theory->optimization

Improving and guaranteeing the robustness of deep learning models has been a topic of intense research. Ensembling, which combines several classifiers to provide a better model, has been shown to be beneficial for generalisation, uncertainty estimation, calibration, and mitigating the effects of concept drift. However, the impact of ensembling on certified robustness is less well understood. In this work, we generalise Lipschitz continuity by introducing S-Lipschitz classifiers, which we use to analyse the theoretical robustness of ensembles. Our results are precise conditions when ensembles of robust classifiers are more robust than any constituent classifier, as well as conditions when they are less robust.

Non-convex optimization plays a key role in a growing number of machine learning applications. This motivates the identification of specialized structure that enables sharper theoretical analysis. One such identified structure is quasar-convexity, a non-convex generalization of convexity that subsumes convex functions. Existing algorithms for minimizing quasar-convex functions in the stochastic setting have either high complexity or slow convergence, which prompts us to derive a new class of stochastic methods for optimizing smooth quasar-convex functions. We demonstrate that our algorithms have fast convergence and outperform existing algorithms on several examples, including the classical problem of learning linear dynamical systems. We also present a unified analysis of our newly proposed algorithms and a previously studied deterministic algorithm.

In this paper, we consider the problem of bit allocation in Neural Video Compression (NVC). First, we reveal a fundamental relationship between bit allocation in NVC and Semi-Amortized Variational Inference (SAVI). Specifically, we show that SAVI with GoP (Group-of-Picture)-level likelihood is equivalent to pixel-level bit allocation with precise rate & quality dependency model. Based on this equivalence, we establish a new paradigm of bit allocation using SAVI. Different from previous bit allocation methods, our approach requires no empirical model and is thus optimal. Moreover, as the original SAVI using gradient ascent only applies to single-level latent, we extend the SAVI to multi-level such as NVC by recursively applying back-propagating through gradient ascent. Finally, we propose a tractable approximation for practical implementation. Our method can be applied to scenarios where performance outweights encoding speed, and serves as an empirical bound on the R-D performance of bit allocation. Experimental results show that current state-of-the-art bit allocation algorithms still have a room of $\approx 0.5$ dB PSNR to improve compared with ours. Code is available at https://github.com/tongdaxu/Bit-Allocation-Using-Optimization.

Two-point zeroth order methods are important in many applications of zeroth-order optimization arising in robotics, wind farms, power systems, online optimization, and adversarial robustness to black-box attacks in deep neural networks, where the problem can be high-dimensional and/or time-varying. Furthermore, such problems may be nonconvex and contain saddle points. While existing works have shown that zeroth-order methods utilizing $\Omega(d)$ function valuations per iteration (with $d$ denoting the problem dimension) can escape saddle points efficiently, it remains an open question if zeroth-order methods based on two-point estimators can escape saddle points. In this paper, we show that by adding an appropriate isotropic perturbation at each iteration, a zeroth-order algorithm based on $2m$ (for any $1 \leq m \leq d$) function evaluations per iteration can not only find $\epsilon$-second order stationary points polynomially fast, but do so using only $\tilde{O}(\frac{d}{m\epsilon^{2}\bar{\psi}})$ function evaluations, where $\bar{\psi} \geq \tilde{\Omega}(\sqrt{\epsilon})$ is a parameter capturing the extent to which the function of interest exhibits the strict saddle property.

Solving large-scale multistage stochastic programming (MSP) problems poses a significant challenge as commonly used stagewise decomposition algorithms, including stochastic dual dynamic programming (SDDP), face growing time complexity as the subproblem size and problem count increase. Traditional approaches approximate the value functions as piecewise linear convex functions by incrementally accumulating subgradient cutting planes from the primal and dual solutions of stagewise subproblems. Recognizing these limitations, we introduce TranSDDP, a novel Transformer-based stagewise decomposition algorithm. This innovative approach leverages the structural advantages of the Transformer model, implementing a sequential method for integrating subgradient cutting planes to approximate the value function. Through our numerical experiments, we affirm TranSDDP's effectiveness in addressing MSP problems. It efficiently generates a piecewise linear approximation for the value function, significantly reducing computation time while preserving solution quality, thus marking a promising progression in the treatment of large-scale multistage stochastic programming problems.

We initiate a principled study of algorithmic collective action on digital platforms that deploy machine learning algorithms. We propose a simple theoretical model of a collective interacting with a firm's learning algorithm. The collective pools the data of participating individuals and executes an algorithmic strategy by instructing participants how to modify their own data to achieve a collective goal. We investigate the consequences of this model in three fundamental learning-theoretic settings: nonparametric optimal learning, parametric risk minimization, and gradient-based optimization. In each setting, we come up with coordinated algorithmic strategies and characterize natural success criteria as a function of the collective's size. Complementing our theory, we conduct systematic experiments on a skill classification task involving tens of thousands of resumes from a gig platform for freelancers. Through more than two thousand model training runs of a BERT-like language model, we see a striking correspondence emerge between our empirical observations and the predictions made by our theory. Taken together, our theory and experiments broadly support the conclusion that algorithmic collectives of exceedingly small fractional size can exert significant control over a platform's learning algorithm.

We study the problem of approximating stationary points of Lipschitz and smooth functions under $(\varepsilon,\delta)$-differential privacy (DP) in both the finite-sum and stochastic settings. A point $\widehat{w}$ is called an $\alpha$-stationary point of a function $F:\mathbb{R}^d\rightarrow\mathbb{R}$ if $\|\nabla F(\widehat{w})\|\leq \alpha$. We give a new construction that improves over the existing rates in the stochastic optimization setting, where the goal is to find approximate stationary points of the population risk given $n$ samples. Our construction finds a $\tilde{O}\big(\frac{1}{n^{1/3}} + \big[\frac{\sqrt{d}}{n\varepsilon}\big]^{1/2}\big)$-stationary point of the population risk in time linear in $n$. We also provide an efficient algorithm that finds an $\tilde{O}\big(\big[\frac{\sqrt{d}}{n\varepsilon}\big]^{2/3}\big)$-stationary point in the finite-sum setting. This improves on the previous best rate of $\tilde{O}\big(\big[\frac{\sqrt{d}}{n\varepsilon}\big]^{1/2}\big)$. Furthermore, under the additional assumption of convexity, we completely characterize the sample complexity of finding stationary points of the population risk (up to polylog factors) and show that the optimal rate on population stationarity is $\tilde \Theta\big(\frac{1}{\sqrt{n}}+\frac{\sqrt{d}}{n\varepsilon}\big)$. Finally, we show that our methods can be used to provide dimension-independent rates of $O\big(\frac{1}{\sqrt{n}}+\min\big(\big[\frac{\sqrt{rank}}{n\varepsilon}\big]^{2/3},\frac{1}{(n\varepsilon)^{2/5}}\big)\big)$ on population stationarity for Generalized Linear Models (GLM), where $rank$ is the rank of the design matrix, which improves upon the previous best known rate.

Zero-order (ZO) optimization is a powerful tool for dealing with realistic constraints. On the other hand, the gradient-tracking (GT) technique proved to be an efficient method for distributed optimization aiming to achieve consensus. However, it is a first-order (FO) method that requires knowledge of the gradient, which is not always possible in practice. In this work, we introduce a zero-order distributed optimization method based on a one-point estimate of the gradient tracking technique. We prove that this new technique converges with a single noisy function query at a time in the non-convex setting. We then establish a convergence rate of $O(\frac{1}{\sqrt[3]{K}})$ after a number of iterations K, which competes with that of $O(\frac{1}{\sqrt[4]{K}})$ of its centralized counterparts. Finally, a numerical example validates our theoretical results.

We consider the problem of global optimization with noisy zeroth order oracles — a well-motivated problem useful for various applications ranging from hyper-parameter tuning for deep learning to new material design. Existing work relies on Gaussian processes or other non-parametric family, which suffers from the curse of dimensionality. In this paper, we propose a new algorithm GO-UCB that leverages a parametric family of functions (e.g., neural networks) instead. Under a realizable assumption and a few other mild geometric conditions, we show that GO-UCB achieves a cumulative regret of $\tilde{O}(\sqrt{T})$ where $T$ is the time horizon. At the core of GO-UCB is a carefully designed uncertainty set over parameters based on gradients that allows optimistic exploration. Synthetic and real-world experiments illustrate GO-UCB works better than popular Bayesian optimization approaches, even if the model is misspecified.

Negative and positive curvatures affect optimization in different ways. However, a lot of existing optimization theories are based on the Lipschitz smoothness assumption, which cannot differentiate between the two. In this paper, we propose to use two separate assumptions for positive and negative curvatures, so that we can study the different implications of the two. We analyze the Lookahead and Local SGD methods as concrete examples. Both of them require multiple copies of model parameters and communication among them for every certain period of time in order to prevent divergence. We show that the minimum communication frequency is inversely proportional to the negative curvature, and when the negative curvature becomes zero, we recover the existing theory results for convex optimization. Finally, both experimentally and theoretically, we demonstrate that modern neural networks have highly unbalanced positive/negative curvatures. Thus, an analysis based on separate positive and negative curvatures is more pertinent.

The notion of omnipredictors (Gopalan, Kalai, Reingold, Sharan and Wieder ITCS 2022), suggested a new paradigm for loss minimization. Rather than learning a predictor based on a known loss function, omnipredictors can easily be post-processed to minimize any one of a rich family of loss functions compared with the loss of hypotheses in a class $\mathcal C$. It has been shown that such omnipredictors exist and are implied (for all convex and Lipschitz loss functions) by the notion of multicalibration from the algorithmic fairness literature. In this paper, we introduce omnipredictors for constrained optimization and study their complexity and implications. The notion that we introduce allows the learner to be unaware of the loss function that will be later assigned *as well as the constraints that will be later imposed*, as long as the subpopulations that are used to define these constraints are known. We show how to obtain omnipredictors for constrained optimization problems, relying on appropriate variants of multicalibration. We also investigate the implications of this notion when the constraints used are so-called group fairness notions.

We study the problem of learning a single neuron with respect to the $L_2^2$-loss in the presence of adversarial label noise. We give an efficient algorithm that, for a broad family of activations including ReLUs, approximates the optimal $L_2^2$-error within a constant factor. Notably, our algorithm succeeds under much milder distributional assumptions compared to prior work. The key ingredient enabling our results is a novel connection to local error bounds from optimization theory.

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.

Additive models have been burgeoning in data analysis due to their flexible representation and desirable interpretability. However, most existing approaches are constructed under empirical risk minimization (ERM), and thus perform poorly in situations where average performance is not a suitable criterion for the problems of interest, e.g., data with complex non-Gaussian noise, imbalanced labels or both of them. In this paper, a novel class of sparse additive models is proposed under tilted empirical risk minimization (TERM), which addresses the deficiencies in ERM by imposing tilted impact on individual losses, and is flexibly capable of achieving a variety of learning objectives, e.g., variable selection, robust estimation, imbalanced classification and multiobjective learning. On the theoretical side, a learning theory analysis which is centered around the generalization bound and function approximation error bound (under some specific data distributions) is conducted rigorously. On the practical side, an accelerated optimization algorithm is designed by integrating Prox-SVRG and random Fourier acceleration technique. The empirical assessments verify the competitive performance of our approach on both synthetic and real data.

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 $\mathcal{O}(L^3 d^{3/2} \epsilon^{-3}+ \Delta L^2 d^{3/2} \delta^{-1} \epsilon^{-3})$.

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.

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.

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.