arXiv subjects

Meir Feder

Publications and source records attributed to Meir Feder.

At least 19 recordsLinked to original sources

arXiv2026

A Layered Simplex Architecture for Large Alphabets

Probability estimation over large alphabets under log loss is a well-studied problem, with celebrated methods such as the Good-Turing estimator. We introduce and study a new Bayesian estimator with four notable properties. First, its construction is exceptionally simple: multiply independent uniform draws from the probability simplex coordinate-wise and renormalize. Depth is the only structural parameter, and averaging over depths eliminates the need to tune it. Second, the regret of the resulting mixture, the excess code length it pays relative to a code that knows the source, admits an explicit and efficiently computable expression. Third, despite its simplicity and lack of tuned constants, the estimator is competitive across a diverse set of synthetic and real-text benchmarks with substantially more specialized methods, including Good-Turing. Fourth, the tractability of its regret allows us to identify scaling laws in data, alphabet size, and depth. For Zipf targets with exponent above one, the regret has a simple reading as long as the sample reveals only a small fraction of the alphabet. It closely matches the description length of the set of discovered symbols, at one bit of code per bit of description, plus a further cost per symbol. The data exponent is therefore the rate at which new symbols are discovered.

cs.IT↗

arXiv2026

One-Shot Information Theory via the Pairwise Error Probability: Lossy, Joint Source-Channel, Erasure, and Multiuser Coding

This paper extends a one-shot (finite-blocklength) information-theoretic framework built on a single primitive: the pairwise error probability (PEP) of a randomized, dither-broken decoding rule, and the error spectrum it induces. A companion paper developed the framework for point-to-point channel coding -- uniformity of the PEP, the error-spectrum representation of achievability and converse, and the linear-programming form of the prior-optimized minimax meta-converse. Here we show that the same primitive, read on an enlarged candidate space, governs four further settings: lossy source coding under average distortion, joint source-channel coding with list decoding, channel coding with an erasure/undetected-error option, and the two-user multiple-access channel. In each case a single spectrum yields a random-coding achievability bound and exact fixed-code identities, and we indicate how the convex -- indeed linear-programming -- prior optimization of the channel-coding case extends under matched decoding. The development recovers the one-shot lossy bound of Matsuta-Uyematsu and the joint source-channel bounds in the Csiszar tradition, complements the lossy bounds of Kostina-Verdu, and connects the multiuser case to the comparable achievability/converse pair through three pairwise error events.

cs.IT↗

arXiv2026

A Pairwise-Error-Probability Framework for One-Shot Information Theory

We develop a one-shot (finite-blocklength) channel-coding framework based on the pairwise error probability (PEP) of a decoder with randomized tie-breaking. The tie-breaking rule yields a probability-integral-transform identity: the induced error spectrum describes both random-coding achievability and exact fixed-code converse statements, for an arbitrary decoding metric. We derive two variational identities for metric-weighted tail functionals of the PEP, one through the Neyman-Pearson $\beta$-functional and one through a reverse channel, valid for an arbitrary metric. Under matched maximum-likelihood decoding they specialize to representations of the spectrum itself, which is then jointly convex in the testing level and the input prior; combined with the reverse-channel representation, this gives a linear program for the prior-optimized minimax meta-converse -- finite-dimensional in general and, for memoryless channels with fixed alphabets, of size polynomial in the blocklength after a type reduction. Prior optimization of the random-coding bound is formulated as a concave program over input distributions with an explicit gradient, solved by a direct first-order method. The framework recovers several classical one-shot bounds, including the random-coding union bound and minimax meta-converse of Polyanskiy-Poor-Verdu, the information-spectrum bounds of Han-Verdu, and the linear-programming converse of Matthews. Numerical examples on the AWGN and binary Z-channels illustrate the achievability-converse comparison and the effect of prior optimization.

cs.IT↗

arXiv2026

Maximum Cut Algorithms and Upper Bounds for Planar and Toroidal Graphs

We demonstrate that the problem of finding the maximum cut of a planar graph with arbitrary weights can be easily mapped to a minimum T-join problem in the absolute dual graph - the dual graph with absolute weights, as opposed to the known mapping to a maximum T-join problem with an empty set in the dual graph. By enabling the use of the shortest paths, this approach allows for the straightforward adaptation of the first efficient Max-Cut algorithm, designed by Hadlock in 1975 for planar graphs with non-negative weights, to handle the general case of planar graphs with arbitrary weights. Furthermore, we prove that applying a planar Max-Cut algorithm to a higher genus graph, such as a toroidal graph, while disregarding its topology, provides an upper bound for its maximum cut. Employing this methodology, we derive upper bounds for the maximum cut across all toroidal graphs within the GSet benchmark. We report that the known maximum cut values for part of those GSet toroidal problems including the three largest instances, which were previously documented in the literature, are the maximum possible because they match their upper bound values. Additionally, we introduce a novel heuristic algorithm for finding Max-Cut of toroidal graphs, which is based on the planar graph algorithm. Applying this algorithm to all seventeen toroidal Max-Cut problems in the GSet benchmark successfully reproduces all the best-known results, and for problem #62, it yields a new, previously unknown best Max-Cut value.

cs.DS↗

arXiv2026

Implicit Binarization via Complex Phase Dynamics in Combinatorial Optimization

We introduce a physics-inspired continuous relaxation framework that yields substantially improved solutions for NP-hard combinatorial optimization problems, including Quadratic Unconstrained Binary Optimization (QUBO), binary sparse coding, and planted-solution Ising models. By parameterizing discrete binary variables as continuous wave-like states on the complex unit circle, we inherently smooth highly non-convex energy landscapes. We show that representing binary variables as complex phases reveals an implicit regularization mechanism that promotes convergence toward discrete states. Extracting this mechanism yields significant improvements even within standard real-valued optimization frameworks, using this regularizer explicitly. Empirically, this regularization yields vastly higher ground-state convergence rates than standard real-valued alternatives. Our models achieved zero error in large-scale 160x160 QUBO tasks under severe noise (sigma=0.25), and outperformed traditional algorithms (OMP and LASSO) in underdefined sparse coding with perfect recovery at sigma=0.15. The solver's robustness was further validated by recovering exact ground-state configurations in 8 out of 11 rigorously engineered planted-solution benchmarks.

cond-mat.stat-mech↗

arXiv2026

Misspecified Universal Learning

This paper addresses the problem of universal learning under model misspecification with log-loss. In this setting, the learner operates with a hypothesis class of models denoted by $\Theta$, while the true data-generating process belongs to a broader class $\Phi \supset \Theta$, and may lie outside the assumed hypothesis space. Classical approaches have characterized the minimax regret and identified optimal universal learners in both the well-specified stochastic and individual deterministic frameworks. The misspecified setting has received comparatively less attention, although several important results have emerged in recent years. Extending these foundations, we analyze the minimax regret in the misspecified setting and derive the corresponding optimal universal learner. We propose this formulation as a unified framework for universal learning, applicable to any form of uncertainty in the data-generating process, across both online and batch data arrival modes, as well as supervised and unsupervised learning tasks.

cs.IT↗

arXiv2025

Leave-One-Out Learning with Log-Loss

We study batch learning with log-loss in the individual setting, where the outcome sequence is deterministic. Because empirical statistics are not directly applicable in this regime, obtaining regret guarantees for batch learning has long posed a fundamental challenge. We propose a natural criterion based on leave-one-out regret and analyze its minimax value for several hypothesis classes. For the multinomial simplex over $m$ symbols, we show that the minimax regret is $\frac{m-1}{N} + o\!\left(\frac{1}{N}\right)$, and compare it to the stochastic realizable case where it is $\frac{m-1}{2N} + o\!\left(\frac{1}{N}\right)$. More generally, we prove that every hypothesis class of VC dimension $d$ is learnable in the individual batch-learning problem, with regret at most $\frac{d\log(N)}{N} + o\!\left(\frac{\log(N)}{N}\right)$, and we establish matching lower bounds for certain classes. We further derive additional upper bounds that depend on structural properties of the hypothesis class. These results establish, for the first time, that universal batch learning with log-loss is possible in the individual setting.

cs.IT↗

arXiv2025

Information-Theoretic Framework for Understanding Modern Machine-Learning

We introduce an information-theoretic framework that views learning as universal prediction under log loss, characterized through regret bounds. Central to the framework is an effective notion of architecture-based model complexity, defined by the probability mass or volume of models in the vicinity of the data-generating process, or its projection on the model class. This volume is related to spectral properties of the expected Hessian or the Fisher Information Matrix, leading to tractable approximations. We argue that successful architectures possess a broad complexity range, enabling learning in highly over-parameterized model classes. The framework sheds light on the role of inductive biases, the effectiveness of stochastic gradient descent, and phenomena such as flat minima. It unifies online, batch, supervised, and generative settings, and applies across the stochastic-realizable and agnostic regimes. Moreover, it provides insights into the success of modern machine-learning architectures, such as deep neural networks and transformers, suggesting that their broad complexity range naturally arises from their layered structure. These insights open the door to the design of alternative architectures with potentially comparable or even superior performance.

cs.LG↗

arXiv2024

Universal Batch Learning Under The Misspecification Setting

In this paper we consider the problem of universal {\em batch} learning in a misspecification setting with log-loss. In this setting the hypothesis class is a set of models $\Theta$. However, the data is generated by an unknown distribution that may not belong to this set but comes from a larger set of models $\Phi \supset \Theta$. Given a training sample, a universal learner is requested to predict a probability distribution for the next outcome and a log-loss is incurred. The universal learner performance is measured by the regret relative to the best hypothesis matching the data, chosen from $\Theta$. Utilizing the minimax theorem and information theoretical tools, we derive the optimal universal learner, a mixture over the set of the data generating distributions, and get a closed form expression for the min-max regret. We show that this regret can be considered as a constrained version of the conditional capacity between the data and its generating distributions set. We present tight bounds for this min-max regret, implying that the complexity of the problem is dominated by the richness of the hypotheses models $\Theta$ and not by the data generating distributions set $\Phi$. We develop an extension to the Arimoto-Blahut algorithm for numerical evaluation of the regret and its capacity achieving prior distribution. We demonstrate our results for the case where the observations come from a $K$-parameters multinomial distributions while the hypothesis class $\Theta$ is only a subset of this family of distributions.

cs.LG↗

arXiv2024

Error Exponent in Agnostic PAC Learning

Statistical learning theory and the Probably Approximately Correct (PAC) criterion are the common approach to mathematical learning theory. PAC is widely used to analyze learning problems and algorithms, and have been studied thoroughly. Uniform worst case bounds on the convergence rate have been well established using, e.g., VC theory or Radamacher complexity. However, in a typical scenario the performance could be much better. In this paper, we consider PAC learning using a somewhat different tradeoff, the error exponent - a well established analysis method in Information Theory - which describes the exponential behavior of the probability that the risk will exceed a certain threshold as function of the sample size. We focus on binary classification and find, under some stability assumptions, an improved distribution dependent error exponent for a wide range of problems, establishing the exponential behavior of the PAC error probability in agnostic learning. Interestingly, under these assumptions, agnostic learning may have the same error exponent as realizable learning. The error exponent criterion can be applied to analyze knowledge distillation, a problem that so far lacks a theoretical analysis.

cs.LG↗

arXiv2023

Batches Stabilize the Minimum Norm Risk in High Dimensional Overparameterized Linear Regression

Learning algorithms that divide the data into batches are prevalent in many machine-learning applications, typically offering useful trade-offs between computational efficiency and performance. In this paper, we examine the benefits of batch-partitioning through the lens of a minimum-norm overparametrized linear regression model with isotropic Gaussian features. We suggest a natural small-batch version of the minimum-norm estimator and derive bounds on its quadratic risk. We then characterize the optimal batch size and show it is inversely proportional to the noise level, as well as to the overparametrization ratio. In contrast to minimum-norm, our estimator admits a stable risk behavior that is monotonically increasing in the overparametrization ratio, eliminating both the blowup at the interpolation point and the double-descent phenomenon. We further show that shrinking the batch minimum-norm estimator by a factor equal to the Weiner coefficient further stabilizes it and results in lower quadratic risk in all settings. Interestingly, we observe that the implicit regularization offered by the batch partition is partially explained by feature overlap between the batches. Our bound is derived via a novel combination of techniques, in particular normal approximation in the Wasserstein metric of noisy projections over random subspaces.

cs.LG↗

arXiv2022

Beyond Ridge Regression for Distribution-Free Data

In supervised batch learning, the predictive normalized maximum likelihood (pNML) has been proposed as the min-max regret solution for the distribution-free setting, where no distributional assumptions are made on the data. However, the pNML is not defined for a large capacity hypothesis class as over-parameterized linear regression. For a large class, a common approach is to use regularization or a model prior. In the context of online prediction where the min-max solution is the Normalized Maximum Likelihood (NML), it has been suggested to use NML with ``luckiness'': A prior-like function is applied to the hypothesis class, which reduces its effective size. Motivated by the luckiness concept, for linear regression we incorporate a luckiness function that penalizes the hypothesis proportionally to its l2 norm. This leads to the ridge regression solution. The associated pNML with luckiness (LpNML) prediction deviates from the ridge regression empirical risk minimizer (Ridge ERM): When the test data reside in the subspace corresponding to the small eigenvalues of the empirical correlation matrix of the training data, the prediction is shifted toward 0. Our LpNML reduces the Ridge ERM error by up to 20% for the PMLB sets, and is up to 4.9% more robust in the presence of distribution shift compared to recent leading methods for UCI sets.

cs.LG↗

arXiv2021

Single Layer Predictive Normalized Maximum Likelihood for Out-of-Distribution Detection

Detecting out-of-distribution (OOD) samples is vital for developing machine learning based models for critical safety systems. Common approaches for OOD detection assume access to some OOD samples during training which may not be available in a real-life scenario. Instead, we utilize the {\em predictive normalized maximum likelihood} (pNML) learner, in which no assumptions are made on the tested input. We derive an explicit expression of the pNML and its generalization error, denoted as the {\em regret}, for a single layer neural network (NN). We show that this learner generalizes well when (i) the test vector resides in a subspace spanned by the eigenvectors associated with the large eigenvalues of the empirical correlation matrix of the training data, or (ii) the test sample is far from the decision boundary. Furthermore, we describe how to efficiently apply the derived pNML regret to any pretrained deep NN, by employing the explicit pNML for the last layer, followed by the softmax function. Applying the derived regret to deep NN requires neither additional tunable parameters nor extra data. We extensively evaluate our approach on 74 OOD detection benchmarks using DenseNet-100, ResNet-34, and WideResNet-40 models trained with CIFAR-100, CIFAR-10, SVHN, and ImageNet-30 showing a significant improvement of up to 15.6\% over recent leading methods.

cs.LG↗

arXiv2021

Utilizing Adversarial Targeted Attacks to Boost Adversarial Robustness

Adversarial attacks have been shown to be highly effective at degrading the performance of deep neural networks (DNNs). The most prominent defense is adversarial training, a method for learning a robust model. Nevertheless, adversarial training does not make DNNs immune to adversarial perturbations. We propose a novel solution by adopting the recently suggested Predictive Normalized Maximum Likelihood. Specifically, our defense performs adversarial targeted attacks according to different hypotheses, where each hypothesis assumes a specific label for the test sample. Then, by comparing the hypothesis probabilities, we predict the label. Our refinement process corresponds to recent findings of the adversarial subspace properties. We extensively evaluate our approach on 16 adversarial attack benchmarks using ResNet-50, WideResNet-28, and a2-layer ConvNet trained with ImageNet, CIFAR10, and MNIST, showing a significant improvement of up to 5.7%, 3.7%, and 0.6% respectively.

cs.CV↗

arXiv2021

Distribution Free Uncertainty for the Minimum Norm Solution of Over-parameterized Linear Regression

A fundamental principle of learning theory is that there is a trade-off between the complexity of a prediction rule and its ability to generalize. Modern machine learning models do not obey this paradigm: They produce an accurate prediction even with a perfect fit to the training set. We investigate over-parameterized linear regression models focusing on the minimum norm solution: This is the solution with the minimal norm that attains a perfect fit to the training set. We utilize the recently proposed predictive normalized maximum likelihood (pNML) learner which is the min-max regret solution for the distribution-free setting. We derive an upper bound of this min-max regret which is associated with the prediction uncertainty. We show that if the test sample lies mostly in a subspace spanned by the eigenvectors associated with the large eigenvalues of the empirical correlation matrix of the training data, the model generalizes despite its over-parameterized nature. We demonstrate the use of the pNML regret as a point-wise learnability measure on synthetic data and successfully observe the double-decent phenomenon of the over-parameterized models on UCI datasets.

cs.LG↗

arXiv2021

Sequential prediction under log-loss and misspecification

We consider the question of sequential prediction under the log-loss in terms of cumulative regret. Namely, given a hypothesis class of distributions, learner sequentially predicts the (distribution of the) next letter in sequence and its performance is compared to the baseline of the best constant predictor from the hypothesis class. The well-specified case corresponds to an additional assumption that the data-generating distribution belongs to the hypothesis class as well. Here we present results in the more general misspecified case. Due to special properties of the log-loss, the same problem arises in the context of competitive-optimality in density estimation, and model selection. For the $d$-dimensional Gaussian location hypothesis class, we show that cumulative regrets in the well-specified and misspecified cases asymptotically coincide. In other words, we provide an $o(1)$ characterization of the distribution-free (or PAC) regret in this case -- the first such result as far as we know. We recall that the worst-case (or individual-sequence) regret in this case is larger by an additive constant ${d\over 2} + o(1)$. Surprisingly, neither the traditional Bayesian estimators, nor the Shtarkov's normalized maximum likelihood achieve the PAC regret and our estimator requires special "robustification" against heavy-tailed data. In addition, we show two general results for misspecified regret: the existence and uniqueness of the optimal estimator, and the bound sandwiching the misspecified regret between well-specified regrets with (asymptotically) close hypotheses classes.

cs.LG↗

arXiv2020

Efficient Data-Dependent Learnability

The predictive normalized maximum likelihood (pNML) approach has recently been proposed as the min-max optimal solution to the batch learning problem where both the training set and the test data feature are individuals, known sequences. This approach has yields a learnability measure that can also be interpreted as a stability measure. This measure has shown some potential in detecting out-of-distribution examples, yet it has considerable computational costs. In this project, we propose and analyze an approximation of the pNML, which is based on influence functions. Combining both theoretical analysis and experiments, we show that when applied to neural networks, this approximation can detect out-of-distribution examples effectively. We also compare its performance to that achieved by conducting a single gradient step for each possible label.

cs.LG↗

arXiv2020

Can Implicit Bias Explain Generalization? Stochastic Convex Optimization as a Case Study

The notion of implicit bias, or implicit regularization, has been suggested as a means to explain the surprising generalization ability of modern-days overparameterized learning algorithms. This notion refers to the tendency of the optimization algorithm towards a certain structured solution that often generalizes well. Recently, several papers have studied implicit regularization and were able to identify this phenomenon in various scenarios. We revisit this paradigm in arguably the simplest non-trivial setup, and study the implicit bias of Stochastic Gradient Descent (SGD) in the context of Stochastic Convex Optimization. As a first step, we provide a simple construction that rules out the existence of a \emph{distribution-independent} implicit regularizer that governs the generalization ability of SGD. We then demonstrate a learning problem that rules out a very general class of \emph{distribution-dependent} implicit regularizers from explaining generalization, which includes strongly convex regularizers as well as non-degenerate norm-based regularizations. Certain aspects of our constructions point out to significant difficulties in providing a comprehensive explanation of an algorithm's generalization performance by solely arguing about its implicit regularization properties.

cs.LG↗