Adaptive stochastic convex optimization (SCO) methods face a fundamental “price of adaptivity” barrier: under the standard set of assumptions, they cannot efficiently adapt to large uncertainty in both the initial distance to optimality and the Lipschitz constant. We circumvent this barrier by requiring a small amount of additional structure common to many learning problems. Specifically, we assume that the objective decomposes into a model and a loss function, enabling us to intervene by modifying the model’s output before it passes to the loss function. Under this assumption, we design a method that clips the learned model output in tail events where it deviates too much from the output of a fixed reference model. Our method matches the optimal bounds for known-parameter SCO up to logarithmic factors in the uncertainty in the distance and Lipschitz parameters, thus efficiently adapting to large uncertainty in both.

We provide a general method to convert a "primal" black-box algorithm for solving regularized convex-concave minimax optimization problems into an algorithm for solving the associated dual maximin optimization problem. Our method adds recursive regularization over a logarithmic number of rounds where each round consists of an approximate regularized primal optimization followed by the computation of a dual best response. We apply this result to obtain new state-of-the-art runtimes for solving matrix games in specific parameter regimes, obtain improved query complexity for solving the dual of the CVaR distributionally robust optimization (DRO) problem, and recover the optimal query complexity for finding a stationary point of a convex function.

We study stochastic gradient descent (SGD) with gradient clipping on convex functions under a generalized smoothness assumption called $(L_0,L_1)$-smoothness. Using gradient clipping, we establish a high probability convergence rate that matches the SGD rate in the $L$ smooth case up to polylogarithmic factors and additive terms. We also propose a variation of adaptive SGD with gradient clipping, which achieves the same guarantee. We perform empirical experiments to examine our theory and algorithmic choices.

We introduce Filter Like You Test (FLYT), an algorithm for curating large-scale vision-language datasets that *learns* the usefulness of each data point as a pretraining example. FLYT trains a scoring model that learns to weigh each example's features using gradient signals from downstream tasks training sets. Based on FLYT, we implement Mixing-FLYT (M-FLYT), which takes the per-example scores generated by different scoring methods as features, and learns to unify them into a single score. FLYT naturally produces a distribution over the training examples, which we leverage through Soft Cap Sampling (SCS), a strategy for obtaining a filtered pretraining dataset from per-example probabilities that samples examples while preventing over-representation through a repetition penalty. Using these methods, we achieve 40.1\% ImageNet zero-shot accuracy on the DataComp medium scale filtering benchmark, a 2\% absolute accuracy increase over all previous results and a 5.5\% increase over results that---like us---use only public resources. Our approach also yields 37.7\% on the average of 38 DataComp evaluation tasks, outperforming previous public-resource approaches by 0.4\%.

We prove impossibility results for adaptivity in non-smooth stochastic convex optimization. Given a set of problem parameters we wish to adapt to, we define a “price of adaptivity” (PoA) that, roughly speaking, measures the multiplicative increase in suboptimality due to uncertainty in these parameters. When the initial distance to the optimum is unknown but a gradient norm bound is known, we show that the PoA is at least logarithmic for expected suboptimality, and double-logarithmic for median suboptimality. When there is uncertainty in both distance and gradient norm, we show that the PoA must be polynomial in the level of uncertainty. Our lower bounds nearly match existing upper bounds, and establish that there is no parameter-free lunch.

We propose a method that achieves near-optimal rates for \emph{smooth} stochastic convex optimization and requires essentially no prior knowledge of problem parameters. This improves on prior work which requires knowing at least the initial distance to optimality $d_0$. Our method, \textsc{U-DoG}, combines \textsc{UniXGrad} (Kavis et al., 2019) and \textsc{DoG} (Ivgi et al., 2023) with novel iterate stabilization techniques. It requires only loose bounds on $d_0$ and the noise magnitude, provides high probability guarantees under sub-Gaussian noise, and is also near-optimal in the non-smooth case. Our experiments show consistent, strong performance on convex problems and mixed results on neural network training.

We introduce DataComp for Language Models, a testbed for controlled dataset experiments with the goal of improving language models.As part of DCLM, we provide a standardized corpus of 240T tokens extracted from Common Crawl, effective pretraining recipes based on the OpenLM framework, and a broad suite of 53 downstream evaluations.Participants in the DCLM benchmark can experiment with data curation strategies such as deduplication, filtering, and data mixing atmodel scales ranging from 412M to 7B parameters.As a baseline for DCLM, we conduct extensive experiments and find that model-based filtering is key to assembling a high-quality training set.The resulting dataset, DCLM-Baseline, enables training a 7B parameter language model from scratch to 63% 5-shot accuracy on MMLU with 2T training tokens.Compared to MAP-Neo, the previous state-of-the-art in open-data language models, DCLM-Baseline represents a 6 percentage point improvement on MMLU while being trained with half the compute.Our results highlight the importance of dataset design for training language models and offer a starting point for further research on data curation. We release the \dclm benchmark, framework, models, and datasets at https://www.datacomp.ai/dclm/

Kaplan et al. and Hoffmann et al. developed influential scaling laws for the optimal model size as a function of the compute budget, but these laws yield substantially different predictions. We explain the discrepancy by reproducing the Kaplan scaling law on two datasets (OpenWebText2 and RefinedWeb) and identifying three factors causing the difference: last layer computational cost, warmup duration, and scale-dependent optimizer tuning. With these factors corrected, we obtain excellent agreement with the Hoffmann et al. (i.e., "Chinchilla") scaling law. Counter to a hypothesis of Hoffmann et al., we find that careful learning rate decay is not essential for the validity of their scaling law. As a secondary result, we derive scaling laws for the optimal learning rate and batch size, finding that tuning the AdamW $\beta_2$ parameter is essential at lower batch sizes.

We design algorithms for minimizing maxi∈[n] fi(x) over a d-dimensional Euclidean or simplex domain. When each fi is 1-Lipschitz and 1-smooth, our method computes an ɛ-approximate solution using Õ(nɛ-1/3 + ɛ-2) gradient and function evaluations, and Õ(nɛ-4/3) additional runtime. For large n, our evaluation complexity is optimal up to polylogarithmic factors. In the special case where each fi is linear—which corresponds to finding a near-optimal primal strategy in a matrix game—our method finds an ɛ-approximate solution in runtime Õ(n(d/ɛ)2/3 +nd+dɛ-2). For n > d and this improves over all existing first-order methods. When additionally d = ω(n8/11) our runtime also improves over all known interior point methods.

We introduce a new tool for stochastic convex optimization (SCO): a Reweighted Stochastic Query (ReSQue) estimator for the gradient of a function convolved with a (Gaussian) probability density. Combining ReSQue with recent advances in ball oracle acceleration [CJJ+20], [ACJ+21], we develop algorithms achieving state-of-the-art complexities for SCO in parallel and private settings. For a SCO objective constrained to the unit ball in $\mathbb{R}^{d}$, we obtain the following results (up to polylogarithmic factors).1)We give a parallel algorithm obtaining optimization error $\epsilon_{\text {opt} }$ with $d^{1 / 3} \epsilon_{\text {opt} }^{-2 / 3}$ gradient oracle query depth and $d^{1 / 3} \epsilon_{\text {opt} }^{-2 / 3}+\epsilon_{\text {opt} }^{-2}$ gradient queries in total, assuming access to a bounded-variance stochastic gradient estimator. For $\epsilon_{\text {opt} } \in\left[d^{-1}, d^{-1 / 4}\right]$, our algorithm matches the state-of-the-art oracle depth of [BJL+19] while maintaining the optimal total work of stochastic gradient descent.2)Given n samples of Lipschitz loss functions, prior works [BFTT19], [BFGT20], [AFKT21], [KLL21] established that if $n \gt rsim d \epsilon_{\mathrm{dp}}^{-2},\left(\epsilon_{\mathrm{dp}}, \delta\right)$-differential privacy is attained at no asymptotic cost to the SCO utility. However, these prior works all required a superlinear number of gradient queries. We close this gap for sufficiently large $n \gt rsim d^{2} \epsilon_{{\bf d p}}^{-3}$, by using ReSQue to design an algorithm with near-linear gradient query complexity in this regime.

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.

Recent research shows that when Gradient Descent (GD) is applied to neural networks, the loss almost never decreases monotonically. Instead, the loss oscillates as gradient descent converges to its “Edge of Stability” (EoS). Here, we find a quantity that does decrease monotonically throughout GD training: the sharpness attained by the gradient flow solution (GFS)—the solution that would be obtained if, from now until convergence, we train with an infinitesimal step size. Theoretically, we analyze scalar neural networks with the squared loss, perhaps the simplest setting where the EoS phenomena still occur. In this model, we prove that the GFS sharpness decreases monotonically. Using this result, we characterize settings where GD provably converges to the EoS in scalar networks. Empirically, we show that GD monotonically decreases the GFS sharpness in a squared regression model as well as practical neural network architectures.

Multimodal datasets are a critical component in recent breakthroughs such as CLIP, Stable Diffusion and GPT-4, yet their design does not receive the same research attention as model architectures or training algorithms. To address this shortcoming in the machine learning ecosystem, we introduce DataComp, a testbed for dataset experiments centered around a new candidate pool of 12.8 billion image-text pairs from Common Crawl. Participants in our benchmark design new filtering techniques or curate new data sources and then evaluate their new dataset by running our standardized CLIP training code and testing the resulting model on 38 downstream test sets. Our benchmark consists of multiple compute scales spanning four orders of magnitude, which enables the study of scaling trends and makes the benchmark accessible to researchers with varying resources. Our baseline experiments show that the DataComp workflow leads to better training sets. Our best baseline, DataComp-1B, enables training a CLIP ViT-L/14 from scratch to 79.2% zero-shot accuracy on ImageNet, outperforming OpenAI's CLIP ViT-L/14 by 3.7 percentage points while using the same training procedure and compute. We release \datanet and all accompanying code at www.datacomp.ai.

We develop an algorithm for parameter-free stochastic convex optimization (SCO) whose rate of convergence is only a double-logarithmic factor larger than the optimal rate for the corresponding known-parameter setting. In contrast, the best previously known rates for parameter-free SCO are based on online parameter-free regret bounds, which contain unavoidable excess logarithmic terms compared to their known-parameter counterparts. Our algorithm is conceptually simple, has high-probability guarantees, and is also partially adaptive to unknown gradient norms, smoothness, and strong convexity. At the heart of our results is a novel parameter-free certificate for SGD step size choice, and a time-uniform concentration result that assumes no a-priori bounds on SGD iterates.

The accelerated proximal point method (APPA), also known as "Catalyst", is a well-established reduction from convex optimization to approximate proximal point computation (i.e., regularized minimization). This reduction is conceptually elegant and yields strong convergence rate guarantees. However, these rates feature an extraneous logarithmic term arising from the need to compute each proximal point to high accuracy. In this work, we propose a novel Relaxed Error Criterion for Accelerated Proximal Point (RECAPP) that eliminates the need for high accuracy subproblem solutions. We apply RECAPP to two canonical problems: finite-sum and max-structured minimization. For finite-sum problems, we match the best known complexity, previously obtained by carefully-designed problem-specific algorithms. For minimizing max_y f(x,y) where f is convex in x and strongly-concave in y, we improve on the best known (Catalyst-based) bound by a logarithmic factor.

The conventional recipe for maximizing model accuracy is to (1) train multiple models with various hyperparameters and (2) pick the individual model which performs best on a held-out validation set, discarding the remainder. In this paper, we revisit the second step of this procedure in the context of fine-tuning large pre-trained models, where fine-tuned models often appear to lie in a single low error basin. We show that averaging the weights of multiple models fine-tuned with different hyperparameter configurations often improves accuracy and robustness. Unlike a conventional ensemble, we may average many models without incurring any additional inference or memory costs—we call the results “model soups.” When fine-tuning large pre-trained models such as CLIP, ALIGN, and a ViT-G pre-trained on JFT, our soup recipe provides significant improvements over the best model in a hyperparameter sweep on ImageNet. The resulting ViT-G model, which attains 90.94% top-1 accuracy on ImageNet, achieved a new state of the art. Furthermore, we show that the model soup approach extends to multiple image classification and natural language processing tasks, improves out-of-distribution performance, and improves zero-shot performance on new downstream tasks. Finally, we analytically relate the performance similarity of weight-averaging and logit-ensembling to flatness of the loss and confidence of the predictions, and validate this relation empirically. Code is available at https://github.com/mlfoundations/model-soups.

We develop and analyze algorithms for distributionally robust optimization (DRO) of convex losses. In particular, we consider group-structured and bounded $f$-divergence uncertainty sets. Our approach relies on an accelerated method that queries a ball optimization oracle, i.e., a subroutine that minimizes the objective within a small ball around the query point. Our main contribution is efficient implementations of this oracle for DRO objectives. For DRO with $N$ non-smooth loss functions, the resulting algorithms find an $\epsilon$-accurate solution with $\widetilde{O}\left(N\epsilon^{-2/3} + \epsilon^{-2}\right)$ first-order oracle queries to individual loss functions. Compared to existing algorithms for this problem, we improve complexity by a factor of up to $\epsilon^{-4/3}$.

We develop a variant of the Monteiro-Svaiter (MS) acceleration framework that removes the need to solve an expensive implicit equation at every iteration. Consequently, for any $p\ge 2$ we improve the complexity of convex optimization with Lipschitz $p$th derivative by a logarithmic factor, matching a lower bound. We also introduce an MS subproblem solver that requires no knowledge of problem parameters, and implement it as either a second- or first-order method by solving linear systems or applying MinRes, respectively. On logistic regression problems our method outperforms previous accelerated second-order methods, but under-performs Newton's method; simply iterating our first-order adaptive subproblem solver is competitive with L-BFGS.

We characterize the complexity of minimizing $\max_{i\in[N]} f_i(x)$ for convex, Lipschitz functions $f_1,\ldots, f_N$. For non-smooth functions, existing methods require $O(N\epsilon^{-2})$ queries to a first-order oracle to compute an $\epsilon$-suboptimal point and $\widetilde{O}(N\epsilon^{-1})$ queries if the $f_i$ are $O(1/\epsilon)$-smooth. We develop methods with improved complexity bounds of $\widetilde{O}(N\epsilon^{-2/3} + \epsilon^{-8/3})$ in the non-smooth case and $\widetilde{O}(N\epsilon^{-2/3} + \sqrt{N}\epsilon^{-1})$ in the $O(1/\epsilon)$-smooth case. Our methods consist of a recently proposed ball optimization oracle acceleration algorithm (which we refine) and a careful implementation of said oracle for the softmax function. We also prove an oracle complexity lower bound scaling as $\Omega(N\epsilon^{-2/3})$, showing that our dependence on $N$ is optimal up to polylogarithmic factors.

For machine learning systems to be reliable, we must understand their performance in unseen, out- of-distribution environments. In this paper, we empirically show that out-of-distribution performance is strongly correlated with in-distribution performance for a wide range of models and distribution shifts. Specifically, we demonstrate strong correlations between in-distribution and out-of- distribution performance on variants of CIFAR- 10 & ImageNet, a synthetic pose estimation task derived from YCB objects, FMoW-WILDS satellite imagery classification, and wildlife classification in iWildCam-WILDS. The correlation holds across model architectures, hyperparameters, training set size, and training duration, and is more precise than what is expected from existing domain adaptation theory. To complete the picture, we also investigate cases where the correlation is weaker, for instance some synthetic distribution shifts from CIFAR-10-C and the tissue classification dataset Camelyon17-WILDS. Finally, we provide a candidate theory based on a Gaussian data model that shows how changes in the data covariance arising from distribution shift can affect the observed correlations.

We study the generalization performance of $\text{\emph{full-batch}}$ optimization algorithms for stochastic convex optimization: these are first-order methods that only access the exact gradient of the empirical risk (rather than gradients with respect to individual data points), that include a wide range of algorithms such as gradient descent, mirror descent, and their regularized and/or accelerated variants. We provide a new separation result showing that, while algorithms such as stochastic gradient descent can generalize and optimize the population risk to within $\epsilon$ after $O(1/\epsilon^2)$ iterations, full-batch methods either need at least $\Omega(1/\epsilon^4)$ iterations or exhibit a dimension-dependent sample complexity.

We develop a new primitive for stochastic optimization: a low-bias, low-cost estimator of the minimizer $x_\star$ of any Lipschitz strongly-convex function $f$. In particular, we use a multilevel Monte-Carlo approach due to Blanchet and Glynn to turn any optimal stochastic gradient method into an estimator of $x_\star$ with bias $\delta$, variance $O(\log(1/\delta))$, and an expected sampling cost of $O(\log(1/\delta))$ stochastic gradient evaluations. As an immediate consequence, we obtain cheap and nearly unbiased gradient estimators for the Moreau envelope of any Lipschitz convex function. We demonstrate the potential of our estimator through four applications. First, we develop a method for minimizing the maximum of $N$ functions, improving on recent results and matching a lower bound up to logarithmic factors. Second and third, we recover state-of-the-art rates for projection-efficient and gradient-efficient optimization using simple algorithms with a transparent analysis. Finally, we show that an improved version of our estimator would yield a nearly linear-time, optimal-utility, differentially-private non-smooth stochastic optimization method.

We design an algorithm which finds an $\epsilon$-approximate stationary point (with $\|\nabla F(x)\|\le \epsilon$) using $O(\epsilon^{-3})$ stochastic gradient and Hessian-vector products, matching guarantees that were previously available only under a stronger assumption of access to multiple queries with the same random seed. We prove a lower bound which establishes that this rate is optimal and—surprisingly—that it cannot be improved using stochastic $p$th order methods for any $p\ge 2$, even when the first $p$ derivatives of the objective are Lipschitz. Together, these results characterize the complexity of non-convex stochastic optimization with second-order methods and beyond. Expanding our scope to the oracle complexity of finding $(\epsilon,\gamma)$-approximate second-order stationary points, we establish nearly matching upper and lower bounds for stochastic second-order methods. Our lower bounds here are novel even in the noiseless case.

We develop primal-dual coordinate methods for solving bilinear saddle-point problems of the form minx ∈ Xmaxy ∈ YyTAx which contain linear programming, classification, and regression as special cases. Our methods push existing fully stochastic sublinear methods and variance-reduced methods towards their limits in terms of per-iteration complexity and sample complexity. We obtain nearly-constant per-iteration complexity by designing efficient data structures leveraging Taylor approximations to the exponential and a binomial heap. We improve sample complexity via low-variance gradient estimators using dynamic sampling distributions that depend on both the iterates and the magnitude of the matrix entries. Our runtime bounds improve upon those of existing primal-dual methods by a factor depending on sparsity measures of the m by n matrix A. For example, when rows and columns have constant l1/l2 norm ratios, we offer improvements by a factor of m+n in the fully stochastic setting and √{m+n} in the variance-reduced setting. We apply our methods to computational geometry problems, i.e. minimum enclosing ball, maximum inscribed ball, and linear regression, and obtain improved complexity bounds. For linear regression with an elementwise nonnegative matrix, our guarantees improve on exact gradient methods by a factor of √{nnz(A)/(m+n)}.

Consider an oracle which takes a point x and returns the minimizer of a convex function f in an l2 ball of radius r around x. It is straightforward to show that roughly r^{-1}\log(1/epsilon) calls to the oracle suffice to find an \epsilon-approximate minimizer of f in an l2 unit ball. Perhaps surprisingly, this is not optimal: we design an accelerated algorithm which attains an epsilon-approximate minimizer with roughly r^{-2/3} \log(1/epsilon) oracle queries, and give a matching lower bound. Further, we implement ball optimization oracles for functions with a locally stable Hessian using a variant of Newton's method and, in certain cases, stochastic first-order methods. The resulting algorithms apply to a number of problems of practical and theoretical import, improving upon previous results for logistic and linfinity regression and achieving guarantees comparable to the state-of-the-art for lp regression.

We propose and analyze algorithms for distributionally robust optimization of convex losses with conditional value at risk (CVaR) and $\chi^2$ divergence uncertainty sets. We prove that our algorithms require a number of gradient evaluations independent of training set size and number of parameters, making them suitable for large-scale applications. For $\chi^2$ uncertainty sets these are the first such guarantees in the literature, and for CVaR our guarantees scale linearly in the uncertainty level rather than quadratically as in previous work. We also provide lower bounds proving the worst-case optimality of our algorithms for CVaR and a penalized version of the $\chi^2$ problem. Our primary technical contributions are novel bounds on the bias of batch robust risk estimation and the variance of a multilevel Monte Carlo gradient estimator due to [Blanchet & Glynn, 2015]. Experiments on MNIST and ImageNet confirm the theoretical scaling of our algorithms, which are 9-36 times more efficient than full-batch methods.

We show that a simple randomized sketch of the matrix multiplicative weight (MMW) update enjoys (in expectation) the same regret bounds as MMW, up to a small constant factor. Unlike MMW, where every step requires full matrix exponentiation, our steps require only a single product of the form $e^A b$, which the Lanczos method approximates efficiently. Our key technique is to view the sketch as a \emph{randomized mirror projection}, and perform mirror descent analysis on the \emph{expected projection}. Our sketch solves the online eigenvector problem, improving the best known complexity bounds by $\Omega(\log^5 n)$. We also apply this sketch to semidefinite programming in saddle-point form, yielding a simple primal-dual scheme with guarantees matching the best in the literature.

We present a randomized primal-dual algorithm that solves the problem minx maxy y^T A x to additive error epsilon in time nnz(A) + sqrt{nnz(A) n} / epsilon, for matrix A with larger dimension n and nnz(A) nonzero entries. This improves the best known exact gradient methods by a factor of sqrt{nnz(A) / n} and is faster than fully stochastic gradient methods in the accurate and/or sparse regime epsilon < sqrt{n / nnz(A)$. Our results hold for x,y in the simplex (matrix games, linear programming) and for x in an \ell_2 ball and y in the simplex (perceptron / SVM, minimum enclosing ball). Our algorithm combines the Nemirovski's "conceptual prox-method" and a novel reduced-variance gradient estimator based on "sampling from the difference" between the current iterate and a reference point.

We demonstrate, theoretically and empirically, that adversarial robustness can significantly benefit from semisupervised learning. Theoretically, we revisit the simple Gaussian model of Schmidt et al. that shows a sample complexity gap between standard and robust classification. We prove that unlabeled data bridges this gap: a simple semisupervised learning procedure (self-training) achieves high robust accuracy using the same number of labels required for achieving high standard accuracy. Empirically, we augment CIFAR-10 with 500K unlabeled images sourced from 80 Million Tiny Images and use robust self-training to outperform state-of-the-art robust accuracies by over 5 points in (i) $\ell_\infty$ robustness against several strong attacks via adversarial training and (ii) certified $\ell_2$ and $\ell_\infty$ robustness via randomized smoothing. On SVHN, adding the dataset's own extra training set with the labels removed provides gains of 4 to 10 points, within 1 point of the gain from using the extra labels.

We provide convergence rates for Krylov subspace solutions to the trust-region and cubic-regularized (nonconvex) quadratic problems. Such solutions may be efficiently computed by the Lanczos method and have long been used in practice. We prove error bounds of the form $1/t^2$ and $e^{-4t/\sqrt{\kappa}}$, where $\kappa$ is a condition number for the problem, and $t$ is the Krylov subspace order (number of Lanczos iterations). We also provide lower bounds showing that our analysis is sharp.

We develop and analyze a variant of Nesterov’s accelerated gradient descent (AGD) for minimization of smooth non-convex functions. We prove that one of two cases occurs: either our AGD variant converges quickly, as if the function was convex, or we produce a certificate that the function is “guilty” of being non-convex. This non-convexity certificate allows us to exploit negative curvature and obtain deterministic, dimension-free acceleration of convergence for non-convex functions. For a function $f$ with Lipschitz continuous gradient and Hessian, we compute a point $x$ with $\|\nabla f(x)\| \le \epsilon$ in $O(\epsilon^{-7/4} \log(1/ \epsilon) )$ gradient and function evaluations. Assuming additionally that the third derivative is Lipschitz, we require only $O(\epsilon^{-5/3} \log(1/ \epsilon) )$ evaluations.

We consider the discrete-time intersymbol interference (ISI) channel model, with additive Gaussian noise and fixed independent identically distributed inputs. In this setting, we investigate the expression put forth by Shamai and Laroia as a conjectured lower bound for the input-output mutual information after application of a minimum mean-square error decision-feedback equalizer receiver. A low-signal to noise ratio (SNR) expansion is used to prove that the conjectured bound does not hold under general conditions, and to characterize inputs for which it is particularly ill-suited. One such input is used to construct a counterexample, indicating that the Shamai-Laroia expression does not always bound even the achievable rate of the channel, thus excluding a natural relaxation of the original conjectured bound. However, this relaxed bound is then shown to hold for any finite entropy input and ISI channel, when the SNR is sufficiently high. We derive two conditions under which the relaxed bound holds, involving compound channel capacity and quasiconvexity arguments. Finally, new simple bounds for the achievable rate are proven, and compared with other known bounds. Information-estimation relations and estimation-theoretic bounds play a key role in establishing our results.

We compare the maximum achievable rates in single-carrier (SC) and orthogonal frequency-division multiplexing (OFDM) modulation schemes, under the practical assumptions of independent identically distributed finite alphabet inputs and linear intersymbol interference with additive Gaussian noise. We show that the Shamai-Laroia approximation serves as a bridge between the two rates: while it is well known that this approximation is often a lower bound on the SC achievable rate, it is revealed to also essentially upper bound the OFDM achievable rate. We apply information-estimation relations in order to rigorously establish this result for both general input distributions and to sharpen it for commonly used pulse-amplitude modulation (PAM) and quadratic-amplitude modulation constellations. To this end, novel bounds on minimum mean-square error estimation of PAM inputs to a scalar Gaussian channel are derived, which may be of general interest. Our results show that, under reasonable assumptions, optimal SC schemes may offer spectral efficiency significantly superior to that of OFDM, motivating further research of such systems.

We consider mean squared estimation of a continuous-time signal corrupted by additive white Gaussian noise. We investigate the trade-off between lookahead and estimation-loss under this model. We study the class of continuous-time stationary Gauss-Markov processes (Ornstein-Uhlenbeck processes) as channel inputs, and explicitly characterize the behavior of the minimum mean squared error (MMSE) with finite lookahead and signal-to-noise ratio (SNR). The MMSE with lookahead is shown to converge exponentially rapidly to the non-causal error, with the exponent being the reciprocal of the non-causal error. We extend our results to mixtures of Ornstein-Uhlenbeck processes, and use the insight gained to present lower and upper bounds on the MMSE with lookahead for a class of stationary Gaussian input processes, whose spectrum can be expressed as a mixture of Ornstein-Uhlenbeck spectra.