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Optimization

Andreas Kontogiannis, Vasilis Pollatos, Sotiris Kanellopoulos, Panayotis Mertikopoulos, Aris Pagourtzis, Ioannis Panageas

Non-convex minimization problems are universally considered hard, and even guaranteeing that a computed solution is locally minimizing is known to be NP-hard. In this general context, our paper focuses on the problem of finding stationary points that satisfy an approximate second-order optimality condition, which serves to exclude strict saddles and other non-minimizing stationary points. Our main result is that the problem of finding approximate second-order stationary points (SOSPs) is PLS-complete, i.e., of the same complexity as the problem of finding first-order stationary points (FOSPs), thus resolving an open question in the field. In particular, our results imply that, under the widely believed complexity conjecture that PLS $\neq$ FNP, finding approximate SOSPs in unconstrained domains is *easier* than in constrained domains, which is known to be NP-hard. This comes in stark contrast with earlier results which implied that, unless PLS = CLS, finding approximate FOSPs in unconstrained domains is *harder* than in constrained domains.

Optimization · Non-Convex

Anastasiia Koloskova, Nikita Doikov, Sebastian Stich, Martin Jaggi

In machine learning and neural network optimization, algorithms like incremental gradient, single shuffle SGD, and random reshuffle SGD are popular due to their cache-mismatch efficiency and good practical convergence behavior. However, their optimization properties in theory, especially for non-convex smooth functions, remain incompletely explored. This paper delves into the convergence properties of SGD algorithms with arbitrary data ordering, within a broad framework for non-convex smooth functions. Our findings show enhanced convergence guarantees for incremental gradient and single shuffle SGD. Particularly if $n$ is the training set size, we improve $n$ times the optimization term of convergence guarantee to reach accuracy $\epsilon$ from $O \left( \frac{n}{\epsilon} \right)$ to $O \left( \frac{1}{\epsilon}\right)$.

Theory

Juno Kim, Taiji Suzuki

Large language models based on the Transformer architecture have demonstrated impressive capabilities to learn in context. However, existing theoretical studies on how this phenomenon arises are limited to the dynamics of a single layer of attention trained on linear regression tasks. In this paper, we study the optimization of a Transformer consisting of a fully connected layer followed by a linear attention layer. The MLP acts as a common nonlinear representation or feature map, greatly enhancing the power of in-context learning. We prove in the mean-field and two-timescale limit that the infinite-dimensional loss landscape for the distribution of parameters, while highly nonconvex, becomes quite benign. We also analyze the second-order stability of mean-field dynamics and show that Wasserstein gradient flow almost always avoids saddle points. Furthermore, we establish novel methods for obtaining concrete improvement rates both away from and near critical points. This represents the first saddle point analysis of mean-field dynamics in general and the techniques are of independent interest.

Deep Learning · Theory

Sungyoon Kim, Mert Pilanci

In this paper, we study the optimality gap between two-layer ReLU networks regularized with weight decay and their convex relaxations. We show that when the training data is random, the relative optimality gap between the original problem and its relaxation can be bounded by a factor of O(√log n), where n is the number of training samples. A simple application leads to a tractable polynomial-time algorithm that is guaranteed to solve the original non-convex problem up to a logarithmic factor. Moreover, under mild assumptions, we show that local gradient methods converge to a point with low training loss with high probability. Our result is an exponential improvement compared to existing results and sheds new light on understanding why local gradient methods work well.

Ahmed Khaled, Chi Jin

Large-scale machine learning problems make the cost of hyperparameter tuning ever more prohibitive. This creates a need for algorithms that can tune themselves on-the-fly. We formalize the notion of *``tuning-free''* algorithms that can match the performance of optimally-tuned optimization algorithms up to polylogarithmic factors given only loose hints on the relevant problem parameters. We consider in particular algorithms that can match optimally-tuned Stochastic Gradient Descent (SGD). When the domain of optimization is bounded, we show tuning-free matching of SGD is possible and achieved by several existing algorithms. We prove that for the task of minimizing a convex and smooth or Lipschitz function over an unbounded domain, tuning-free optimization is impossible. We discuss conditions under which tuning-free optimization is possible even over unbounded domains. In particular, we show that the recently proposed DoG and DoWG algorithms are tuning-free when the noise distribution is sufficiently well-behaved. For the task of finding a stationary point of a smooth and potentially nonconvex function, we give a variant of SGD that matches the best-known high-probability convergence rate for tuned SGD at only an additional polylogarithmic cost. However, we also give an impossibility result that shows no algorithm can hope to match the optimal expected convergence rate for tuned SGD with high probability.

Optimization · Convex

Phillip Kerger, Marco Molinaro, Hongyi Jiang, Amitabh Basu

Given *any* algorithm for convex optimization that uses exact first-order information (i.e., function values and subgradients), we show how to use such an algorithm to solve the problem with access to *inexact* first-order information. This is done in a ``black-box'' manner without knowledge of the internal workings of the algorithm. This complements previous work that considers the performance of specific algorithms like (accelerated) gradient descent with inexact information. In particular, our results apply to a wider range of algorithms beyond variants of gradient descent, e.g., projection-free methods, cutting-plane methods, or any other first-order methods formulated in the future. Further, they also apply to algorithms that handle structured nonconvexities like mixed-integer decision variables.

Optimization · Stochastic

Wei Jiang, Sifan Yang, Wenhao Yang, Yibo Wang, Yuanyu Wan, Lijun Zhang

This paper investigates projection-free algorithms for stochastic constrained multi-level optimization. In this context, the objective function is a nested composition of several smooth functions, and the decision set is closed and convex. Existing projection-free algorithms for solving this problem suffer from two limitations: 1) they solely focus on the gradient mapping criterion and fail to match the optimal sample complexities in unconstrained settings; 2) their analysis is exclusively applicable to non-convex functions, without considering convex and strongly convex objectives. To address these issues, we introduce novel projection-free variance reduction algorithms and analyze their complexities under different criteria. For gradient mapping, our complexities improve existing results and match the optimal rates for unconstrained problems. For the widely-used Frank-Wolfe gap criterion, we provide theoretical guarantees that align with those for single-level problems. Additionally, by using a stage-wise adaptation, we further obtain complexities for convex and strongly convex functions. Finally, numerical experiments on different tasks demonstrate the effectiveness of our methods.

Theory · Online Learning and Bandits

Yu-Hu Yan, Peng Zhao, Zhi-Hua Zhou

Online control with non-stochastic disturbances and adversarially chosen convex cost functions, referred to as online non-stochastic control, has recently attracted increasing attention. We study online non-stochastic control with partial feedback, where learners can only access partially observed states and partially informed (bandit) costs. The problem setting arises naturally in real-world decision-making applications and strictly generalizes exceptional cases studied disparately by previous works. We propose the first online algorithm for this problem, with an $\tilde{O}(T^{3/4})$ regret competing with the best policy in hindsight, where $T$ denotes the time horizon and the $\tilde{O}(\cdot)$-notation omits the poly-logarithmic factors in $T$. To further enhance the algorithms' robustness to changing environments, we then design a novel method with a two-layer structure to optimize the dynamic regret, a more challenging measure that competes with time-varying policies. Our method is based on the online ensemble framework by treating the controller above as the base learner. On top of that, we design two different meta-combiners to simultaneously handle the unknown variation of environments and the memory issue arising from the online control. We prove that the two resulting algorithms enjoy $\tilde{O}(T^{3/4}(1+P_T)^{1/2})$ and $\tilde{O}(T^{3/4}(1+P_T)^{1/4}+T^{5/6})$ dynamic regret respectively, where $P_T$ measures the environmental non-stationarity. Our results are further extended to unknown transition matrices. Finally, empirical studies in both synthetic linear and simulated nonlinear tasks validate our method's effectiveness, thus supporting the theoretical findings.

Optimization · Large Scale, Parallel and Distributed

Feihu Huang, jianyu zhao

Decentralized learning recently has received increasing attention in machine learning due to its advantages in implementation simplicity and system robustness, data privacy. Meanwhile, the adaptive gradient methods show superior performances in many machine learning tasks such as training neural networks. Although some works focus on studying decentralized optimization algorithms with adaptive learning rates, these adaptive decentralized algorithms still suffer from high sample complexity. To fill these gaps, we propose a class of faster adaptive decentralized algorithms (i.e., AdaMDOS and AdaMDOF) for distributed nonconvex stochastic and finite-sum optimization, respectively. Moreover, we provide a solid convergence analysis framework for our methods. In particular, we prove that our AdaMDOS obtains a near-optimal sample complexity of $\tilde{O}(\epsilon^{-3})$ for finding an $\epsilon$-stationary solution of nonconvex stochastic optimization. Meanwhile, our AdaMDOF obtains a near-optimal sample complexity of $O(\sqrt{n}\epsilon^{-2})$ for finding an $\epsilon$-stationary solution of for nonconvex finite-sum optimization, where $n$ denotes the sample size. To the best of our knowledge, our AdaMDOF algorithm is the first adaptive decentralized algorithm for nonconvex finite-sum optimization. Some experimental results demonstrate efficiency of our algorithms.

Deep Learning · Sequential Models, Time series

Ignacio Hounie, Javier Porras-Valenzuela, Alejandro Ribeiro

Several applications in time series forecasting require predicting multiple steps ahead. Despite the vast amount of literature in the topic, both classical and recent deep learning based approaches have mostly focused on minimising performance averaged over the predicted window. We observe that this can lead to disparate distributions of errors across forecasting steps, especially for recent transformer architectures trained on popular forecasting benchmarks. That is, optimising performance on average can lead to undesirably large errors at specific time-steps. In this work, we present a Constrained Learning approach for long-term time series forecasting that aims to find the best model in terms of average performance that respects a user-defined upper bound on the loss at each time-step. We call our approach loss shaping constraints because it imposes constraints on the loss at each time step, and leverage recent duality results to show that despite its non-convexity, the resulting problem has a bounded duality gap. We propose a practical primal-dual algorithm to tackle it, and demonstrate that the proposed approach exhibits competitive average performance in time series forecasting benchmarks, while shaping the distribution of errors across the predicted window.

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

Yifei He, Shiji Zhou, Guojun Zhang, Hyokun Yun, Yi Xu, Belinda Zeng, Trishul Chilimbi, Han Zhao

Multi-task learning (MTL) considers learning a joint model for multiple tasks by optimizing a convex combination of all task losses. To solve the optimization problem, existing methods use an adaptive weight updating scheme, where task weights are dynamically adjusted based on their respective losses to prioritize difficult tasks. However, these algorithms face a great challenge whenever *label noise* is present, in which case excessive weights tend to be assigned to noisy tasks that have relatively large Bayes optimal errors, thereby overshadowing other tasks and causing performance to drop across the board. To overcome this limitation, we propose **M**ulti-**T**ask **L**earning with **Excess** Risks (ExcessMTL), an excess risk-based task balancing method that updates the task weights by their distances to convergence instead. Intuitively, ExcessMTL assigns higher weights to worse-trained tasks that are further from convergence. To estimate the excess risks, we develop an efficient and accurate method with Taylor approximation. Theoretically, we show that our proposed algorithm achieves convergence guarantees and Pareto stationarity. Empirically, we evaluate our algorithm on various MTL benchmarks and demonstrate its superior performance over existing methods in the presence of label noise. Our code is available at https://github.com/yifei-he/ExcessMTL.

Miscellaneous Aspects of Machine Learning · Representation Learning

Christopher Harker, Aditya Bhaskara

Graph embeddings have emerged as a powerful tool for understanding the structure of graphs. Unlike classical spectral methods, recent methods such as DeepWalk, Node2Vec, etc. are based on solving nonlinear optimization problems on the graph, using local information obtained by performing random walks. These techniques have empirically been shown to produce ``better'' embeddings than their classical counterparts. However, due to their reliance on solving a nonconvex optimization problem, obtaining theoretical guarantees on the properties of the solution has remained a challenge, even for simple classes of graphs. In this work, we show convergence properties for the DeepWalk algorithm on graphs obtained from the Stochastic Block Model (SBM). Despite being simplistic, the SBM has proved to be a classic model for analyzing the behavior of algorithms on large graphs. Our results mirror the existing ones for spectral embeddings on SBMs, showing that even in the case of one-dimensional embeddings, the output of the DeepWalk algorithm provably recovers the cluster structure with high probability.

Optimization · Non-Convex

Andi Han, Pratik Kumar Jawanpuria, Bamdev Mishra

Many machine learning applications are naturally formulated as optimization problems on Riemannian manifolds. The main idea behind Riemannian optimization is to maintain the feasibility of the variables while moving along a descent direction on the manifold. This results in updating all the variables at every iteration. In this work, we provide a general framework for developing computationally efficient coordinate descent (CD) algorithms on matrix manifolds that allows updating only a few variables at every iteration while adhering to the manifold constraint. In particular, we propose CD algorithms for various manifolds such as Stiefel, Grassmann, (generalized) hyperbolic, symplectic, and symmetric positive (semi)definite. While the cost per iteration of the proposed CD algorithms is low, we further develop a more efficient variant via a first-order approximation of the objective function. We analyze their convergence and complexity, and empirically illustrate their efficacy in several applications.

Theory · Online Learning and Bandits

Seok-Ju Hahn, Gi-Soo Kim, Junghye Lee

In traditional federated learning, a single global model cannot perform equally well for all clients. Therefore, the need to achieve the *client-level fairness* in federated system has been emphasized, which can be realized by modifying the static aggregation scheme for updating the global model to an adaptive one, in response to the local signals of the participating clients. Our work reveals that existing fairness-aware aggregation strategies can be unified into an online convex optimization framework, in other words, a central server's *sequential decision making* process. To enhance the decision making capability, we propose simple and intuitive improvements for suboptimal designs within existing methods, presenting $\texttt{AAggFF}$. Considering practical requirements, we further subdivide our method tailored for the *cross-device* and the *cross-silo* settings, respectively. Theoretical analyses guarantee sublinear regret upper bounds for both settings: $\mathcal{O}(\sqrt{T \log{K}})$ for the cross-device setting, and $\mathcal{O}(K \log{T})$ for the cross-silo setting, with $K$ clients and $T$ federation rounds. Extensive experiments demonstrate that the federated system equipped with $\texttt{AAggFF}$ achieves better degree of client-level fairness than existing methods in both practical settings.

Theory · Game Theory

Mukesh Ghimire, Lei Zhang, Zhe Xu, Yi Ren

We study zero-sum differential games with state constraints and one-sided information, where the informed player (Player 1) has a categorical payoff type unknown to the uninformed player (Player 2). The goal of Player 1 is to minimize his payoff without violating the constraints, while that of Player 2 is to violate the state constraints if possible, or to maximize the payoff otherwise. One example of the game is a man-to-man matchup in football. Without state constraints, Cardaliaguet (2007) showed that the value of such a game exists and is convex to the common belief of players. Our theoretical contribution is an extension of this result to games with state constraints and the derivation of the primal and dual subdynamic principles necessary for computing behavioral strategies. Different from existing works that are concerned about the scalability of no-regret learning in games with discrete dynamics, our study reveals the underlying structure of strategies for belief manipulation resulting from information asymmetry and state constraints. This structure will be necessary for scalable learning on games with continuous actions and long time windows. We use a simplified football game to demonstrate the utility of this work, where we reveal player positions and belief states in which the attacker should (or should not) play specific random deceptive moves to take advantage of information asymmetry, and compute how the defender should respond.

Theory · Online Learning and Bandits

Dan Garber, Ben Kretzu

We consider the setting of online convex optimization with adversarial time-varying constraints in which actions must be feasible w.r.t. a fixed constraint set, and are also required on average to approximately satisfy additional time-varying constraints. Motivated by scenarios in which the fixed feasible set (hard constraint) is difficult to project on, we consider projection-free algorithms that access this set only through a linear optimization oracle (LOO). We present an algorithm that, on a sequence of length $T$ and using overall $T$ calls to the LOO, guarantees $\tilde{O}(T^{3/4})$ regret w.r.t. the losses and $O(T^{7/8})$ constraints violation (ignoring all quantities except for $T$). In particular, these bounds hold w.r.t. any interval of the sequence. This algorithm however also requires access to an oracle for minimizing a strongly convex nonsmooth function over a Euclidean ball. We present a more efficient algorithm that does not require the latter optimization oracle but only first-order access to the time-varying constraints, and achieves similar bounds w.r.t. the entire sequence. We extend the latter to the setting of bandit feedback and obtain similar bounds (as a function of $T$) in expectation.

Optimization · Stochastic

Yuan Gao, Anton Rodomanov, Sebastian Stich

The stochastic proximal gradient method is a powerful generalization of the widely used stochastic gradient descent (SGD) method and has found numerous applications in Machine Learning. However, it is notoriously known that this method fails to converge in non-convex settings where the stochastic noise is significant (i.e. when only small or bounded batch sizes are used). In this paper, we focus on the stochastic proximal gradient method with Polyak momentum. We prove this method attains an optimal convergence rate for non-convex composite optimization problems, regardless of batch size. Additionally, we rigorously analyze the variance reduction effect of the Polyak momentum in the composite optimization setting and we show the method also converges when the proximal step can only be solved inexactly. Finally, we provide numerical experiments to validate our theoretical results.

Theory · Optimization

Wenzhi Gao, Qi Deng

This paper considers stochastic weakly convex optimization without the standard Lipschitz continuity assumption. Based on new adaptive regularization (stepsize) strategies, we show that a wide class of stochastic algorithms, including the stochastic subgradient method, preserve the $\mathcal{O} ( 1 / \sqrt{K})$ convergence rate with constant failure rate. Our analyses rest on rather weak assumptions: the Lipschitz parameter can be either bounded by a general growth function of $\\|x\\|$ or locally estimated through independent random samples. Numerical experiments demonstrate the efficiency and robustness of our proposed stepsize policies.

Optimization · Large Scale, Parallel and Distributed

Hongchang Gao

Federated compositional optimization has been actively studied in the past few years. However, existing methods mainly focus on the two-level compositional optimization problem, which cannot be directly applied to the multi-level counterparts. Moreover, the convergence rate of existing federated two-level compositional optimization learning algorithms fails to achieve linear speedup with respect to the number of workers under heterogeneous settings. After identifying the reason for this failure, we developed a novel federated stochastic multi-level compositional optimization algorithm by introducing a novel Jacobian-vector product estimator. This innovation mitigates both the heterogeneity issue and the communication efficiency issue simultaneously. We then theoretically proved that our algorithm can achieve the level-independent and linear speedup convergence rate for nonconvex problems. To our knowledge, this is the first time that a federated learning algorithm can achieve such a favorable convergence rate for multi-level compositional problems. Moreover, experimental results confirm the efficacy of our algorithm.

Deep Learning · Everything Else

Mateusz Gabor, Tomasz Piotrowski, Renato L. G. Cavalcante

Deep equilibrium (DEQ) models are widely recognized as a memory efficient alternative to standard neural networks, achieving state-of-the-art performance in language modeling and computer vision tasks. These models solve a fixed point equation instead of explicitly computing the output, which sets them apart from standard neural networks. However, existing DEQ models often lack formal guarantees of the existence and uniqueness of the fixed point, and the convergence of the numerical scheme used for computing the fixed point is not formally established. As a result, DEQ models are potentially unstable in practice. To address these drawbacks, we introduce a novel class of DEQ models called positive concave deep equilibrium (pcDEQ) models. Our approach, which is based on nonlinear Perron-Frobenius theory, enforces nonnegative weights and activation functions that are concave on the positive orthant. By imposing these constraints, we can easily ensure the existence and uniqueness of the fixed point without relying on additional complex assumptions commonly found in the DEQ literature, such as those based on monotone operator theory in convex analysis. Furthermore, the fixed point can be computed with the standard fixed point algorithm, and we provide theoretical guarantees of its geometric convergence, which, in particular, simplifies the training process. Experiments demonstrate the competitiveness of our pcDEQ models against other implicit models.