Lifted Primal-Dual Method for Bilinearly Coupled Smooth Minimax Optimization

We study the bilinearly coupled minimax problem: $\min_{x} \max_{y} f(x) + y^\top A x - h(y)$, where $f$ and $h$ are both strongly convex smooth functions and admit first-order gradient oracles. Surprisingly, no known first-order algorithms have hitherto achieved the lower complexity bound of $\Omega((\sqrt{\frac{L_x}{\mu_x}} + \frac{\|A\|}{\sqrt{\mu_x \mu_y}} + \sqrt{\frac{L_y}{\mu_y}}) \log(\frac1{\varepsilon}))$ for solving this problem up to an $\varepsilon$ primal-dual gap in the general parameter regime, where $L_x, L_y,\mu_x,\mu_y$ are the corresponding smoothness and strongly convexity constants. We close this gap by devising the first optimal algorithm, the Lifted Primal-Dual (LPD) method. Our method lifts the objective into an extended form that allows both the smooth terms and the bilinear term to be handled optimally and seamlessly with the same primal-dual framework. Besides optimality, our method yields a desirably simple single-loop algorithm that uses only one gradient oracle call per iteration. Moreover, when $f$ is just convex, the same algorithm applied to a smoothed objective achieves the nearly optimal iteration complexity. We also provide a direct single-loop algorithm, using the LPD method, that achieves the iteration complexity of $O(\sqrt{\frac{L_x}{\varepsilon}} + \frac{\|A\|}{\sqrt{\mu_y \varepsilon}} + \sqrt{\frac{L_y}{\varepsilon}})$. Numerical experiments on quadratic minimax problems and policy evaluation problems further demonstrate the fast convergence of our algorithm in practice.

Gradient flows on graphons: existence, convergence, continuity equations

Wasserstein gradient flows on probability measures have found a host of applications in various optimization problems. They typically arise as the continuum limit of exchangeable particle systems evolving by some mean-field interaction involving a gradient-type potential. However, in many problems, such as in multi-layer neural networks, the so-called particles are edge weights on large graphs whose nodes are exchangeable. Such large graphs are known to converge to continuum limits called graphons as their size grow to infinity. We show that the Euclidean gradient flow of a suitable function of the edge-weights converges to a novel continuum limit given by a curve on the space of graphons that can be appropriately described as a gradient flow or, more technically, a curve of maximal slope. Several natural functions on graphons, such as homomorphism functions and the scalar entropy, are covered by our set-up, and the examples have been worked out in detail.

Differential privacy and robust statistics in high dimensions

We introduce a universal framework for characterizing the statistical efficiency of a statistical estimation problem with differential privacy guarantees. Our framework, which we call High-dimensional Propose-Test-Release (HPTR), builds upon three crucial components: the exponential mechanism, robust statistics, and the Propose-Test-Release mechanism. Gluing all these together is the concept of resilience, which is central to robust statistical estimation. Resilience guides the design of the algorithm, the sensitivity analysis, and the success probability analysis of the test step in Propose-Test-Release. The key insight is that if we design an exponential mechanism that accesses the data only via one-dimensional robust statistics, then the resulting local sensitivity can be dramatically reduced. Using resilience, we can provide tight local sensitivity bounds. These tight bounds readily translate into near-optimal utility guarantees in several cases. We give a general recipe for applying HPTR to a given instance of a statistical estimation problem and demonstrate it on canonical problems of mean estimation, linear regression, covariance estimation, and principal component analysis. We introduce a general utility analysis technique that proves that HPTR nearly achieves the optimal sample complexity under several scenarios studied in the literature.

Gradient Inversion with Generative Image Prior

Federated Learning (FL) is a distributed learning framework, in which the local data never leaves clients devices to preserve privacy, and the server trains models on the data via accessing only the gradients of those local data. Without further privacy mechanisms such as differential privacy, this leaves the system vulnerable against an attacker who inverts those gradients to reveal clients sensitive data. However, a gradient is often insufficient to reconstruct the user data without any prior knowledge. By exploiting a generative model pretrained on the data distribution, we demonstrate that data privacy can be easily breached. Further, when such prior knowledge is unavailable, we investigate the possibility of learning the prior from a sequence of gradients seen in the process of FL training. We experimentally show that the prior in a form of generative model is learnable from iterative interactions in FL. Our findings strongly suggest that additional mechanisms are necessary to prevent privacy leakage in FL.

KO codes: Inventing Nonlinear Encoding and Decoding for Reliable Wireless Communication via Deep-learning

Landmark codes underpin reliable physical layer communication, e.g., Reed-Muller, BCH, Convolution, Turbo, LDPC and Polar codes: each is a linear code and represents a mathematical breakthrough. The impact on humanity is huge: each of these codes has been used in global wireless communication standards (satellite, WiFi, cellular). Reliability of communication over the classical additive white Gaussian noise (AWGN) channel enables benchmarking and ranking of the different codes. In this paper, we construct KO codes, a computationaly efficient family of deep-learning driven (encoder, decoder) pairs that outperform the state-of-the-art reliability performance on the standardized AWGN channel. KO codes beat state-of-the-art Reed-Muller and Polar codes, under the low-complexity successive cancellation decoding, in the challenging short-to-medium block length regime on the AWGN channel. We show that the gains of KO codes are primarily due to the nonlinear mapping of information bits directly to transmit real symbols (bypassing modulation) and yet possess an efficient, high performance decoder. The key technical innovation that renders this possible is design of a novel family of neural architectures inspired by the computation tree of the {\bf K}ronecker {\bf O}peration (KO) central to Reed-Muller and Polar codes. These architectures pave way for the discovery of a much richer class of hitherto unexplored nonlinear algebraic structures. The code is available at \href{https://github.com/deepcomm/KOcodes}{https://github.com/deepcomm/KOcodes}

Reducing the Communication Cost of Federated Learning through Multistage Optimization

A central question in federated learning (FL) is how to design optimization algorithms that minimize the communication cost of training a model over heterogeneous data distributed across many clients. A popular technique for reducing communication is the use of local steps, where clients take multiple optimization steps over local data before communicating with the server (e.g., FedAvg, SCAFFOLD). This contrasts with centralized methods, where clients take one optimization step per communication round (e.g., Minibatch SGD). A recent lower bound on the communication complexity of first-order methods shows that centralized methods are optimal over highly-heterogeneous data, whereas local methods are optimal over purely homogeneous data [Woodworth et al., 2020]. For intermediate heterogeneity levels, no algorithm is known to match the lower bound. In this paper, we propose a multistage optimization scheme that nearly matches the lower bound across all heterogeneity levels. The idea is to first run a local method up to a heterogeneity-induced error floor; next, we switch to a centralized method for the remaining steps. Our analysis may help explain empirically-successful stepsize decay methods in FL [Charles et al., 2020; Reddi et al., 2020]. We demonstrate the scheme's practical utility in image classification tasks.

Divergence Frontiers for Generative Models: Sample Complexity, Quantization Level, and Frontier Integral

The spectacular success of deep generative models calls for quantitative tools to measure their statistical performance. Divergence frontiers have recently been proposed as an evaluation framework for generative models, due to their ability to measure the quality-diversity trade-off inherent to deep generative modeling. However, the statistical behavior of divergence frontiers estimated from data remains unknown to this day. In this paper, we establish non-asymptotic bounds on the sample complexity of the plug-in estimator of divergence frontiers. Along the way, we introduce a novel integral summary of divergence frontiers. We derive the corresponding non-asymptotic bounds and discuss the choice of the quantization level by balancing the two types of approximation errors arisen from its computation. We also augment the divergence frontier framework by investigating the statistical performance of smoothed distribution estimators such as the Good-Turing estimator. We illustrate the theoretical results with numerical examples from natural language processing and computer vision.

Sample Efficient Linear Meta-Learning by Alternating Minimization

Meta-learning synthesizes and leverages the knowledge from a given set of tasks to rapidly learn new tasks using very little data. Meta-learning of linear regression tasks, where the regressors lie in a low-dimensional subspace, is an extensively-studied fundamental problem in this domain. However, existing results either guarantee highly suboptimal estimation errors, or require $\Omega(d)$ samples per task (where $d$ is the data dimensionality) thus providing little gain over separately learning each task. In this work, we study a simple alternating minimization method (MLLAM), which alternately learns the low-dimensional subspace and the regressors. We show that, for a constant subspace dimension MLLAM obtains nearly-optimal estimation error, despite requiring only $\Omega(\log d)$ samples per task. However, the number of samples required per task grows logarithmically with the number of tasks. To remedy this in the low-noise regime, we propose a novel task subset selection scheme that ensures the same strong statistical guarantee as MLLAM, even with bounded number of samples per task for arbitrarily large number of tasks.

SPECTRE: Defending Against Backdoor Attacks Using Robust Statistics

Modern machine learning increasingly requires training on a large collection of data from multiple sources, not all of which can be trusted. A particularly concerning scenario is when a small fraction of poisoned data changes the behavior of the trained model when triggered by an attacker-specified watermark. Such a compromised model will be deployed unnoticed as the model is accurate otherwise. There have been promising attempts to use the intermediate representations of such a model to separate corrupted examples from clean ones. However, these defenses work only when a certain spectral signature of the poisoned examples is large enough for detection. There is a wide range of attacks that cannot be protected against by the existing defenses. We propose a novel defense algorithm using robust covariance estimation to amplify the spectral signature of corrupted data. This defense provides a clean model, completely removing the backdoor, even in regimes where previous methods have no hope of detecting the poisoned examples. Code and pre-trained models are available at https://github.com/SewoongLab/spectre-defense .

Robust and Differentially Private Mean Estimation

Differential privacy has emerged as a standard requirement in a variety of applications ranging from the U.S. Census to data collected in commercial devices, initiating an extensive line of research in accurately and privately releasing statistics of a database. An increasing number of such databases consist of data from multiple sources, not all of which can be trusted. This leaves existing private analyses vulnerable to attacks by an adversary who injects corrupted data. Despite the significance of designing algorithms that guarantee privacy and robustness (to a fraction of data being corrupted) simultaneously, even the simplest questions remain open. For the canonical problem of estimating the mean from i.i.d. samples, we introduce the first efficient algorithm that achieves both privacy and robustness for a wide range of distributions. This achieves optimal accuracy matching the known lower bounds for robustness, but the sample complexity has a factor of $d^{1/2}$ gap from known lower bounds. We further show that this gap is due to the computational efficiency; we introduce the first family of algorithms that close this gap but takes exponential time. The innovation is in exploiting resilience (a key property in robust estimation) to adaptively bound the sensitivity and improve privacy.