of the closest feasible QP solutions in a classical $\ell_2$ sense. We then harness this approach to enrich the expressive capabilities of existing QP layers. More precisely, we show how differentiating through infeasible QPs during training enables to drive towards feasibility at test time a new range of QP layers. These layers notably demonstrate superior predictive performance in some conventional learning tasks. Additionally, we present alternative formulations that enhance numerical robustness, speed, and accuracy for training such layers. Along with these contributions, we provide an open-source C++ software package called QPLayer for differentiating feasible and infeasible convex QPs and which can be interfaced with modern learning frameworks.

Optimization · Non-Convex

Aaron Zweig, Joan Bruna

Few neural architectures lend themselves to provable learning with gradient based methods. One popular model is the single-index model, in which labels are produced by composing an unknown linear projection with a possibly unknown scalar link function. Learning this model with SGD is relatively well-understood, whereby the so-called information exponent of the link function governs a polynomial sample complexity rate. However, extending this analysis to deeper or more complicated architectures remains challenging.In this work, we consider single index learning in the setting of symmetric neural networks. Under analytic assumptions on the activation and maximum degree assumptions on the link function, we prove that gradient flow recovers the hidden planted direction, represented as a finitely supported vector in the feature space of power sum polynomials. We characterize a notion of information exponent adapted to our setting that controls the efficiency of learning.

Optimization

Lizhang Chen, Bo Liu, Kaizhao Liang, Qiang Liu

Lion (Evolved Sign Momentum), a new optimizer discovered through program search, has shown promising results in training large AI models. It achieves results comparable to AdamW but with greater memory efficiency. As what we can expect from the result of the random search, Lion blends a number of elements from existing algorithms, including signed momentum, decoupled weight decay, Polayk and Nesterov momentum, but doesn't fit into any existing category of theoretically grounded optimizers. Thus, even though Lion appears to perform well as a general-purpose optimizer for a wide range of tasks, its theoretical basis remains uncertain. This absence of theoretical clarity limits opportunities to further enhance and expand Lion's efficacy. This work aims to demystify Lion. Using both continuous-time and discrete-time analysis, we demonstrate that Lion is a novel and theoretically grounded approach for minimizing a general loss function $f(x)$ while enforcing a bound constraint $||x||_\infty \leq 1/\lambda$. Lion achieves this through the incorporation of decoupled weight decay, where $\lambda$ represents the weight decay coefficient. Our analysis is facilitated by the development of a new Lyapunov function for the Lion updates. It applies to a wide range of Lion-$\phi$ algorithms, where the $sign(\cdot)$ operator in Lion is replaced by the subgradient of a convex function $\phi$, leading to the solution of the general composite optimization problem $\min_x f(x) + \phi^*(x)$. Our findings provide valuable insights into the dynamics of Lion and pave the way for further enhancements and extensions of Lion-related algorithms.

Deep Learning · Large Language Models

Lingfeng Shen, Sihao Chen, Linfeng Song, Lifeng Jin, Baolin Peng, Haitao Mi, Daniel Khashabi, Dong Yu

Standard practice within Reinforcement Learning from Human Feedback (RLHF) involves optimizing against a Reward Model (RM), which itself is trained to reflect human preferences for desirable generations. A notable subject that is understudied is the (in-)consistency of RMs --- whether they can recognize the semantic changes to different prompts and appropriately adapt their reward assignments--- and their impact on the downstream RLHF model.In this paper, we visit a series of research questions relevant to RM inconsistency:(1) How can we measure the consistency of reward models? (2) How consistent are the existing RMs and how can we improve them? (3) In what ways does reward inconsistency influence the chatbots resulting from the RLHF model training?We propose **Contrast Instruction** -- a benchmarking strategy for the consistency of RM. Each example in **Contrast Instruction** features a pair of lexically similar instructions with different ground truth responses. A consistent RM is expected to rank the corresponding instruction and response higher than other combinations. We observe that current RMs trained with the standard ranking objective fail miserably on \contrast{} compared to average humans. To show that RM consistency can be improved efficiently without using extra training budget, we propose two techniques **ConvexDA** and **RewardFusion**, which enhance reward consistency through extrapolation during the RM training and inference stage, respectively.We show that RLHF models trained with a more consistent RM yield more useful responses, suggesting that reward inconsistency exhibits a trickle-down effect on the downstream RLHF process.

Probabilistic Methods · Structure Learning

Seyed Saman Saboksayr, Gonzalo Mateos, Mariano Tepper

We deal with the combinatorial problem of learning directed acyclic graph (DAG) structure from observational data adhering to a linear structural equation model (SEM). Leveraging advances in differentiable, nonconvex characterizations of acyclicity, recent efforts have advocated a continuous constrained optimization paradigm to efficiently explore the space of DAGs. Most existing methods employ lasso-type score functions to guide this search, which (i) require expensive penalty parameter retuning when the $\textit{unknown}$ SEM noise variances change across problem instances; and (ii) implicitly rely on limiting homoscedasticity assumptions. In this work, we propose a new convex score function for sparsity-aware learning of linear DAGs, which incorporates concomitant estimation of scale and thus effectively decouples the sparsity parameter from noise levels. Regularization via a smooth, nonconvex acyclicity penalty term yields CoLiDE ($\textbf{Co}$ncomitant $\textbf{Li}$near $\textbf{D}$AG $\textbf{E}$stimation), a regression-based criterion amenable to efficient gradient computation and closed-form estimation of exogenous noise levels in heteroscedastic scenarios. Our algorithm outperforms state-of-the-art methods without incurring added complexity, especially when the DAGs are larger and the noise level profile is heterogeneous. We also find CoLiDE exhibits enhanced stability manifested via reduced standard deviations in several domain-specific metrics, underscoring the robustness of our novel linear DAG estimator.

Deep Learning · Theory

Quinn Fisher, Haoming Meng, Vardan Papyan

Mixup is a data augmentation strategy that employs convex combinations of training instances and their respective labels to improve the robustness and calibration of deep neural networks. Despite its widespread adoption, the nuanced mechanisms that underpin its success are not entirely understood. The observed phenomenon of Neural Collapse, where the last-layer activations and classifier of deep networks converge to a simplex equiangular tight frame (ETF), provides a compelling motivation to explore whether mixup induces alternative geometric configurations and whether those could explain its success. In this study, we delve into the last-layer activations of training data for deep networks subjected to mixup, aiming to uncover insights into its operational efficacy. Our investigation, spanning various architectures and dataset pairs, reveals that mixup's last-layer activations predominantly converge to a distinctive configuration different than one might expect. In this configuration, activations from mixed-up examples of identical classes align with the classifier, while those from different classes delineate channels along the decision boundary. These findings are unexpected, as mixed-up features are not simple convex combinations of feature class means (as one might get, for example, by training mixup with the mean squared error loss). By analyzing this distinctive geometric configuration, we elucidate the mechanisms by which mixup enhances model calibration. To further validate our empirical observations, we conduct a theoretical analysis under the assumption of an unconstrained features model, utilizing the mixup loss. Through this, we characterize and derive the optimal last-layer features under the assumption that the classifier forms a simplex ETF.

Social Aspects · Trustworthy Machine Learning

Xiaosen Zheng, Tianyu Pang, Chao Du, Jing Jiang, Min Lin

Data attribution seeks to trace model outputs back to training data. With the recent development of diffusion models, data attribution has become a desired module to properly assign valuations for high-quality or copyrighted training samples, ensuring that data contributors are fairly compensated or credited. Several theoretically motivated methods have been proposed to implement data attribution, in an effort to improve the trade-off between computational scalability and effectiveness. In this work, we conduct extensive experiments and ablation studies on attributing diffusion models, specifically focusing on DDPMs trained on CIFAR-10 and CelebA, as well as a Stable Diffusion model LoRA-finetuned on ArtBench. Intriguingly, we report counter-intuitive observations that theoretically unjustified design choices for attribution empirically outperform previous baselines by a large margin, in terms of both linear datamodeling score and counterfactual evaluation. Our work presents a significantly more efficient approach for attributing diffusion models, while the unexpected findings suggest that at least in non-convex settings, constructions guided by theoretical assumptions may lead to inferior attribution performance. The code is available at https://github.com/sail-sg/D-TRAK.