arXiv subjects

Christos Tzamos

Publications and source records attributed to Christos Tzamos.

At least 19 recordsLinked to original sources

arXiv2026

A Derandomization Framework for Structure Discovery: Applications in Neural Networks and Beyond

Understanding the dynamics of feature learning in neural networks (NNs) remains a significant challenge. The work of (Mousavi-Hosseini et al., 2023) analyzes a multiple index teacher-student setting and shows that a two-layer student attains a low-rank structure in its first-layer weights when trained with stochastic gradient descent (SGD) and a strong regularizer. This structural property is known to reduce sample complexity of generalization. Indeed, in a second step, the same authors establish algorithm-specific learning guarantees under additional assumptions. In this paper, we focus exclusively on the structure discovery aspect and study it under weaker assumptions, more specifically: we allow (a) NNs of arbitrary size and depth, (b) with all parameters trainable, (c) under any smooth loss function, (d) tiny regularization, and (e) trained by any method that attains a second-order stationary point (SOSP), e.g.\ perturbed gradient descent (PGD). At the core of our approach is a key $\textit{derandomization}$ lemma, which states that optimizing the function $\mathbb{E}_{\mathbf{x}} \left[g_θ(\mathbf{W}\mathbf{x} + \mathbf{b})\right]$ converges to a point where $\mathbf{W} = \mathbf{0}$, under mild conditions. The fundamental nature of this lemma directly explains structure discovery and has immediate applications in other domains including an end-to-end approximation for MAXCUT, and computing Johnson-Lindenstrauss embeddings.

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arXiv2026

MaxSketch: Robust Distinct Counting in Streams via Random Projections

Estimating the number of distinct elements in a data stream is well understood when repeated elements are identical. In modern settings, however, observations are high-dimensional and noisy, so repeated instances of the same object are only approximately similar -- for example, different images of the same individual may vary significantly at the pixel level. Classical sketches such as HyperLogLog rely on consistent hash values for identical elements and break down in this regime. Recent work on robust distinct counting in general metric spaces achieves $\widetildeΘ(\sqrt{n})$ memory, which is tight in the worst case. We show that substantially improved memory guarantees are possible under geometric structure common in learned representations. We introduce MaxSketch, a simple max-linear sketch built from random Gaussian projections, and prove that it succeeds in estimating the number of distinct latent objects. Concretely, we show that under this assumption $m = \widetilde{O} (\log n / \varepsilon^2)$ random projections (and hence $\widetilde{O} (\log n/\varepsilon^2)$ memory) suffice to recover the true distinct count within a $(1+\varepsilon)$ factor. Experiments on image streams confirm that MaxSketch accurately estimates distinct counts and generalizes beyond the training regime. Our results bridge classical streaming algorithms and modern representation learning, showing how geometric structure can fundamentally reduce the complexity of distinct counting.

stat.ML↗

arXiv2026

Efficient Parameter Estimation of Truncated Boolean Product Distributions

We study the problem of estimating the parameters of a Boolean product distribution in $d$ dimensions, when the samples are truncated by a set $S \subset \{0, 1\}^d$ accessible through a membership oracle. This is the first time that the computational and statistical complexity of learning from truncated samples is considered in a discrete setting. We introduce a natural notion of fatness of the truncation set $S$, under which truncated samples reveal enough information about the true distribution. We show that if the truncation set is sufficiently fat, samples from the true distribution can be generated from truncated samples. A stunning consequence is that virtually any statistical task (e.g., learning in total variation distance, parameter estimation, uniformity or identity testing) that can be performed efficiently for Boolean product distributions, can also be performed from truncated samples, with a small increase in sample complexity. We generalize our approach to ranking distributions over $d$ alternatives, where we show how fatness implies efficient parameter estimation of Mallows models from truncated samples. Exploring the limits of learning discrete models from truncated samples, we identify three natural conditions that are necessary for efficient identifiability: (i) the truncation set $S$ should be rich enough; (ii) $S$ should be accessible through membership queries; and (iii) the truncation by $S$ should leave enough randomness in all directions. By carefully adapting the Stochastic Gradient Descent approach of (Daskalakis et al., FOCS 2018), we show that these conditions are also sufficient for efficient learning of truncated Boolean product distributions.

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arXiv2026

Teaching Transformers to Solve Combinatorial Problems through Efficient Trial & Error

Despite their proficiency in various language tasks, Large Language Models (LLMs) struggle with combinatorial problems like Satisfiability, Traveling Salesman Problem, or even basic arithmetic. We address this gap through a novel trial & error approach for solving problems in the class NP, where candidate solutions are iteratively generated and efficiently validated using verifiers. We focus on the paradigmatic task of Sudoku and achieve state-of-the-art accuracy (99%) compared to prior neuro-symbolic approaches. Unlike prior work that used custom architectures, our method employs a vanilla decoder-only Transformer (GPT-2) without external tools or function calling. Our method integrates imitation learning of simple Sudoku rules with an explicit Depth-First Search (DFS) exploration strategy involving informed guessing and backtracking. Moving beyond imitation learning, we seek to minimize the number of guesses until reaching a solution. This is achieved using depth-1 guessing, showing empirically that almost all Sudoku can be solved using the puzzle's rules with at most one guess. We provide a rigorous analysis of this setup formalizing its connection to a contextual variant of Min-Sum Set Cover, a well-studied problem in algorithms and stochastic optimization.

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arXiv2025

Learning Intersections of Two Margin Halfspaces under Factorizable Distributions

Learning intersections of halfspaces is a central problem in Computational Learning Theory. Even for just two halfspaces, it remains a major open question whether learning is possible in polynomial time with respect to the margin $γ$ of the data points and their dimensionality $d$. The best-known algorithms run in quasi-polynomial time $d^{O(\log(1/γ))}$, and it has been shown that this complexity is unavoidable for any algorithm relying solely on correlational statistical queries (CSQ). In this work, we introduce a novel algorithm that provably circumvents the CSQ hardness barrier. Our approach applies to a broad class of distributions satisfying a natural, previously studied, factorizability assumption. Factorizable distributions lie between distribution-specific and distribution-free settings, and significantly extend previously known tractable cases. Under these distributions, we show that CSQ-based methods still require quasipolynomial time even for weakly learning, whereas our algorithm achieves $poly(d,1/γ)$ time by leveraging more general statistical queries (SQ), establishing a strong separation between CSQ and SQ for this simple realizable PAC learning problem. Our result is grounded in a rigorous analysis utilizing a novel duality framework that characterizes the moment tensor structure induced by the marginal distributions. Building on these structural insights, we propose new, efficient learning algorithms. These algorithms combine a refined variant of Jennrich's Algorithm with PCA over random projections of the moment tensor, along with a gradient-descent-based non-convex optimization framework.

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arXiv2025

Statistical Query Hardness of Multiclass Linear Classification with Random Classification Noise

We study the task of Multiclass Linear Classification (MLC) in the distribution-free PAC model with Random Classification Noise (RCN). Specifically, the learner is given a set of labeled examples $(x, y)$, where $x$ is drawn from an unknown distribution on $R^d$ and the labels are generated by a multiclass linear classifier corrupted with RCN. That is, the label $y$ is flipped from $i$ to $j$ with probability $H_{ij}$ according to a known noise matrix $H$ with non-negative separation $σ: = \min_{i \neq j} H_{ii}-H_{ij}$. The goal is to compute a hypothesis with small 0-1 error. For the special case of two labels, prior work has given polynomial-time algorithms achieving the optimal error. Surprisingly, little is known about the complexity of this task even for three labels. As our main contribution, we show that the complexity of MLC with RCN becomes drastically different in the presence of three or more labels. Specifically, we prove super-polynomial Statistical Query (SQ) lower bounds for this problem. In more detail, even for three labels and constant separation, we give a super-polynomial lower bound on the complexity of any SQ algorithm achieving optimal error. For a larger number of labels and smaller separation, we show a super-polynomial SQ lower bound even for the weaker goal of achieving any constant factor approximation to the optimal loss or even beating the trivial hypothesis.

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arXiv2024

Optimization Can Learn Johnson Lindenstrauss Embeddings

Embeddings play a pivotal role across various disciplines, offering compact representations of complex data structures. Randomized methods like Johnson-Lindenstrauss (JL) provide state-of-the-art and essentially unimprovable theoretical guarantees for achieving such representations. These guarantees are worst-case and in particular, neither the analysis, nor the algorithm, takes into account any potential structural information of the data. The natural question is: must we randomize? Could we instead use an optimization-based approach, working directly with the data? A first answer is no: as we show, the distance-preserving objective of JL has a non-convex landscape over the space of projection matrices, with many bad stationary points. But this is not the final answer. We present a novel method motivated by diffusion models, that circumvents this fundamental challenge: rather than performing optimization directly over the space of projection matrices, we use optimization over the larger space of random solution samplers, gradually reducing the variance of the sampler. We show that by moving through this larger space, our objective converges to a deterministic (zero variance) solution, avoiding bad stationary points. This method can also be seen as an optimization-based derandomization approach and is an idea and method that we believe can be applied to many other problems.

stat.ML↗

arXiv2024

First Order Stochastic Optimization with Oblivious Noise

We initiate the study of stochastic optimization with oblivious noise, broadly generalizing the standard heavy-tailed noise setup. In our setting, in addition to random observation noise, the stochastic gradient may be subject to independent oblivious noise, which may not have bounded moments and is not necessarily centered. Specifically, we assume access to a noisy oracle for the stochastic gradient of $f$ at $x$, which returns a vector $\nabla f(γ, x) + ξ$, where $γ$ is the bounded variance observation noise and $ξ$ is the oblivious noise that is independent of $γ$ and $x$. The only assumption we make on the oblivious noise $ξ$ is that $\mathbf{Pr}[ξ= 0] \ge α$ for some $α\in (0, 1)$. In this setting, it is not information-theoretically possible to recover a single solution close to the target when the fraction of inliers $α$ is less than $1/2$. Our main result is an efficient list-decodable learner that recovers a small list of candidates, at least one of which is close to the true solution. On the other hand, if $α= 1-ε$, where $0< ε< 1/2$ is sufficiently small constant, the algorithm recovers a single solution. Along the way, we develop a rejection-sampling-based algorithm to perform noisy location estimation, which may be of independent interest.

cs.DS↗

arXiv2024

Active Learning with Simple Questions

We consider an active learning setting where a learner is presented with a pool S of n unlabeled examples belonging to a domain X and asks queries to find the underlying labeling that agrees with a target concept h^* \in H. In contrast to traditional active learning that queries a single example for its label, we study more general region queries that allow the learner to pick a subset of the domain T \subset X and a target label y and ask a labeler whether h^*(x) = y for every example in the set T \cap S. Such more powerful queries allow us to bypass the limitations of traditional active learning and use significantly fewer rounds of interactions to learn but can potentially lead to a significantly more complex query language. Our main contribution is quantifying the trade-off between the number of queries and the complexity of the query language used by the learner. We measure the complexity of the region queries via the VC dimension of the family of regions. We show that given any hypothesis class H with VC dimension d, one can design a region query family Q with VC dimension O(d) such that for every set of n examples S \subset X and every h^* \in H, a learner can submit O(d log n) queries from Q to a labeler and perfectly label S. We show a matching lower bound by designing a hypothesis class H with VC dimension d and a dataset S \subset X of size n such that any learning algorithm using any query class with VC dimension less than O(d) must make poly(n) queries to label S perfectly. Finally, we focus on well-studied hypothesis classes including unions of intervals, high-dimensional boxes, and d-dimensional halfspaces, and obtain stronger results. In particular, we design learning algorithms that (i) are computationally efficient and (ii) work even when the queries are not answered based on the learner's pool of examples S but on some unknown superset L of S

cs.LG↗

arXiv2024

Online Learning of Halfspaces with Massart Noise

We study the task of online learning in the presence of Massart noise. Instead of assuming that the online adversary chooses an arbitrary sequence of labels, we assume that the context $\mathbf{x}$ is selected adversarially but the label $y$ presented to the learner disagrees with the ground-truth label of $\mathbf{x}$ with unknown probability at most $η$. We study the fundamental class of $γ$-margin linear classifiers and present a computationally efficient algorithm that achieves mistake bound $ηT + o(T)$. Our mistake bound is qualitatively tight for efficient algorithms: it is known that even in the offline setting achieving classification error better than $η$ requires super-polynomial time in the SQ model. We extend our online learning model to a $k$-arm contextual bandit setting where the rewards -- instead of satisfying commonly used realizability assumptions -- are consistent (in expectation) with some linear ranking function with weight vector $\mathbf{w}^\ast$. Given a list of contexts $\mathbf{x}_1,\ldots \mathbf{x}_k$, if $\mathbf{w}^*\cdot \mathbf{x}_i > \mathbf{w}^* \cdot \mathbf{x}_j$, the expected reward of action $i$ must be larger than that of $j$ by at least $Δ$. We use our Massart online learner to design an efficient bandit algorithm that obtains expected reward at least $(1-1/k)~ ΔT - o(T)$ bigger than choosing a random action at every round.

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arXiv2024

Buy-Many Mechanisms for Many Unit-Demand Buyers

A recent line of research has established a novel desideratum for designing approximately-revenue-optimal multi-item mechanisms, namely the buy-many constraint. Under this constraint, prices for different allocations made by the mechanism must be subadditive, implying that the price of a bundle cannot exceed the sum of prices of individual items it contains. This natural constraint has enabled several positive results in multi-item mechanism design bypassing well-established impossibility results. Our work addresses the main open question from this literature of extending the buy-many constraint to multiple buyer settings and developing an approximation. We propose a new revenue benchmark for multi-buyer mechanisms via an ex-ante relaxation that captures several different ways of extending the buy-many constraint to the multi-buyer setting. Our main result is that a simple sequential item pricing mechanism with buyer-specific prices can achieve an $O(\log m)$ approximation to this revenue benchmark when all buyers have unit-demand or additive preferences over m items. This is the best possible as it directly matches the previous results for the single-buyer setting where no simple mechanism can obtain a better approximation. From a technical viewpoint we make two novel contributions. First, we develop a supply-constrained version of buy-many approximation for a single buyer. Second, we develop a multi-dimensional online contention resolution scheme for unit-demand buyers that may be of independent interest in mechanism design.

cs.GT↗

arXiv2024

Contextual Pandora's Box

Pandora's Box is a fundamental stochastic optimization problem, where the decision-maker must find a good alternative while minimizing the search cost of exploring the value of each alternative. In the original formulation, it is assumed that accurate distributions are given for the values of all the alternatives, while recent work studies the online variant of Pandora's Box where the distributions are originally unknown. In this work, we study Pandora's Box in the online setting, while incorporating context. At every round, we are presented with a number of alternatives each having a context, an exploration cost and an unknown value drawn from an unknown distribution that may change at every round. Our main result is a no-regret algorithm that performs comparably well to the optimal algorithm which knows all prior distributions exactly. Our algorithm works even in the bandit setting where the algorithm never learns the values of the alternatives that were not explored. The key technique that enables our result is a novel modification of the realizability condition in contextual bandits that connects a context to a sufficient statistic of each alternative's distribution (its "reservation value") rather than its mean.

cs.LG↗

arXiv2023

Agnostically Learning Multi-index Models with Queries

We study the power of query access for the task of agnostic learning under the Gaussian distribution. In the agnostic model, no assumptions are made on the labels and the goal is to compute a hypothesis that is competitive with the {\em best-fit} function in a known class, i.e., it achieves error $\mathrm{opt}+ε$, where $\mathrm{opt}$ is the error of the best function in the class. We focus on a general family of Multi-Index Models (MIMs), which are $d$-variate functions that depend only on few relevant directions, i.e., have the form $g(\mathbf{W} \mathbf{x})$ for an unknown link function $g$ and a $k \times d$ matrix $\mathbf{W}$. Multi-index models cover a wide range of commonly studied function classes, including constant-depth neural networks with ReLU activations, and intersections of halfspaces. Our main result shows that query access gives significant runtime improvements over random examples for agnostically learning MIMs. Under standard regularity assumptions for the link function (namely, bounded variation or surface area), we give an agnostic query learner for MIMs with complexity $O(k)^{\mathrm{poly}(1/ε)} \; \mathrm{poly}(d) $. In contrast, algorithms that rely only on random examples inherently require $d^{\mathrm{poly}(1/ε)}$ samples and runtime, even for the basic problem of agnostically learning a single ReLU or a halfspace. Our algorithmic result establishes a strong computational separation between the agnostic PAC and the agnostic PAC+Query models under the Gaussian distribution. Prior to our work, no such separation was known -- even for the special case of agnostically learning a single halfspace, for which it was an open problem first posed by Feldman. Our results are enabled by a general dimension-reduction technique that leverages query access to estimate gradients of (a smoothed version of) the underlying label function.

cs.LG↗

arXiv2023

Optimizing Solution-Samplers for Combinatorial Problems: The Landscape of Policy-Gradient Methods

Deep Neural Networks and Reinforcement Learning methods have empirically shown great promise in tackling challenging combinatorial problems. In those methods a deep neural network is used as a solution generator which is then trained by gradient-based methods (e.g., policy gradient) to successively obtain better solution distributions. In this work we introduce a novel theoretical framework for analyzing the effectiveness of such methods. We ask whether there exist generative models that (i) are expressive enough to generate approximately optimal solutions; (ii) have a tractable, i.e, polynomial in the size of the input, number of parameters; (iii) their optimization landscape is benign in the sense that it does not contain sub-optimal stationary points. Our main contribution is a positive answer to this question. Our result holds for a broad class of combinatorial problems including Max- and Min-Cut, Max-$k$-CSP, Maximum-Weight-Bipartite-Matching, and the Traveling Salesman Problem. As a byproduct of our analysis we introduce a novel regularization process over vanilla gradient descent and provide theoretical and experimental evidence that it helps address vanishing-gradient issues and escape bad stationary points.

cs.LG↗

arXiv2023

Distribution-Independent Regression for Generalized Linear Models with Oblivious Corruptions

We demonstrate the first algorithms for the problem of regression for generalized linear models (GLMs) in the presence of additive oblivious noise. We assume we have sample access to examples $(x, y)$ where $y$ is a noisy measurement of $g(w^* \cdot x)$. In particular, \new{the noisy labels are of the form} $y = g(w^* \cdot x) + ξ+ ε$, where $ξ$ is the oblivious noise drawn independently of $x$ \new{and satisfies} $\Pr[ξ= 0] \geq o(1)$, and $ε\sim \mathcal N(0, σ^2)$. Our goal is to accurately recover a \new{parameter vector $w$ such that the} function $g(w \cdot x)$ \new{has} arbitrarily small error when compared to the true values $g(w^* \cdot x)$, rather than the noisy measurements $y$. We present an algorithm that tackles \new{this} problem in its most general distribution-independent setting, where the solution may not \new{even} be identifiable. \new{Our} algorithm returns \new{an accurate estimate of} the solution if it is identifiable, and otherwise returns a small list of candidates, one of which is close to the true solution. Furthermore, we \new{provide} a necessary and sufficient condition for identifiability, which holds in broad settings. \new{Specifically,} the problem is identifiable when the quantile at which $ξ+ ε= 0$ is known, or when the family of hypotheses does not contain candidates that are nearly equal to a translated $g(w^* \cdot x) + A$ for some real number $A$, while also having large error when compared to $g(w^* \cdot x)$. This is the first \new{algorithmic} result for GLM regression \new{with oblivious noise} which can handle more than half the samples being arbitrarily corrupted. Prior work focused largely on the setting of linear regression, and gave algorithms under restrictive assumptions.

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arXiv2023

Self-Directed Linear Classification

In online classification, a learner is presented with a sequence of examples and aims to predict their labels in an online fashion so as to minimize the total number of mistakes. In the self-directed variant, the learner knows in advance the pool of examples and can adaptively choose the order in which predictions are made. Here we study the power of choosing the prediction order and establish the first strong separation between worst-order and random-order learning for the fundamental task of linear classification. Prior to our work, such a separation was known only for very restricted concept classes, e.g., one-dimensional thresholds or axis-aligned rectangles. We present two main results. If $X$ is a dataset of $n$ points drawn uniformly at random from the $d$-dimensional unit sphere, we design an efficient self-directed learner that makes $O(d \log \log(n))$ mistakes and classifies the entire dataset. If $X$ is an arbitrary $d$-dimensional dataset of size $n$, we design an efficient self-directed learner that predicts the labels of $99\%$ of the points in $X$ with mistake bound independent of $n$. In contrast, under a worst- or random-ordering, the number of mistakes must be at least $Ω(d \log n)$, even when the points are drawn uniformly from the unit sphere and the learner only needs to predict the labels for $1\%$ of them.

cs.LG↗

arXiv2023

Approximating Pandora's Box with Correlations

We revisit the classic Pandora's Box (PB) problem under correlated distributions on the box values. Recent work of arXiv:1911.01632 obtained constant approximate algorithms for a restricted class of policies for the problem that visit boxes in a fixed order. In this work, we study the complexity of approximating the optimal policy which may adaptively choose which box to visit next based on the values seen so far. Our main result establishes an approximation-preserving equivalence of PB to the well studied Uniform Decision Tree (UDT) problem from stochastic optimization and a variant of the Min-Sum Set Cover ($\text{MSSC}_f$) problem. For distributions of support $m$, UDT admits a $\log m$ approximation, and while a constant factor approximation in polynomial time is a long-standing open problem, constant factor approximations are achievable in subexponential time (arXiv:1906.11385). Our main result implies that the same properties hold for PB and $\text{MSSC}_f$. We also study the case where the distribution over values is given more succinctly as a mixture of $m$ product distributions. This problem is again related to a noisy variant of the Optimal Decision Tree which is significantly more challenging. We give a constant-factor approximation that runs in time $n^{ \tilde O( m^2/\varepsilon^2 ) }$ when the mixture components on every box are either identical or separated in TV distance by $\varepsilon$.

cs.DS↗

arXiv2023

Buying Information for Stochastic Optimization

Stochastic optimization is one of the central problems in Machine Learning and Theoretical Computer Science. In the standard model, the algorithm is given a fixed distribution known in advance. In practice though, one may acquire at a cost extra information to make better decisions. In this paper, we study how to buy information for stochastic optimization and formulate this question as an online learning problem. Assuming the learner has an oracle for the original optimization problem, we design a $2$-competitive deterministic algorithm and a $e/(e-1)$-competitive randomized algorithm for buying information. We show that this ratio is tight as the problem is equivalent to a robust generalization of the ski-rental problem, which we call super-martingale stopping. We also consider an adaptive setting where the learner can choose to buy information after taking some actions for the underlying optimization problem. We focus on the classic optimization problem, Min-Sum Set Cover, where the goal is to quickly find an action that covers a given request drawn from a known distribution. We provide an $8$-competitive algorithm running in polynomial time that chooses actions and decides when to buy information about the underlying request.

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