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Optimization

Charles Guille-Escuret, Hiroki Naganuma, Kilian Fatras, Ioannis Mitliagkas

Understanding the optimization dynamics of neural networks is necessary for closing the gap between theory and practice. Stochastic first-order optimization algorithms are known to efficiently locate favorable minima in deep neural networks. This efficiency, however, contrasts with the non-convex and seemingly complex structure of neural loss landscapes. In this study, we delve into the fundamental geometric properties of sampled gradients along optimization paths. We focus on two key quantities, the restricted secant inequality and error bound, as well as their ratio γ, which hold high significance for first-order optimization. Our analysis reveals that these quantities exhibit predictable, consistent behavior throughout training, despite the stochasticity induced by sampling minibatches. Our findings suggest that not only do optimization trajectories never encounter significant obstacles, but they also maintain stable dynamics during the majority of training. These observed properties are sufficiently expressive to theoretically guarantee linear convergence and prescribe learning rate schedules mirroring empirical practices. We conduct our experiments on image classification, semantic segmentation and language modeling across different batch sizes, network architectures, datasets, optimizers, and initialization seeds. We discuss the impact of each factor. Our work provides novel insights into the properties of neural network loss functions, and opens the door to theoretical frameworks more relevant to prevalent practice.

Optimization · Stochastic

Hongcheng Liu, Jindong Tong

This paper studies sample average approximation (SAA) and its simple regularized variation in solving convex or strongly convex stochastic programming problems. Under heavy-tailed assumptions and comparable regularity conditions as in the typical SAA literature, we show --- perhaps for the first time --- that the sample complexity can be completely free from any complexity measure (e.g., logarithm of the covering number) of the feasible region. As a result, our new bounds can be more advantageous than the state-of-the-art in terms of the dependence on the problem dimensionality.

Kazusato Oko, Shunta Akiyama, Denny Wu, Tomoya Murata, Taiji Suzuki

Most variance reduction methods require multiple times of full gradient computation, which is time-consuming and hence a bottleneck in application to distributed optimization. We present a single-loop variance-reduced gradient estimator named SILVER (SIngle-Loop VariancE-Reduction) for the finite-sum non-convex optimization, which does not require multiple full gradients but nevertheless achieves the optimal gradient complexity. Notably, unlike existing methods, SILVER provably reaches second-order optimality, with exponential convergence in the Polyak-Łojasiewicz (PL) region, and achieves further speedup depending on the data heterogeneity. Owing to these advantages, SILVER serves as a new base method to design communication-efficient federated learning algorithms: we combine SILVER with local updates which gives the best communication rounds and number of communicated gradients across all range of Hessian heterogeneity, and, at the same time, guarantees second-order optimality and exponential convergence in the PL region.

Optimization · Non-Convex

Ganzhao Yuan

Nonsmooth sparsity constrained optimization encompasses a broad spectrum of applications in machine learning. This problem is generally non-convex and NP-hard. Existing solutions to this problem exhibit several notable limitations, including their inability to address general nonsmooth problems, tendency to yield weaker optimality conditions, and lack of comprehensive convergence analysis. This paper considers Smoothing Proximal Gradient Methods (SPGM) as solutions to nonsmooth sparsity constrained optimization problems. Two specific variants of SPGM are explored: one based on Iterative Hard Thresholding (SPGM-IHT) and the other on Block Coordinate Decomposition (SPGM-BCD). It is shown that the SPGM-BCD algorithm finds stronger stationary points compared to previous methods. Additionally, novel theories for analyzing the convergence rates to approximate global optimal solutions of both the SPGM-IHT and SPGM-BCD algorithms are developed. Our theoretical bounds, capitalizing on the intrinsic sparsity of the optimization problem, are on par with the best-known error bounds available to date. Finally, numerical experiments reveal that SPGM-IHT performs comparably to current IHT-style methods, while SPGM-BCD consistently surpasses them.

Theory · Learning Theory

Idan Attias, Gintare Karolina Dziugaite, Mahdi Haghifam, Roi Livni, Daniel Roy

In this work, we investigate the interplay between memorization and learning in the context of *stochastic convex optimization* (SCO). We define memorization via the information a learning algorithm reveals about its training data points. We then quantify this information using the framework of conditional mutual information (CMI) proposed by Steinke and Zakynthinou (2020). Our main result is a precise characterization of the tradeoff between the accuracy of a learning algorithm and its CMI, answering an open question posed by Livni (2023). We show that, in the $L^2$ Lipschitz--bounded setting and under strong convexity, every learner with an excess error $\epsilon$ has CMI bounded below by $\Omega(1/\epsilon^2)$ and $\Omega(1/\epsilon)$, respectively. We further demonstrate the essential role of memorization in learning problems in SCO by designing an adversary capable of accurately identifying a significant fraction of the training samples in specific SCO problems. Finally, we enumerate several implications of our results, such as a limitation of generalization bounds based on CMI and the incompressibility of samples in SCO problems.

Khashayar Gatmiry, Nikunj Saunshi, Sashank J. Reddi, Stefanie Jegelka, Sanjiv Kumar

Transformers to do reasoning and few-shot learning, without any fine-tuning, is widely conjectured to stem from their ability to implicitly simulate a multi-step algorithms -- such as gradient descent -- with their weights in a single forward pass. Recently, there has been progress in understanding this complex phenomenon from an expressivity point of view, by demonstrating that Transformers can express such multi-step algorithms. However, our knowledge about the more fundamental aspect of its learnability, beyond single layer models, is very limited. In particular, *can training Transformers enable convergence to algorithmic solutions*? In this work we resolve this for in context linear regression with linear looped Transformers -- a multi-layer model with weight sharing that is conjectured to have an inductive bias to learn fix-point iterative algorithms. More specifically, for this setting we show that the global minimizer of the population training loss implements multi-step preconditioned gradient descent, with a preconditioner that adapts to the data distribution. Furthermore, we show a fast convergence for gradient flow on the regression loss, despite the non-convexity of the landscape, by proving a novel gradient dominance condition. To our knowledge, this is the first theoretical analysis for multi-layer Transformer in this setting. We further validate our theoretical findings through synthetic experiments.

Theory · Probabilistic Methods

Emile van Krieken, Pasquale Minervini, Edoardo Ponti, Antonio Vergari

State-of-the-art neurosymbolic learning systems use probabilistic reasoning to guide neural networks towards predictions that conform to logical constraints. Many such systems assume that the probabilities of the considered symbols are conditionally independent given the input to simplify learning and reasoning. We study and criticise this assumption, highlighting how it can hinder optimisation and prevent uncertainty quantification. We prove that loss functions bias conditionally independent neural networks to become overconfident in their predictions. As a result, they are unable to represent uncertainty over multiple valid options. Furthermore, we prove that the minima of such loss functions are usually highly disconnected and non-convex, and thus difficult to optimise. Our theoretical analysis gives the foundation for replacing the conditional independence assumption and designing more expressive neurosymbolic probabilistic models.

Theory · Optimization

Weijia Shao

This work proposes an algorithm improving the dimensionality dependence for gradient-free optimisation over cross-polytopes, which has many applications such as adversarial attacks, explainable AI and sparse regression. For bandit convex optimisation with two-point feedback over cross-polytopes, the state-of-the-art algorithms have a dimensionality dependence of $\mathcal{O}(\sqrt{d\log d})$, while the known lower bound is of the form $\Omega(\sqrt{d(\log d)^{-1}})$. We propose a mirror descent algorithm equipped with a symmetric version of the negative $\frac{1}{2}$-Tsallis entropy. Combined with an $\ell_1$-ellipsoidal smoothing-based gradient estimator, the proposed algorithm guarantees a dimensionality dependence on $\mathcal{O}(\sqrt{d})$, which improves the state-of-the-art algorithms by a factor of $\sqrt{\log d}$. The idea can be further applied to optimising non-smooth and non-convex functions. We propose an algorithm with a convergence depending on $\mathcal{O}(d)$, which is the best-known dimensionality dependence.

Reinforcement Learning · Deep RL

Amrit Singh Bedi, Anjaly Parayil, Junyu Zhang, Mengdi Wang, Alec Koppel

Reinforcement learning is a framework for interactive decision-making with incentives sequentially revealed across time without a system dynamics model. Due to its scaling to continuous spaces, we focus on policy search where one iteratively improves a parameterized policy with stochastic policy gradient (PG) updates. In tabular Markov Decision Problems (MDPs), under persistent exploration and suitable parameterization, global optimality may be obtained. By contrast, in continuous space, the non-convexity poses a pathological challenge as evidenced by existing convergence results being mostly limited to stationarity or arbitrary local extrema. To close this gap, we step towards persistent exploration in continuous space through policy parameterizations defined by distributions of heavier tails defined by tail-index parameter $\alpha$, which increases the likelihood of jumping in state space. Doing so invalidates smoothness conditions of the score function common to PG. Thus, we establish how the convergence rate to stationarity depends on the policy's tail index $\alpha$, a H{{\"o}}lder continuity parameter, integrability conditions, and an exploration tolerance parameter introduced here for the first time. Further, we characterize the dependence of the set of local maxima on the tail index through an exit and transition time analysis of a suitably defined Markov chain, identifying that policies associated with L{{\'e}}vy Processes of a heavier tail converge to wider peaks. This phenomenon yields improved stability to perturbations in supervised learning, which we corroborate also manifests in improved performance of policy search, especially when myopic and farsighted incentives are misaligned.

Optimization · Stochastic

Haitong LIU, Qiang Li, Hoi To Wai

This paper studies the performative prediction problem where a learner aims to minimize the expected loss with a decision-dependent data distribution. Such setting is motivated when outcomes can be affected by the prediction model, e.g., in strategic classification. We consider a state-dependent setting where the data distribution evolves according to an underlying controlled Markov chain. We focus on stochastic derivative free optimization (DFO) where the learner is given access to a loss function evaluation oracle with the above Markovian data. We propose a two-timescale DFO($\lambda$) algorithm that features (i) a sample accumulation mechanism that utilizes every observed sample to estimate the overall gradient of performative risk, and (ii) a two-timescale diminishing step size that balances the rates of DFO updates and bias reduction. Under a general non-convex optimization setting, we show that DFO($\lambda$) requires ${\cal O}( 1 /\epsilon^3)$ samples (up to a log factor) to attain a near-stationary solution with expected squared gradient norm less than $\epsilon > 0$. Numerical experiments verify our analysis.

Deep Learning · Theory

Artem Vysogorets, Anna Dawid, Julia Kempe

The second-order properties of the training loss have a massive impact on the optimization dynamics of deep learning models. Fort & Scherlis (2019) discovered that a large excess of positive curvature and local convexity of the loss Hessian is associated with highly trainable initial points located in a region coined the "Goldilocks zone". Only a handful of subsequent studies touched upon this relationship, so it remains largely unexplained. In this paper, we present a rigorous and comprehensive analysis of the Goldilocks zone for homogeneous neural networks. In particular, we derive the fundamental condition resulting in excess of positive curvature of the loss, explaining and refining its conventionally accepted connection to the initialization norm. Further, we relate the excess of positive curvature to model confidence, low initial loss, and a previously unknown type of vanishing cross-entropy loss gradient. To understand the importance of excessive positive curvature for trainability of deep networks, we optimize fully-connected and convolutional architectures outside the Goldilocks zone and analyze the emergent behaviors. We find that strong model performance is not perfectly aligned with the Goldilocks zone, calling for further research into this relationship.

Hossein Esfandiari, Praneeth Kacham, Vahab Mirrokni, David Woodruff, Peilin Zhong

We study streaming algorithms for the $\ell_p$ subspace approximation problem. Given points $a_1, \ldots, a_n$ as an insertion-only stream and a rank parameter $k$, the $\ell_p$ subspace approximation problem is to find a $k$-dimensional subspace $V$ such that $(\sum_{i=1}^n d(a_i, V)^p)^{1/p}$ is minimized, where $d(a, V)$ denotes the Euclidean distance between $a$ and $V$ defined as $\min_{v \in V} ||a - v||$. When $p = \infty$, we need to find a subspace $V$ that minimizes $\max_i d(a_i, V)$. For $\ell_{\infty}$ subspace approximation, we give a deterministic strong coreset construction algorithm and show that it can be used to compute a $\mathrm{poly}(k, \log n)$ approximate solution. We show that the distortion obtained by our coreset is nearly tight for any sublinear space algorithm. For $\ell_p$ subspace approximation, we show that suitably scaling the points and then using our $\ell_{\infty}$ coreset construction, we can compute a $\mathrm{poly}(k, \log n)$ approximation. Our algorithms are easy to implement and run very fast on large datasets. We also use our strong coreset construction to improve the results in a recent work of Woodruff and Yasuda (FOCS 2022) which gives streaming algorithms for high-dimensional geometric problems such as width estimation, convex hull estimation, and volume estimation.

Social Aspects · Privacy-preserving Statistics and Machine Learning

Changyu Gao, Andrew Lowy, Xingyu Zhou, Stephen Wright

We revisit the problem of federated learning (FL) with private data from people who do not trust the server or other silos/clients. In this context, every silo (e.g. hospital) has data from several people (e.g. patients) and needs to protect the privacy of each person's data (e.g. health records), even if the server and/or other silos try to uncover this data. Inter-Silo Record-Level Differential Privacy (ISRL-DP) prevents each silo's data from being leaked, by requiring that silo $i$'s *communications* satisfy item-level differential privacy. Prior work (Lowy & Razaviyayn, 2023a) characterized the optimal excess risk bounds for ISRL-DP algorithms with *homogeneous* (i.i.d.) silo data and convex loss functions. However, two important questions were left open: 1) Can the same excess risk bounds be achieved with *heterogeneous* (non-i.i.d.) silo data? 2) Can the optimal risk bounds be achieved with *fewer communication rounds*? In this paper, we give positive answers to both questions. We provide novel ISRL-DP FL algorithms that achieve the optimal excess risk bounds in the presence of heterogeneous silo data. Moreover, our algorithms are more *communication-efficient* than the prior state-of-the-art. For smooth loss functions, our algorithm achieves the *optimal* excess risk bound and has *communication complexity that matches the non-private lower bound*. Additionally, our algorithms are more *computationally efficient* than the previous state-of-the-art.

Social Aspects · Privacy-preserving Statistics and Machine Learning

Zhenlong Liu, Lei Feng, HUIPING ZHUANG, Xiaofeng Cao, Hongxin Wei

Machine learning models are susceptible to membership inference attacks (MIAs), which aim to infer whether a sample is in the training set. Existing work utilizes gradient ascent to enlarge the loss variance of training data, alleviating the privacy risk. However, optimizing toward a reverse direction may cause the model parameters to oscillate near local minima, leading to instability and suboptimal performance. In this work, we propose a novel method -- Convex Concave Loss (CCL), which enables a high variance of training loss distribution by gradient descent. Our method is motivated by the theoretical analysis that convex losses tend to decrease the loss variance during training. Thus, our key idea behind CCL is to reduce the convexity of loss functions with a concave term. Trained with CCL, neural networks produce losses with high variance for training data, reinforcing the defense against MIAs. Extensive experiments demonstrate the superiority of CCL, achieving a state-of-the-art balance in the privacy-utility trade-off.

Probabilistic Methods · Monte Carlo and Sampling Methods

Haoyang Zheng, Hengrong Du, Qi Feng, Wei Deng, Guang Lin

Replica exchange stochastic gradient Langevin dynamics (reSGLD) is an effective sampler for non-convex learning in large-scale datasets. However, the simulation may encounter stagnation issues when the high-temperature chain delves too deeply into the distribution tails. To tackle this issue, we propose reflected reSGLD (r2SGLD): an algorithm tailored for constrained non-convex exploration by utilizing reflection steps within a bounded domain. Theoretically, we observe that reducing the diameter of the domain enhances mixing rates, exhibiting a *quadratic* behavior. Empirically, we test its performance through extensive experiments, including identifying dynamical systems with physical constraints, simulations of constrained multi-modal distributions, and image classification tasks. The theoretical and empirical findings highlight the crucial role of constrained exploration in improving the simulation efficiency.

Optimization · Non-Convex

Luis Scoccola, Siddharth Setlur, David Loiseaux, Mathieu Carrière, Steve Oudot

Real-valued functions on geometric data---such as node attributes on a graph---can be optimized using descriptors from persistent homology, allowing the user to incorporate topological terms in the loss function. When optimizing a single real-valued function (the one-parameter setting), there is a canonical choice of descriptor for persistent homology: the barcode. The operation mapping a real-valued function to its barcode is differentiable almost everywhere, and the convergence of gradient descent for losses using barcodes is relatively well understood. When optimizing a vector-valued function (the multiparameter setting), there is no unique choice of descriptor for multiparameter persistent homology, and many distinct descriptors have been proposed. This calls for the development of a general framework for differentiability and optimization that applies to a wide range of multiparameter homological descriptors. In this article, we develop such a framework and show that it encompasses well-known descriptors of different flavors, such as signed barcodes and the multiparameter persistence landscape. We complement the theory with numerical experiments supporting the idea that optimizing multiparameter homological descriptors can lead to improved performances compared to optimizing one-parameter descriptors, even when using the simplest and most efficiently computable multiparameter descriptors.

Optimization · Large Scale, Parallel and Distributed

Yuxing Liu, Lesi Chen, Luo Luo

This paper studies decentralized optimization problem, where the local objective on each node is an average of a finite set of convex functions and the global function is strongly convex. We propose an efficient stochastic variance reduced first-order method that allows the different nodes to establish their stochastic local gradient estimator with different mini-batch sizes per iteration. We prove the upper bound on the computation time of the proposed method contains the dependence on the global condition number, which is sharper than the previous results that only depend on the local condition numbers. Compared with the state-of-the-art methods, we also show that our method requires less local incremental first-order oracle calls and comparable communication cost. We further perform numerical experiments to validate the advantage of our method.

Tianshu Chu, Dachuan Xu, Wei Yao, Jin Zhang

While stochastic bilevel optimization methods have been extensively studied for addressing large-scale nested optimization problems in machine learning, it remains an open question whether the optimal complexity bounds for solving bilevel optimization are the same as those in single-level optimization. Our main result resolves this question: SPABA, an adaptation of the PAGE method for nonconvex optimization in (Li et al., 2021) to the bilevel setting, can achieve optimal sample complexity in both the finite-sum and expectation settings. We show the optimality of SPABA by proving that there is no gap in complexity analysis between stochastic bilevel and single-level optimization when implementing PAGE. Notably, as indicated by the results of (Dagréou et al., 2022), there might exist a gap in complexity analysis when implementing other stochastic gradient estimators, like SGD and SAGA. In addition to SPABA, we propose several other single-loop stochastic bilevel algorithms, that either match or improve the state-of-the-art sample complexity results, leveraging our convergence rate and complexity analysis. Numerical experiments demonstrate the superior practical performance of the proposed methods.

Reinforcement Learning

Siqiao Mu, Diego Klabjan

Since the objective functions of reinforcement learning problems are typically highly nonconvex, it is desirable that policy gradient, the most popular algorithm, escapes saddle points and arrives at second-order stationary points. Existing results only consider vanilla policy gradient algorithms with unbiased gradient estimators, but practical implementations under the infinite-horizon discounted reward setting are biased due to finite-horizon sampling. Moreover, actor-critic methods, whose second-order convergence has not yet been established, are also biased due to the critic approximation of the value function. We provide a novel second-order analysis of biased policy gradient methods, including the vanilla gradient estimator computed from Monte-Carlo sampling of trajectories as well as the double-loop actor-critic algorithm, where in the inner loop the critic improves the approximation of the value function via TD(0) learning. Separately, we also establish the convergence of TD(0) on Markov chains irrespective of initial state distribution.

Optimization

Fabian Schaipp, Ruben Ohana, Michael Eickenberg, Aaron Defazio, Robert Gower

Training a modern machine learning architecture on a new task requires extensive learning-rate tuning, which comes at a high computational cost. Here we develop new Polyak-type adaptive learning rates that can be used on top of any momentum method, and require less tuning to perform well. We first develop MoMo, a **Mo**mentum **Mo**del based adaptive learning rate for SGD-M (stochastic gradient descent with momentum). MoMo uses momentum estimates of the batch losses and gradients sampled at each iteration to build a model of the loss function. Our model also makes use of any known lower bound of the loss function by using truncation, e.g. most losses are lower-bounded by zero. The models is then approximately minimized at each iteration to compute the next step. We show how MoMo can be used in combination with any momentum-based method, and showcase this by developing MoMo-Adam - which is Adam with our new model-based adaptive learning rate. We show that MoMo attains a $\mathcal{O}(1/\sqrt{K})$ convergence rate for convex problems with interpolation, needing knowledge of no problem-specific quantities other than the optimal value. Additionally, for losses with unknown lower bounds, we develop on-the-fly estimates of a lower bound, that are incorporated in our model. We demonstrate that MoMo and MoMo-Adam improve over SGD-M and Adam in terms of robustness to hyperparameter tuning for training image classifiers on MNIST, CIFAR, and Imagenet, for recommender systems on the Criteo dataset, for a transformer model on the translation task IWSLT14, and for a diffusion model.