SearcharxivSearch

arXiv subjects

Colin Sandon

Publications and source records attributed to Colin Sandon.

At least 19 recordsLinked to original sources

Soft Guidance Starts to Outperform CoT Prompting as LLMs Improve

Chain-of-Thought (CoT) prompting remains the standard baseline for evaluating models' reasoning abilities. Originally, this technique was introduced to elicit step-by-step reasoning from large language models (LLMs), which would otherwise tend to directly output the final answer. However, many modern LLMs produce CoT-style responses \textit{natively} when presented with reasoning tasks, which made us revisit the effectiveness of standard CoT prompting. We evaluate several modern mid-sized language models on a math problem-solving task and find that models specialized for reasoning achieve better performance in a simple zero-shot setting than when using few-shot CoT examples - significantly surpassing officially reported results at no additional cost (e.g., from $\sim$77\% to $\sim$84\% for Mathstral on GSM8K). For the tested general-purpose model, a zero-shot CoT prompt is also sufficient to outperform a few-shot CoT baseline. We attribute this to a `guidance-distraction' tradeoff: standard CoT prompting also demands style adaptation, formatting compliance, and potentially undesired contextualization, which can distract models from the core reasoning task. Our findings suggest that using standard CoT prompting increasingly acts as a source of distraction as models grow stronger.

cs.AI

Recovering Assignments with One-Sided Noise

We study the query complexity of recovering a planted assignment from a random constraint-satisfaction instance with one-sided noise. We consider the following 1-CNF recovery problem: an unknown binary string with $n/2$ ones and $n/2$ zeros is queried at individual variables. A query to a $1$-variable returns "$1$" with probability $p$ and "$0$" otherwise, while a $0$-variable always returns "$0$" (each query is a fresh noisy draw). The goal is to recover the binary string with probability at least $1 - \delta$. While the naive counting argument may suggest a query complexity of $\log_2 \binom{n}{n/2}=\Theta(n)$, we show that the query complexity is $(1+o(1))c(p) \frac{n}{2} \left( \log_2 n + \log_2(1/\delta)\right)$, where $c(p) = \tfrac{1}{-\log_2(1-p)}$. We then study planted $k$-CNF satisfaction with one-sided noise. Each $k$-set containing a $1$-variable is included as a clause independently with probability $p$, and an algorithm may ask whether any given $k$-set is a clause. Unlike the $1$-CNF case, a clause-existence query is one-shot: each $k$-set either is or is not a clause, so repeating yields no new information. The model is one-sided because an observed clause certifies that at least one queried variable is assigned 1, whereas its absence does not certify all are assigned 0. The goal is to recover the planted assignment with probability at least $1 - \delta$. The counting baseline is $\Theta(n)$, yet we prove a query complexity of $(1+o(1))\,c(p,k)\, \frac{n}{2}\left( \log_2 n + \log_2(1/\delta)\right)$, where $c(p,k) = \tfrac{1}{k(-\log_2(1-p))}$. These bounds are for adaptive algorithms. We also prove bounds for nonadaptive algorithms, showing that for fixed $p$, adaptivity gives a factor $\exp(\Theta(k))$ improvement. Our results also imply lower bounds for noisy sorting of $\{0,1\}$-valued strings, and we study a variant of the model with negations.

cs.DS

Tensor Reed-Muller Codes: Achieving Capacity with Quasilinear Decoding Time

Define the codewords of the Tensor Reed-Muller code $\mathsf{TRM}(r_1,m_1;r_2,m_2;\dots;r_t,m_t)$ to be the evaluation vectors of all multivariate polynomials in the variables $\left\{x_{ij}\right\}_{i=1,\dots,t}^{j=1,\dots m_i}$ with degree at most $r_i$ in the variables $x_{i1},x_{i2},\dots,x_{im_i}$. The generator matrix of $\mathsf{TRM}(r_1,m_1;\dots;r_t,m_t)$ is thus the tensor product of the generator matrices of the Reed-Muller codes $\mathsf{RM}(r_1,m_1),\dots, \mathsf{RM}(r_t,m_t)$. We show that for any constant rate $R$ below capacity, one can construct a Tensor Reed-Muller code $\mathsf{TRM}(r_1,m_1;\dotsc;r_t,m_t)$ of rate $R$ that is decodable in quasilinear time. For any blocklength $n$, we provide two constructions of such codes: 1) Our first construction (with $t=3$) has error probability $n^{-\omega(\log n)}$ and decoding time $O(n\log\log n)$. 2) Our second construction, for any $t\geq 4$, has error probability $2^{-n^{\frac{1}{2}-\frac{1}{2(t-2)}-o(1)}}$ and decoding time $O(n\log n)$. One of our main tools is a polynomial-time algorithm for decoding an arbitrary tensor code $C=C_1\otimes\dotsc\otimes C_t$ from $\frac{d_{\min}(C)}{2\max\{d_{\min}(C_1),\dotsc,d_{\min}(C_t) \}}-1$ adversarial errors. Crucially, this algorithm does not require the codes $C_1,\dotsc,C_t$ to themselves be decodable in polynomial time.

cs.IT

Finding Planted Cycles in a Random Graph

In this paper, we study the problem of finding a collection of planted cycles in an \ER random graph $G \sim \mathcal{G}(n, \lambda/n)$, in analogy to the famous Planted Clique Problem. When the cycles are planted on a uniformly random subset of $\delta n$ vertices, we show that almost-exact recovery (that is, recovering all but a vanishing fraction of planted-cycle edges as $n \to \infty$) is information-theoretically possible if $\lambda < \frac{1}{(\sqrt{2 \delta} + \sqrt{1-\delta})^2}$ and impossible if $\lambda > \frac{1}{(\sqrt{2 \delta} + \sqrt{1-\delta})^2}$. Moreover, despite the worst-case computational hardness of finding long cycles, we design a polynomial-time algorithm that attains almost exact recovery when $\lambda < \frac{1}{(\sqrt{2 \delta} + \sqrt{1-\delta})^2}$. This stands in stark contrast to the Planted Clique Problem, where a significant computational-statistical gap is widely conjectured.

math.ST

Reed-Muller Codes for Quantum Pauli and Multiple Access Channels

Reed-Muller (RM) codes have undergone significant analytical advancements over the past decade, particularly for binary memoryless symmetric (BMS) channels. We extend the scope of RM codes development and analysis to multiple-access channels (MACs) and quantum Pauli channels, leveraging a unified approach. Specifically, we first derive the achievable rate region for RM codes on so-called Q-MACs, a class of MACs with additive correlated noise. This is achieved via a generalization of the bending and boosting arguments defined in arXiv:2304.02509. We then put forward a connection between the rate region of these QMACs and quantum RM codes designed for Pauli noise channels. This connection highlights a universality property of quantum RM codes, demonstrating their rate-optimal performance across a range of channel parameters, rather than for a single Pauli channel.

cs.IT

Phase Transitions in Planted k-Factor Recovery

This paper studies the problem of inferring a $k$-factor, specifically a spanning $k$-regular graph, planted within an Erdos-Renyi random graph $G(n,\lambda/n)$. We show that as the average degree $\lambda$ surpasses the critical threshold of $1/k$, the inference problem undergoes a transition from almost exact recovery to partial recovery. Moreover, as $\lambda$ tends to infinity, the accuracy of recovery diminishes to zero. In addition, we characterize the recovery accuracy of a linear-time iterative pruning algorithm and show that it achieves almost exact recovery when $\lambda < 1/k$. A key component of our analysis is a two-step cycle construction: we first build trees through local neighborhood exploration and then connect them by sprinkling using reserved edges. Interestingly, for proving impossibility of almost exact recovery, we construct $\Theta(n)$ many small trees of size $\Theta(1)$, whereas for establishing the algorithmic lower bound, a single large tree of size $\Theta(\sqrt{n\log n})$ suffices.

math.PR

Polynomial Freiman-Ruzsa, Reed-Muller codes and Shannon capacity

In 1948, Shannon used a probabilistic argument to show the existence of codes achieving a maximal rate defined by the channel capacity. In 1954, Muller and Reed introduced a simple deterministic code construction based on polynomial evaluations, which was conjectured and eventually proven to achieve capacity. Meanwhile, polarization theory emerged as an analytic framework to prove capacity results for a variation of RM codes - the polar codes. Polarization theory further gave a powerful framework for various other code constructions, but it remained unfulfilled for RM codes. In this paper, we settle the establishment of a polarization theory for RM codes, which implies in particular that RM codes have a vanishing local error below capacity. Our proof puts forward a striking connection with the recent proof of the Polynomial Freiman-Ruzsa conjecture [40] and an entropy extraction approach related to [2]. It further puts forward a small orbit localization lemma of potential broader applicability in combinatorial number theory. Finally, a new additive combinatorics conjecture is put forward, with potentially broader applications to coding theory.

cs.IT

How Far Can Transformers Reason? The Globality Barrier and Inductive Scratchpad

Can Transformers predict new syllogisms by composing established ones? More generally, what type of targets can be learned by such models from scratch? Recent works show that Transformers can be Turing-complete in terms of expressivity, but this does not address the learnability objective. This paper puts forward the notion of 'globality degree' of a target distribution to capture when weak learning is efficiently achievable by regular Transformers. This measure shows a contrast with the expressivity results of Transformers captured by $TC^0/TC^1$ classes (further studied here), since the globality relates to correlations with the more limited $NC^0$ class. We show here experimentally and theoretically under additional assumptions that distributions with high globality cannot be learned efficiently. In particular, syllogisms cannot be composed on long chains. Further, we develop scratchpad techniques and show that: (i) agnostic scratchpads cannot break the globality barrier, (ii) educated scratchpads can break the globality with intermediate steps, although not all such scratchpads can generalize out-of-distribution (OOD), (iii) a notion of 'inductive scratchpad', that composes the prior information more efficiently, can both break the globality barrier and improve the OOD generalization. In particular, some of our inductive scratchpads can achieve length generalizations of up to $6\times$ for some arithmetic tasks depending on the input formatting.

cs.LG

Reed-Muller codes have vanishing bit-error probability below capacity: a simple tighter proof via camellia boosting

This paper shows that a class of codes such as Reed-Muller (RM) codes have vanishing bit-error probability below capacity on symmetric channels. The proof relies on the notion of `camellia codes': a class of symmetric codes decomposable into `camellias', i.e., set systems that differ from sunflowers by allowing for scattered petal overlaps. The proof then follows from a boosting argument on the camellia petals with second moment Fourier analysis. For erasure channels, this gives a self-contained proof of the bit-error result in Kudekar et al.'17, without relying on sharp thresholds for monotone properties Friedgut-Kalai'96. For error channels, this gives a shortened proof of Reeves-Pfister'23 with an exponentially tighter bound, and a proof variant of the bit-error result in Abbe-Sandon'23. The control of the full (block) error probability still requires Abbe-Sandon'23 for RM codes.

cs.IT

A proof that Reed-Muller codes achieve Shannon capacity on symmetric channels

Reed-Muller codes were introduced in 1954, with a simple explicit construction based on polynomial evaluations, and have long been conjectured to achieve Shannon capacity on symmetric channels. Major progress was made towards a proof over the last decades; using combinatorial weight enumerator bounds, a breakthrough on the erasure channel from sharp thresholds, hypercontractivity arguments, and polarization theory. Another major progress recently established that the bit error probability vanishes slowly below capacity. However, when channels allow for errors, the results of Bourgain-Kalai do not apply for converting a vanishing bit to a vanishing block error probability, neither do the known weight enumerator bounds. The conjecture that RM codes achieve Shannon capacity on symmetric channels, with high probability of recovering the codewords, has thus remained open. This paper closes the conjecture's proof. It uses a new recursive boosting framework, which aggregates the decoding of codeword restrictions on `subspace-sunflowers', handling their dependencies via an $L_p$ Boolean Fourier analysis, and using a list-decoding argument with a weight enumerator bound from Sberlo-Shpilka. The proof does not require a vanishing bit error probability for the base case, but only a non-trivial probability, obtained here for general symmetric codes. This gives in particular a shortened and tightened argument for the vanishing bit error probability result of Reeves-Pfister, and with prior works, it implies the strong wire-tap secrecy of RM codes on pure-state classical-quantum channels.

cs.IT

The Power of an Adversary in Glauber Dynamics

Glauber dynamics are a natural model of dynamics of dependent systems. While originally introduced in statistical physics, they have found important applications in the study of social networks, computer vision and other domains. In this work, we introduce a model of corrupted Glauber dynamics whereby instead of updating according to the prescribed conditional probabilities, some of the vertices and their updates are controlled by an adversary. We study the effect of such corruptions on global features of the system. Among the questions we study are: How many nodes need to be controlled in order to change the average statistics of the system in polynomial time? And how many nodes are needed to obstruct approximate convergence of the dynamics? Our results can be viewed as studying the robustness of classical sampling methods and are thus related to robust inference. The proofs connect to classical theory of Glauber dynamics from statistical physics.

math.PR

The Power of Two Matrices in Spectral Algorithms for Community Recovery

Spectral algorithms are some of the main tools in optimization and inference problems on graphs. Typically, the graph is encoded as a matrix and eigenvectors and eigenvalues of the matrix are then used to solve the given graph problem. Spectral algorithms have been successfully used for graph partitioning, hidden clique recovery and graph coloring. In this paper, we study the power of spectral algorithms using two matrices in a graph partitioning problem. We use two different matrices resulting from two different encodings of the same graph and then combine the spectral information coming from these two matrices. We analyze a two-matrix spectral algorithm for the problem of identifying latent community structure in large random graphs. In particular, we consider the problem of recovering community assignments exactly in the censored stochastic block model, where each edge status is revealed independently with some probability. We show that spectral algorithms based on two matrices are optimal and succeed in recovering communities up to the information theoretic threshold. Further, we show that for most choices of the parameters, any spectral algorithm based on one matrix is suboptimal. The latter observation is in contrast to our prior works (2022a, 2022b) which showed that for the symmetric Stochastic Block Model and the Planted Dense Subgraph problem, a spectral algorithm based on one matrix achieves the information theoretic threshold. We additionally provide more general geometric conditions for the (sub)-optimality of spectral algorithms.

math.ST

Spectral Algorithms Optimally Recover Planted Sub-structures

Spectral algorithms are an important building block in machine learning and graph algorithms. We are interested in studying when such algorithms can be applied directly to provide optimal solutions to inference tasks. Previous works by Abbe, Fan, Wang and Zhong (2020) and by Dhara, Gaudio, Mossel and Sandon (2022) showed the optimality for community detection in the Stochastic Block Model (SBM), as well as in a censored variant of the SBM. Here we show that this optimality is somewhat universal as it carries over to other planted substructures such as the planted dense subgraph problem and submatrix localization problem, as well as to a censored version of the planted dense subgraph problem.

cs.DS

On the Power of Differentiable Learning versus PAC and SQ Learning

We study the power of learning via mini-batch stochastic gradient descent (SGD) on the population loss, and batch Gradient Descent (GD) on the empirical loss, of a differentiable model or neural network, and ask what learning problems can be learnt using these paradigms. We show that SGD and GD can always simulate learning with statistical queries (SQ), but their ability to go beyond that depends on the precision $\rho$ of the gradient calculations relative to the minibatch size $b$ (for SGD) and sample size $m$ (for GD). With fine enough precision relative to minibatch size, namely when $b \rho$ is small enough, SGD can go beyond SQ learning and simulate any sample-based learning algorithm and thus its learning power is equivalent to that of PAC learning; this extends prior work that achieved this result for $b=1$. Similarly, with fine enough precision relative to the sample size $m$, GD can also simulate any sample-based learning algorithm based on $m$ samples. In particular, with polynomially many bits of precision (i.e. when $\rho$ is exponentially small), SGD and GD can both simulate PAC learning regardless of the mini-batch size. On the other hand, when $b \rho^2$ is large enough, the power of SGD is equivalent to that of SQ learning.

cs.LG

Spectral Recovery of Binary Censored Block Models

Community detection is the problem of identifying community structure in graphs. Often the graph is modeled as a sample from the Stochastic Block Model, in which each vertex belongs to a community. The probability that two vertices are connected by an edge depends on the communities of those vertices. In this paper, we consider a model of {\em censored} community detection with two communities, where most of the data is missing as the status of only a small fraction of the potential edges is revealed. In this model, vertices in the same community are connected with probability $p$ while vertices in opposite communities are connected with probability $q$. The connectivity status of a given pair of vertices $\{u,v\}$ is revealed with probability $\alpha$, independently across all pairs, where $\alpha = \frac{t \log(n)}{n}$. We establish the information-theoretic threshold $t_c(p,q)$, such that no algorithm succeeds in recovering the communities exactly when $t < t_c(p,q)$. We show that when $t > t_c(p,q)$, a simple spectral algorithm based on a weighted, signed adjacency matrix succeeds in recovering the communities exactly. While spectral algorithms are shown to have near-optimal performance in the symmetric case, we show that they may fail in the asymmetric case where the connection probabilities inside the two communities are allowed to be different. In particular, we show the existence of a parameter regime where a simple two-phase algorithm succeeds but any algorithm based on the top two eigenvectors of the weighted, signed adjacency matrix fails.

math.PR

Spoofing Generalization: When Can't You Trust Proprietary Models?

In this work, we study the computational complexity of determining whether a machine learning model that perfectly fits the training data will generalizes to unseen data. In particular, we study the power of a malicious agent whose goal is to construct a model g that fits its training data and nothing else, but is indistinguishable from an accurate model f. We say that g strongly spoofs f if no polynomial-time algorithm can tell them apart. If instead we restrict to algorithms that run in $n^c$ time for some fixed $c$, we say that g c-weakly spoofs f. Our main results are 1. Under cryptographic assumptions, strong spoofing is possible and 2. For any c> 0, c-weak spoofing is possible unconditionally While the assumption of a malicious agent is an extreme scenario (hopefully companies training large models are not malicious), we believe that it sheds light on the inherent difficulties of blindly trusting large proprietary models or data.

cs.LG

Learning to Sample from Censored Markov Random Fields

We study learning Censor Markov Random Fields (abbreviated CMRFs). These are Markov Random Fields where some of the nodes are censored (not observed). We present an algorithm for learning high-temperature CMRFs within o(n) transportation distance. Crucially our algorithm makes no assumption about the structure of the graph or the number or location of the observed nodes. We obtain stronger results for high girth high-temperature CMRFs as well as computational lower bounds indicating that our results can not be qualitatively improved.

cs.LG

Poly-time universality and limitations of deep learning

The goal of this paper is to characterize function distributions that deep learning can or cannot learn in poly-time. A universality result is proved for SGD-based deep learning and a non-universality result is proved for GD-based deep learning; this also gives a separation between SGD-based deep learning and statistical query algorithms: (1) {\it Deep learning with SGD is efficiently universal.} Any function distribution that can be learned from samples in poly-time can also be learned by a poly-size neural net trained with SGD on a poly-time initialization with poly-steps, poly-rate and possibly poly-noise. Therefore deep learning provides a universal learning paradigm: it was known that the approximation and estimation errors could be controlled with poly-size neural nets, using ERM that is NP-hard; this new result shows that the optimization error can also be controlled with SGD in poly-time. The picture changes for GD with large enough batches: (2) {\it Result (1) does not hold for GD:} Neural nets of poly-size trained with GD (full gradients or large enough batches) on any initialization with poly-steps, poly-range and at least poly-noise cannot learn any function distribution that has super-polynomial {\it cross-predictability,} where the cross-predictability gives a measure of ``average'' function correlation -- relations and distinctions to the statistical dimension are discussed. In particular, GD with these constraints can learn efficiently monomials of degree $k$ if and only if $k$ is constant. Thus (1) and (2) point to an interesting contrast: SGD is universal even with some poly-noise while full GD or SQ algorithms are not (e.g., parities).

cs.LG