ICML 2023 Papers

In this work, we address the problem of Hessian inversion bias in distributed second-order optimization algorithms. We introduce a novel shrinkage-based estimator for the resolvent of gram matrices which is asymptotically unbiased, and characterize its non-asymptotic convergence rate in the isotropic case. We apply this estimator to bias correction of Newton steps in distributed second-order optimization algorithms, as well as randomized sketching based methods. We examine the bias present in the naive averaging-based distributed Newton's method using analytical expressions and contrast it with our proposed biasfree approach. Our approach leads to significant improvements in convergence rate compared to standard baselines and recent proposals, as shown through experiments on both real and synthetic datasets.

Policy optimization methods are popular reinforcement learning algorithms in practice and recent works have build theoretical foundation for them by proving $\sqrt{T}$ regret bounds even when the losses are adversarial. Such bounds are tight in the worst case but often overly pessimistic. In this work, we show that by carefully designing the regularizer, bonus terms, and learning rates, one can achieve a more favorable $\text{polylog}(T)$ regret bound when the losses are stochastic, without sacrificing the worst-case guarantee in the adversarial regime. Specifically, we show the first best of both worlds guarantee for policy optimization in tabular MDPs by leveraging either a Tsallis entropy or a Shannon entropy regularizer. Then we show that under known transitions, we can further obtain a first-order regret bound in the adversarial regime by leveraging the log barrier regularizer.

Sign Stochastic Gradient Descent (signSGD) is a communication-efficient stochastic algorithm that only uses the sign information of the stochastic gradient to update the model's weights. However, the existing convergence theory of signSGD either requires increasing batch sizes during training or assumes the gradient noise is symmetric and unimodal. Error feedback has been used to guarantee the convergence of signSGD under weaker assumptions at the cost of communication overhead. This paper revisits the convergence of signSGD and proves that momentum can remedy signSGD under weaker assumptions than previous techniques; in particular, our convergence theory does not require the assumption of bounded stochastic gradient or increased batch size. Our results resonate with echoes of previous empirical results where, unlike signSGD, signSGD with momentum maintains good performance even with small batch sizes. Another new result is that signSGD with momentum can achieve an improved convergence rate when the objective function is second-order smooth. We further extend our theory to signSGD with major vote and federated learning.

We propose a tuning-free dynamic SGD step size formula, which we call Distance over Gradients (DoG). The DoG step sizes depend on simple empirical quantities (distance from the initial point and norms of gradients) and have no ``learning rate'' parameter. Theoretically, we show that, for stochastic convex optimization, a slight variation of the DoG formula enjoys strong, high-probability parameter-free convergence guarantees and iterate movement bounds. Empirically, we consider a broad range of vision and language transfer learning tasks, and show that DoG's performance is close to that of SGD with tuned learning rate. We also propose a per-layer variant of DoG that generally outperforms tuned SGD, approaching the performance of tuned Adam. A PyTorch implementation of our algorithms is available at https://github.com/formll/dog.

Selecting hyperparameters in deep learning greatly impacts its effectiveness but requires manual effort and expertise. Recent works show that Bayesian model selection with Laplace approximations can allow to optimize such hyperparameters just like standard neural network parameters using gradients and on the training data. However, estimating a single hyperparameter gradient requires a pass through the entire dataset, limiting the scalability of such algorithms. In this work, we overcome this issue by introducing lower bounds to the linearized Laplace approximation of the marginal likelihood. In contrast to previous estimators, these bounds are amenable to stochastic-gradient-based optimization and allow to trade off estimation accuracy against computational complexity. We derive them using the function-space form of the linearized Laplace, which can be estimated using the neural tangent kernel. Experimentally, we show that the estimators can significantly accelerate gradient-based hyperparameter optimization.

Sampling approaches like Markov chain Monte Carlo were once popular for combinatorial optimization, but the inefficiency of classical methods and the need for problem-specific designs curtailed ongoing development. Recent work has favored data-driven approaches that mitigate the need for hand-craft heuristics, but these are often not usable as out-of-the-box solvers due to dependence on in-distribution training and limited scalability to large instances. In this paper, we revisit the idea of using sampling for combinatorial optimization, motivated by the significant recent advances of gradient-based discrete MCMC and new techniques for parallel neighborhood exploration on accelerators. Remarkably, we find that modern sampling strategies can leverage landscape information to provide general-purpose solvers that require no training and yet are competitive with state of the art combinatorial solvers. In particular, experiments on cover vertex selection, graph partition and routing demonstrate better speed-quality trade-offs over current learning based approaches, and sometimes even superior performance to commercial solvers and specialized algorithms.

We consider a fundamental class of assortment optimization problems in an offline data-driven setting. The firm does not know the underlying customer choice model but has access to an offline dataset consisting of the historically offered assortment set, customer choice, and revenue. The objective is to use the offline dataset to find an optimal assortment. Due to the combinatorial nature of assortment optimization, the problem of insufficient data coverage is likely to occur in the offline dataset. Therefore, designing a provably efficient offline learning algorithm becomes a significant challenge. To this end, based on the principle of pessimism, we propose a novel algorithm called Pessimistic ASsortment opTimizAtion (PASTA for short), which can correctly identify the optimal assortment by only requiring the offline data to cover the optimal assortment under general settings. In particular, we establish the first regret bound for the offline assortment optimization problem under the celebrated multinomial logit model (MNL). We also propose an efficient computational procedure to solve our pessimistic assortment optimization problem. Our numerical studies demonstrate the superiority of the proposed method over the existing baseline method.

When fitting the learning data of an individual to algorithm-like learning models, the observations are so dependent and non-stationary that one may wonder what the classical Maximum Likelihood Estimator (MLE) could do, even if it is the usual tool applied to experimental cognition. Our objective in this work is to show that the estimation of the learning rate cannot be efficient if the learning rate is constant in the classical Exp3 (Exponential weights for Exploration and Exploitation) algorithm. Secondly, we show that if the learning rate decreases polynomially with the sample size, then the prediction error and in some cases the estimation error of the MLE satisfy bounds in probability that decrease at a polynomial rate.

Auditing large language models for unexpected behaviors is critical to preempt catastrophic deployments, yet remains challenging. In this work, we cast auditing as an optimization problem, where we automatically search for input-output pairs that match a desired target behavior. For example, we might aim to find a non-toxic input that starts with ``Barack Obama'' that a model maps to a toxic output. This optimization problem is difficult to solve as the set of feasible points is sparse, the space is discrete, and the language models we audit are non-linear and high-dimensional. To combat these challenges, we introduce a discrete optimization algorithm, ARCA, that jointly and efficiently optimizes over inputs and outputs. Our approach automatically uncovers derogatory completions about celebrities (e.g. "Barack Obama is a legalized unborn" --> "child murderer"), produces French inputs that complete to English outputs, and finds inputs that generate a specific name. Our work offers a promising new tool to uncover models' failure-modes before deployment. Content Warning: This paper contains examples that may be offensive in nature.

This work develops analysis and algorithms for solving a class of bilevel optimization problems where the lower-level (LL) problems have linear constraints. Most of the existing approaches for constrained bilevel problems rely on value function-based approximate reformulations, which suffer from issues such as non-convex and non-differentiable constraints. In contrast, in this work, we develop an implicit gradient-based approach, which is easy to implement, and is suitable for machine learning applications. We first provide an in-depth understanding of the problem, by showing that the implicit objective for such problems is in general non-differentiable. However, if we add some small (linear) perturbation to the LL objective, the resulting implicit objective becomes differentiable almost surely. This key observation opens the door for developing (deterministic and stochastic) gradient-based algorithms similar to the state-of-the-art ones for unconstrained bi-level problems. We show that when the implicit function is assumed to be strongly-convex, convex, and weakly-convex, the resulting algorithms converge with guaranteed rate. Finally, we experimentally corroborate the theoretical findings and evaluate the performance of the proposed framework on numerical and adversarial learning problems.

We consider the optimization problem of the form $\min_{x \in \mathbb{R}^d} f(x) \triangleq \mathbb{E}[F(x;\xi)]$ , where the component $F(x;\xi)$ is $L$-mean-squared Lipschitz but possibly nonconvex and nonsmooth.The recently proposed gradient-free method requires at most $\mathcal{O}( L^4 d^{3/2} \epsilon^{-4} + \Delta L^3 d^{3/2} \delta^{-1} \epsilon^{-4})$ stochastic zeroth-order oracle complexity to find a $(\delta,\epsilon)$-Goldstein stationary point of objective function, where $\Delta = f(x_0) - \inf_{x \in \mathbb{R}^d} f(x)$ and $x_0$ is the initial point of the algorithm. This paper proposes a more efficient algorithm using stochastic recursive gradient estimators, which improves the complexity to $\mathcal{O}(L^3 d^{3/2} \epsilon^{-3}+ \Delta L^2 d^{3/2} \delta^{-1} \epsilon^{-3})$.

The success of deep learning is greatly attributed to stochastic gradient descent (SGD), yet it remains unclear how SGD finds well-generalized models. We demonstrate that SGD has an implicit regularization effect on the logit-weight Jacobian norm of neural networks. This regularization effect is weighted with the *impurity* of the probability output, and thus it is active in a certain phase of training. Moreover, based on these findings, we propose a novel optimization method that explicitly regularizes the Jacobian norm, which leads to similar performance as other state-of-the-art sharpness-aware optimization methods.

We focus on analyzing the classical stochastic projected gradient methods under a general dependent data sampling scheme for constrained smooth nonconvex optimization. We show the worst-case rate of convergence $\tilde{O}(t^{-1/4})$ and complexity $\tilde{O}(\varepsilon^{-4})$ for achieving an $\varepsilon$-near stationary point in terms of the norm of the gradient of Moreau envelope and gradient mapping. While classical convergence guarantee requires i.i.d. data sampling from the target distribution, we only require a mild mixing condition of the conditional distribution, which holds for a wide class of Markov chain sampling algorithms. This improves the existing complexity for the constrained smooth nonconvex optimization with dependent data from $\tilde{O}(\varepsilon^{-8})$ to $\tilde{O}(\varepsilon^{-4})$ with a significantly simpler analysis. We illustrate the generality of our approach by deriving convergence results with dependent data for stochastic proximal gradient methods, adaptive stochastic gradient algorithm AdaGrad and stochastic gradient algorithm with heavy ball momentum. As an application, we obtain first online nonnegative matrix factorization algorithms for dependent data based on stochastic projected gradient methods with adaptive step sizes and optimal rate of convergence.

Bilevel optimization has gained significant popularity in recent years due to its ability to formulate various machine learning problems. For instance, in meta-learning, the upper-level (UL) problem offers a good initialization for the lower-level (LL) model to facilitate adaptation. However, the decision variables can impact data features and outcomes, leading to the phenomenon known as performativity. In this work, we investigate the inclusion of decision-dependent distributions in bilevel optimization. Specifically, we consider the scenarios where the UL data distribution depends on the LL optimization variable, and the LL data distribution also depends on the UL decision variable. We first establish sufficient conditions for the existence of performatively stable (PS) solutions in this class of bilevel problems. Also, we propose efficient stochastic algorithms to find the PS point with theoretical convergence rate analysis and discuss the theoretical optimality of the obtained solution. Our theoretical analysis is corroborated through a series of numerical experiments, wherein we evaluate the performance of the bilevel performative prediction algorithms alongside non-performative counterparts in the context of meta strategic learning problems.

Incorporating a deep generative model as the prior distribution in inverse problems has established substantial success in reconstructing images from corrupted observations. Notwithstanding, the existing optimization approaches use gradient descent largely without adapting to the non-convex nature of the problem and can be sensitive to initial values, impeding further performance improvement. In this paper, we propose an efficient amortized optimization scheme for inverse problems with a deep generative prior. Specifically, the optimization task with high degrees of difficulty is decomposed into optimizing a sequence of much easier ones. We provide a theoretical guarantee of the proposed algorithm and empirically validate it on different inverse problems. As a result, our approach outperforms baseline methods qualitatively and quantitatively by a large margin.

Adversarial training is well-known to produce high-quality neural network models that are empirically robust against adversarial perturbations. Nevertheless, once a model has been adversarially trained, one often desires a certification that the model is truly robust against all future attacks. Unfortunately, when faced with adversarially trained models, all existing approaches have significant trouble making certifications that are strong enough to be practically useful. Linear programming (LP) techniques in particular face a ``convex relaxation barrier'' that prevent them from making high-quality certifications, even after refinement with mixed-integer linear programming (MILP) and branch-and-bound (BnB) techniques. In this paper, we propose a nonconvex certification technique, based on a low-rank restriction of a semidefinite programming (SDP) relaxation. The nonconvex relaxation makes strong certifications comparable to much more expensive SDP methods, while optimizing over dramatically fewer variables comparable to much weaker LP methods. Despite nonconvexity, we show how off-the-shelf local optimization algorithms can be used to achieve and to certify global optimality in polynomial time. Our experiments find that the nonconvex relaxation almost completely closes the gap towards exact certification of adversarially trained models.

SARSA, a classical on-policy control algorithm for reinforcement learning, is known to chatter when combined with linear function approximation: SARSA does not diverge but oscillates in a bounded region. However, little is known about how fast SARSA converges to that region and how large the region is. In this paper, we make progress towards this open problem by showing the convergence rate of projected SARSA to a bounded region. Importantly, the region is much smaller than the region that we project into, provided that the the magnitude of the reward is not too large. Existing works regarding the convergence of linear SARSA to a fixed point all require the Lipschitz constant of SARSA's policy improvement operator to be sufficiently small; our analysis instead applies to arbitrary Lipschitz constants and thus characterizes the behavior of linear SARSA for a new regime.

We consider online learning problems in the realizable setting, where there is a zero-loss solution, and propose new Differentially Private (DP) algorithms that obtain near-optimal regret bounds. For the problem of online prediction from experts, we design new algorithms that obtain near-optimal regret $O \big( \varepsilon^{-1} \mathsf{poly}(\log{d}) \big)$ where $d$ is the number of experts. This significantly improves over the best existing regret bounds for the DP non-realizable setting which are $O \big( \varepsilon^{-1} \min\big\{d, \sqrt{T\log d}\big\} \big)$. We also develop an adaptive algorithm for the small-loss setting with regret $(L^\star+ \varepsilon^{-1}) \cdot O(\mathsf{poly}(\log{d}))$ where $L^\star$ is the total loss of the best expert. Additionally, we consider DP online convex optimization in the realizable setting and propose an algorithm with near-optimal regret $O \big(\varepsilon^{-1} \mathsf{poly}(d) \big)$, as well as an algorithm for the smooth case with regret $O \big( (\sqrt{Td}/\varepsilon)^{2/3} \big)$, both significantly improving over existing bounds in the non-realizable regime.

We study variants of the online linear optimization (OLO) problem with bandit feedback, where the algorithm has access to external information about the unknown cost vector. Our motivation is the recent body of work on using such ``hints'' towards improving regret bounds for OLO problems in the full-information setting. Unlike in the full-information OLO setting, with bandit feedback, we first show that one cannot improve the standard regret bounds of $\tilde{O}(\sqrt{T})$ by using hints, even if they are always well-correlated with the cost vector. In contrast, if the algorithm is empowered to issue queries and if all the responses are correct, then we show $O(\log T)$ regret is achievable. We then show how to make this result more robust---when some of the query responses can be adversarial---by using a little feedback on the quality of the responses.