Nishanth Dikkala

Kiran Vodrahalli, Santiago Ontanon, Nilesh Tripuraneni et al. · deepmind

We introduce Michelangelo: a minimal, synthetic, and unleaked long-context reasoning evaluation for large language models which is also easy to automatically score. This evaluation is derived via a novel, unifying framework for evaluations over arbitrarily long contexts which measure the model's ability to do more than retrieve a single piece of information from its context. The central idea of the Latent Structure Queries framework (LSQ) is to construct tasks which require a model to ``chisel away'' the irrelevant information in the context, revealing a latent structure in the context. To verify a model's understanding of this latent structure, we query the model for details of the structure. Using LSQ, we produce three diagnostic long-context evaluations across code and natural-language domains intended to provide a stronger signal of long-context language model capabilities. We perform evaluations on several state-of-the-art models and demonstrate both that a) the proposed evaluations are high-signal and b) that there is significant room for improvement in synthesizing long-context information.

Pranjal Awasthi, Nishanth Dikkala, Pritish Kamath

Recent investigations in noise contrastive estimation suggest, both empirically as well as theoretically, that while having more "negative samples" in the contrastive loss improves downstream classification performance initially, beyond a threshold, it hurts downstream performance due to a "collision-coverage" trade-off. But is such a phenomenon inherent in contrastive learning? We show in a simple theoretical setting, where positive pairs are generated by sampling from the underlying latent class (introduced by Saunshi et al. (ICML 2019)), that the downstream performance of the representation optimizing the (population) contrastive loss in fact does not degrade with the number of negative samples. Along the way, we give a structural characterization of the optimal representation in our framework, for noise contrastive estimation. We also provide empirical support for our theoretical results on CIFAR-10 and CIFAR-100 datasets.

Cenk Baykal, Nishanth Dikkala, Rina Panigrahy et al.

Deep and wide neural networks successfully fit very complex functions today, but dense models are starting to be prohibitively expensive for inference. To mitigate this, one promising direction is networks that activate a sparse subgraph of the network. The subgraph is chosen by a data-dependent routing function, enforcing a fixed mapping of inputs to subnetworks (e.g., the Mixture of Experts (MoE) paradigm in Switch Transformers). However, prior work is largely empirical, and while existing routing functions work well in practice, they do not lead to theoretical guarantees on approximation ability. We aim to provide a theoretical explanation for the power of sparse networks. As our first contribution, we present a formal model of data-dependent sparse networks that captures salient aspects of popular architectures. We then introduce a routing function based on locality sensitive hashing (LSH) that enables us to reason about how well sparse networks approximate target functions. After representing LSH-based sparse networks with our model, we prove that sparse networks can match the approximation power of dense networks on Lipschitz functions. Applying LSH on the input vectors means that the experts interpolate the target function in different subregions of the input space. To support our theory, we define various datasets based on Lipschitz target functions, and we show that sparse networks give a favorable trade-off between number of active units and approximation quality.

Cenk Baykal, Dylan Cutler, Nishanth Dikkala et al.

It has been well established that increasing scale in deep transformer networks leads to improved quality and performance. However, this increase in scale often comes with prohibitive increases in compute cost and inference latency. We introduce Alternating Updates (AltUp), a simple-to-implement method to increase a model's capacity without the computational burden. AltUp enables the widening of the learned representation, i.e., the token embedding, while only incurring a negligible increase in latency. AltUp achieves this by working on a subblock of the widened representation at each layer and using a predict-and-correct mechanism to update the inactivated blocks. We present extensions of AltUp, such as its applicability to the sequence dimension, and demonstrate how AltUp can be synergistically combined with existing approaches, such as Sparse Mixture-of-Experts models, to obtain efficient models with even higher capacity. Our experiments on benchmark transformer models and language tasks demonstrate the consistent effectiveness of AltUp on a diverse set of scenarios. Notably, on SuperGLUE and SQuAD benchmarks, AltUp enables up to $87\%$ speedup relative to the dense baselines at the same accuracy.

Kulin Shah, Nishanth Dikkala, Xin Wang et al.

Causal language modeling using the Transformer architecture has yielded remarkable capabilities in Large Language Models (LLMs) over the last few years. However, the extent to which fundamental search and reasoning capabilities emerged within LLMs remains a topic of ongoing debate. In this work, we study if causal language modeling can learn a complex task such as solving Sudoku puzzles. To solve a Sudoku, the model is first required to search over all empty cells of the puzzle to decide on a cell to fill and then apply an appropriate strategy to fill the decided cell. Sometimes, the application of a strategy only results in thinning down the possible values in a cell rather than concluding the exact value of the cell. In such cases, multiple strategies are applied one after the other to fill a single cell. We observe that Transformer models trained on this synthetic task can indeed learn to solve Sudokus (our model solves $94.21\%$ of the puzzles fully correctly) when trained on a logical sequence of steps taken by a solver. We find that training Transformers with the logical sequence of steps is necessary and without such training, they fail to learn Sudoku. We also extend our analysis to Zebra puzzles (known as Einstein puzzles) and show that the model solves $92.04 \%$ of the puzzles fully correctly. In addition, we study the internal representations of the trained Transformer and find that through linear probing, we can decode information about the set of possible values in any given cell from them, pointing to the presence of a strong reasoning engine implicit in the Transformer weights.

Cenk Baykal, Dylan J Cutler, Nishanth Dikkala et al.

One way of introducing sparsity into deep networks is by attaching an external table of parameters that is sparsely looked up at different layers of the network. By storing the bulk of the parameters in the external table, one can increase the capacity of the model without necessarily increasing the inference time. Two crucial questions in this setting are then: what is the lookup function for accessing the table and how are the contents of the table consumed? Prominent methods for accessing the table include 1) using words/wordpieces token-ids as table indices, 2) LSH hashing the token vector in each layer into a table of buckets, and 3) learnable softmax style routing to a table entry. The ways to consume the contents include adding/concatenating to input representation, and using the contents as expert networks that specialize to different inputs. In this work, we conduct rigorous experimental evaluations of existing ideas and their combinations. We also introduce a new method, alternating updates, that enables access to an increased token dimension without increasing the computation time, and demonstrate its effectiveness in language modeling.

Mehran Kazemi, Bahare Fatemi, Hritik Bansal et al. · deepmind

Large language models (LLMs) are increasingly deployed in everyday applications, demanding robust general reasoning capabilities and diverse reasoning skillset. However, current LLM reasoning benchmarks predominantly focus on mathematical and coding abilities, leaving a gap in evaluating broader reasoning proficiencies. One particular exception is the BIG-Bench dataset, which has served as a crucial benchmark for evaluating the general reasoning capabilities of LLMs, thanks to its diverse set of challenging tasks that allowed for a comprehensive assessment of general reasoning across various skills within a unified framework. However, recent advances in LLMs have led to saturation on BIG-Bench, and its harder version BIG-Bench Hard (BBH). State-of-the-art models achieve near-perfect scores on many tasks in BBH, thus diminishing its utility. To address this limitation, we introduce BIG-Bench Extra Hard (BBEH), a new benchmark designed to push the boundaries of LLM reasoning evaluation. BBEH replaces each task in BBH with a novel task that probes a similar reasoning capability but exhibits significantly increased difficulty. We evaluate various models on BBEH and observe a (harmonic) average accuracy of 9.8\% for the best general-purpose model and 44.8\% for the best reasoning-specialized model, indicating substantial room for improvement and highlighting the ongoing challenge of achieving robust general reasoning in LLMs. We release BBEH publicly at: https://github.com/google-deepmind/bbeh.

Gheorghe Comanici, Eric Bieber, Mike Schaekermann et al. · amazon-science, baidu

In this report, we introduce the Gemini 2.X model family: Gemini 2.5 Pro and Gemini 2.5 Flash, as well as our earlier Gemini 2.0 Flash and Flash-Lite models. Gemini 2.5 Pro is our most capable model yet, achieving SoTA performance on frontier coding and reasoning benchmarks. In addition to its incredible coding and reasoning skills, Gemini 2.5 Pro is a thinking model that excels at multimodal understanding and it is now able to process up to 3 hours of video content. Its unique combination of long context, multimodal and reasoning capabilities can be combined to unlock new agentic workflows. Gemini 2.5 Flash provides excellent reasoning abilities at a fraction of the compute and latency requirements and Gemini 2.0 Flash and Flash-Lite provide high performance at low latency and cost. Taken together, the Gemini 2.X model generation spans the full Pareto frontier of model capability vs cost, allowing users to explore the boundaries of what is possible with complex agentic problem solving.

Mehran Kazemi, Nishanth Dikkala, Ankit Anand et al.

With the continuous advancement of large language models (LLMs), it is essential to create new benchmarks to effectively evaluate their expanding capabilities and identify areas for improvement. This work focuses on multi-image reasoning, an emerging capability in state-of-the-art LLMs. We introduce ReMI, a dataset designed to assess LLMs' ability to Reason with Multiple Images. This dataset encompasses a diverse range of tasks, spanning various reasoning domains such as math, physics, logic, code, table/chart understanding, and spatial and temporal reasoning. It also covers a broad spectrum of characteristics found in multi-image reasoning scenarios. We have benchmarked several cutting-edge LLMs using ReMI and found a substantial gap between their performance and human-level proficiency. This highlights the challenges in multi-image reasoning and the need for further research. Our analysis also reveals the strengths and weaknesses of different models, shedding light on the types of reasoning that are currently attainable and areas where future models require improvement. To foster further research in this area, we are releasing ReMI publicly: https://huggingface.co/datasets/mehrankazemi/ReMI.

Nikunj Saunshi, Nishanth Dikkala, Zhiyuan Li et al.

Large language models have shown remarkable reasoning abilities and scaling laws suggest that large parameter count, especially along the depth axis, is the primary driver. In this work, we make a stronger claim -- many reasoning problems require a large depth but not necessarily many parameters. This unlocks a novel application of looped models for reasoning. Firstly, we show that for many synthetic reasoning problems like addition, $p$-hop induction, and math problems, a $k$-layer transformer looped $L$ times nearly matches the performance of a $kL$-layer non-looped model, and is significantly better than a $k$-layer model. This is further corroborated by theoretical results showing that many such reasoning problems can be solved via iterative algorithms, and thus, can be solved effectively using looped models with nearly optimal depth. Perhaps surprisingly, these benefits also translate to practical settings of language modeling -- on many downstream reasoning tasks, a language model with $k$-layers looped $L$ times can be competitive to, if not better than, a $kL$-layer language model. In fact, our empirical analysis reveals an intriguing phenomenon: looped and non-looped models exhibit scaling behavior that depends on their effective depth, akin to the inference-time scaling of chain-of-thought (CoT) reasoning. We further elucidate the connection to CoT reasoning by proving that looped models implicitly generate latent thoughts and can simulate $T$ steps of CoT with $T$ loops. Inspired by these findings, we also present an interesting dichotomy between reasoning and memorization, and design a looping-based regularization that is effective on both fronts.

Dylan Cutler, Arun Kandoor, Nishanth Dikkala et al.

Decoding in a Transformer based language model is inherently sequential as a token's embedding needs to pass through all the layers in the network before the generation of the next token can begin. In this work, we propose a new architecture StagFormer (Staggered Transformer), which staggers execution along the sequence axis and thereby enables parallelizing the decoding process along the depth of the model. We achieve this by breaking the dependency of the token representation at time step $i$ in layer $l$ upon the representations of tokens until time step $i$ from layer $l-1$. Instead, we stagger the execution and only allow a dependency on token representations until time step $i-1$. The later sections of the Transformer still get access to the "rich" representations from the prior section but only from those token positions which are one time step behind. StagFormer allows for different sections of the model to be executed in parallel yielding a potential speedup in decoding while being quality neutral in our simulations. We also explore many natural extensions of this idea. We present how weight-sharing across the different sections being staggered can be more practical in settings with limited memory. We explore the efficacy of using a bounded window attention to pass information from one section to another which helps drive further latency gains for some applications. We also explore the scalability of the staggering idea over more than 2 sections of the Transformer. Finally, we show how one can approximate a recurrent model during inference using weight-sharing. This variant can lead to substantial gains in quality for short generations while being neutral in its latency impact.

Guan Zhe Hong, Nishanth Dikkala, Enming Luo et al.

Due to the size and complexity of modern large language models (LLMs), it has proven challenging to uncover the underlying mechanisms that models use to solve reasoning problems. For instance, is their reasoning for a specific problem localized to certain parts of the network? Do they break down the reasoning problem into modular components that are then executed as sequential steps as we go deeper in the model? To better understand the reasoning capability of LLMs, we study a minimal propositional logic problem that requires combining multiple facts to arrive at a solution. By studying this problem on Mistral and Gemma models, up to 27B parameters, we illuminate the core components the models use to solve such logic problems. From a mechanistic interpretability point of view, we use causal mediation analysis to uncover the pathways and components of the LLMs' reasoning processes. Then, we offer fine-grained insights into the functions of attention heads in different layers. We not only find a sparse circuit that computes the answer, but we decompose it into sub-circuits that have four distinct and modular uses. Finally, we reveal that three distinct models -- Mistral-7B, Gemma-2-9B and Gemma-2-27B -- contain analogous but not identical mechanisms.

Pranjal Awasthi, Nishanth Dikkala, Pritish Kamath et al.

A core component present in many successful neural network architectures, is an MLP block of two fully connected layers with a non-linear activation in between. An intriguing phenomenon observed empirically, including in transformer architectures, is that, after training, the activations in the hidden layer of this MLP block tend to be extremely sparse on any given input. Unlike traditional forms of sparsity, where there are neurons/weights which can be deleted from the network, this form of {\em dynamic} activation sparsity appears to be harder to exploit to get more efficient networks. Motivated by this we initiate a formal study of PAC learnability of MLP layers that exhibit activation sparsity. We present a variety of results showing that such classes of functions do lead to provable computational and statistical advantages over their non-sparse counterparts. Our hope is that a better theoretical understanding of {\em sparsely activated} networks would lead to methods that can exploit activation sparsity in practice.

Yuval Dagan, Constantinos Daskalakis, Nishanth Dikkala et al.

We consider a general statistical estimation problem wherein binary labels across different observations are not independent conditioned on their feature vectors, but dependent, capturing settings where e.g. these observations are collected on a spatial domain, a temporal domain, or a social network, which induce dependencies. We model these dependencies in the language of Markov Random Fields and, importantly, allow these dependencies to be substantial, i.e do not assume that the Markov Random Field capturing these dependencies is in high temperature. As our main contribution we provide algorithms and statistically efficient estimation rates for this model, giving several instantiations of our bounds in logistic regression, sparse logistic regression, and neural network settings with dependent data. Our estimation guarantees follow from novel results for estimating the parameters (i.e. external fields and interaction strengths) of Ising models from a {\em single} sample. {We evaluate our estimation approach on real networked data, showing that it outperforms standard regression approaches that ignore dependencies, across three text classification datasets: Cora, Citeseer and Pubmed.}

Nishanth Dikkala, Gal Kaplun, Rina Panigrahy

It is well established that training deep neural networks gives useful representations that capture essential features of the inputs. However, these representations are poorly understood in theory and practice. In the context of supervised learning an important question is whether these representations capture features informative for classification, while filtering out non-informative noisy ones. We explore a formalization of this question by considering a generative process where each class is associated with a high-dimensional manifold and different classes define different manifolds. Under this model, each input is produced using two latent vectors: (i) a "manifold identifier" $γ$ and; (ii)~a "transformation parameter" $θ$ that shifts examples along the surface of a manifold. E.g., $γ$ might represent a canonical image of a dog, and $θ$ might stand for variations in pose, background or lighting. We provide theoretical and empirical evidence that neural representations can be viewed as LSH-like functions that map each input to an embedding that is a function of solely the informative $γ$ and invariant to $θ$, effectively recovering the manifold identifier $γ$. An important consequence of this behavior is one-shot learning to unseen classes.

Nishanth Dikkala, Greg Lewis, Lester Mackey et al.

We develop an approach for estimating models described via conditional moment restrictions, with a prototypical application being non-parametric instrumental variable regression. We introduce a min-max criterion function, under which the estimation problem can be thought of as solving a zero-sum game between a modeler who is optimizing over the hypothesis space of the target model and an adversary who identifies violating moments over a test function space. We analyze the statistical estimation rate of the resulting estimator for arbitrary hypothesis spaces, with respect to an appropriate analogue of the mean squared error metric, for ill-posed inverse problems. We show that when the minimax criterion is regularized with a second moment penalty on the test function and the test function space is sufficiently rich, then the estimation rate scales with the critical radius of the hypothesis and test function spaces, a quantity which typically gives tight fast rates. Our main result follows from a novel localized Rademacher analysis of statistical learning problems defined via minimax objectives. We provide applications of our main results for several hypothesis spaces used in practice such as: reproducing kernel Hilbert spaces, high dimensional sparse linear functions, spaces defined via shape constraints, ensemble estimators such as random forests, and neural networks. For each of these applications we provide computationally efficient optimization methods for solving the corresponding minimax problem (e.g. stochastic first-order heuristics for neural networks). In several applications, we show how our modified mean squared error rate, combined with conditions that bound the ill-posedness of the inverse problem, lead to mean squared error rates. We conclude with an extensive experimental analysis of the proposed methods.

Yuval Dagan, Constantinos Daskalakis, Nishanth Dikkala et al.

There have been two separate lines of work on estimating Ising models: (1) estimating them from multiple independent samples under minimal assumptions about the model's interaction matrix; and (2) estimating them from one sample in restrictive settings. We propose a unified framework that smoothly interpolates between these two settings, enabling significantly richer estimation guarantees from one, a few, or many samples. Our main theorem provides guarantees for one-sample estimation, quantifying the estimation error in terms of the metric entropy of a family of interaction matrices. As corollaries of our main theorem, we derive bounds when the model's interaction matrix is a (sparse) linear combination of known matrices, or it belongs to a finite set, or to a high-dimensional manifold. In fact, our main result handles multiple independent samples by viewing them as one sample from a larger model, and can be used to derive estimation bounds that are qualitatively similar to those obtained in the afore-described multiple-sample literature. Our technical approach benefits from sparsifying a model's interaction network, conditioning on subsets of variables that make the dependencies in the resulting conditional distribution sufficiently weak. We use this sparsification technique to prove strong concentration and anti-concentration results for the Ising model, which we believe have applications beyond the scope of this paper.

Constantinos Daskalakis, Nishanth Dikkala, Ioannis Panageas

Spin glass models, such as the Sherrington-Kirkpatrick, Hopfield and Ising models, are all well-studied members of the exponential family of discrete distributions, and have been influential in a number of application domains where they are used to model correlation phenomena on networks. Conventionally these models have quadratic sufficient statistics and consequently capture correlations arising from pairwise interactions. In this work we study extensions of these to models with higher-order sufficient statistics, modeling behavior on a social network with peer-group effects. In particular, we model binary outcomes on a network as a higher-order spin glass, where the behavior of an individual depends on a linear function of their own vector of covariates and some polynomial function of the behavior of others, capturing peer-group effects. Using a {\em single}, high-dimensional sample from such model our goal is to recover the coefficients of the linear function as well as the strength of the peer-group effects. The heart of our result is a novel approach for showing strong concavity of the log pseudo-likelihood of the model, implying statistical error rate of $\sqrt{d/n}$ for the Maximum Pseudo-Likelihood Estimator (MPLE), where $d$ is the dimensionality of the covariate vectors and $n$ is the size of the network (number of nodes). Our model generalizes vanilla logistic regression as well as the peer-effect models studied in recent works, and our results extend these results to accommodate higher-order interactions.

Yuval Dagan, Constantinos Daskalakis, Nishanth Dikkala et al.

Statistical learning theory has largely focused on learning and generalization given independent and identically distributed (i.i.d.) samples. Motivated by applications involving time-series data, there has been a growing literature on learning and generalization in settings where data is sampled from an ergodic process. This work has also developed complexity measures, which appropriately extend the notion of Rademacher complexity to bound the generalization error and learning rates of hypothesis classes in this setting. Rather than time-series data, our work is motivated by settings where data is sampled on a network or a spatial domain, and thus do not fit well within the framework of prior work. We provide learning and generalization bounds for data that are complexly dependent, yet their distribution satisfies the standard Dobrushin's condition. Indeed, we show that the standard complexity measures of Gaussian and Rademacher complexities and VC dimension are sufficient measures of complexity for the purposes of bounding the generalization error and learning rates of hypothesis classes in our setting. Moreover, our generalization bounds only degrade by constant factors compared to their i.i.d. analogs, and our learnability bounds degrade by log factors in the size of the training set.

Constantinos Daskalakis, Nishanth Dikkala, Ioannis Panageas

The standard linear and logistic regression models assume that the response variables are independent, but share the same linear relationship to their corresponding vectors of covariates. The assumption that the response variables are independent is, however, too strong. In many applications, these responses are collected on nodes of a network, or some spatial or temporal domain, and are dependent. Examples abound in financial and meteorological applications, and dependencies naturally arise in social networks through peer effects. Regression with dependent responses has thus received a lot of attention in the Statistics and Economics literature, but there are no strong consistency results unless multiple independent samples of the vectors of dependent responses can be collected from these models. We present computationally and statistically efficient methods for linear and logistic regression models when the response variables are dependent on a network. Given one sample from a networked linear or logistic regression model and under mild assumptions, we prove strong consistency results for recovering the vector of coefficients and the strength of the dependencies, recovering the rates of standard regression under independent observations. We use projected gradient descent on the negative log-likelihood, or negative log-pseudolikelihood, and establish their strong convexity and consistency using concentration of measure for dependent random variables.

Constantinos Daskalakis, Nishanth Dikkala, Siddhartha Jayanti

Asynchronous Gibbs sampling has been recently shown to be fast-mixing and an accurate method for estimating probabilities of events on a small number of variables of a graphical model satisfying Dobrushin's condition~\cite{DeSaOR16}. We investigate whether it can be used to accurately estimate expectations of functions of {\em all the variables} of the model. Under the same condition, we show that the synchronous (sequential) and asynchronous Gibbs samplers can be coupled so that the expected Hamming distance between their (multivariate) samples remains bounded by $O(τ\log n),$ where $n$ is the number of variables in the graphical model, and $τ$ is a measure of the asynchronicity. A similar bound holds for any constant power of the Hamming distance. Hence, the expectation of any function that is Lipschitz with respect to a power of the Hamming distance, can be estimated with a bias that grows logarithmically in $n$. Going beyond Lipschitz functions, we consider the bias arising from asynchronicity in estimating the expectation of polynomial functions of all variables in the model. Using recent concentration of measure results, we show that the bias introduced by the asynchronicity is of smaller order than the standard deviation of the function value already present in the true model. We perform experiments on a multi-processor machine to empirically illustrate our theoretical findings.

Ran Canetti, Aloni Cohen, Nishanth Dikkala et al.

A popular methodology for building binary decision-making classifiers in the presence of imperfect information is to first construct a non-binary "scoring" classifier that is calibrated over all protected groups, and then to post-process this score to obtain a binary decision. We study the feasibility of achieving various fairness properties by post-processing calibrated scores, and then show that deferring post-processors allow for more fairness conditions to hold on the final decision. Specifically, we show: 1. There does not exist a general way to post-process a calibrated classifier to equalize protected groups' positive or negative predictive value (PPV or NPV). For certain "nice" calibrated classifiers, either PPV or NPV can be equalized when the post-processor uses different thresholds across protected groups, though there exist distributions of calibrated scores for which the two measures cannot be both equalized. When the post-processing consists of a single global threshold across all groups, natural fairness properties, such as equalizing PPV in a nontrivial way, do not hold even for "nice" classifiers. 2. When the post-processing is allowed to `defer' on some decisions (that is, to avoid making a decision by handing off some examples to a separate process), then for the non-deferred decisions, the resulting classifier can be made to equalize PPV, NPV, false positive rate (FPR) and false negative rate (FNR) across the protected groups. This suggests a way to partially evade the impossibility results of Chouldechova and Kleinberg et al., which preclude equalizing all of these measures simultaneously. We also present different deferring strategies and show how they affect the fairness properties of the overall system. We evaluate our post-processing techniques using the COMPAS data set from 2016.

Constantinos Daskalakis, Nishanth Dikkala, Gautam Kamath

We prove near-tight concentration of measure for polynomial functions of the Ising model under high temperature. For any degree $d$, we show that a degree-$d$ polynomial of a $n$-spin Ising model exhibits exponential tails that scale as $\exp(-r^{2/d})$ at radius $r=\tildeΩ_d(n^{d/2})$. Our concentration radius is optimal up to logarithmic factors for constant $d$, improving known results by polynomial factors in the number of spins. We demonstrate the efficacy of polynomial functions as statistics for testing the strength of interactions in social networks in both synthetic and real world data.

Constantinos Daskalakis, Nishanth Dikkala, Nick Gravin

Classical distribution testing assumes access to i.i.d. samples from the distribution that is being tested. We initiate the study of Markov chain testing, assuming access to a single trajectory of a Markov Chain. In particular, we observe a single trajectory X0,...,Xt,... of an unknown, symmetric, and finite state Markov Chain M. We do not control the starting state X0, and we cannot restart the chain. Given our single trajectory, the goal is to test whether M is identical to a model Markov Chain M0 , or far from it under an appropriate notion of difference. We propose a measure of difference between two Markov chains, motivated by the early work of Kazakos [Kaz78], which captures the scaling behavior of the total variation distance between trajectories sampled from the Markov chains as the length of these trajectories grows. We provide efficient testers and information-theoretic lower bounds for testing identity of symmetric Markov chains under our proposed measure of difference, which are tight up to logarithmic factors if the hitting times of the model chain M0 is O(n) in the size of the state space n.

Constantinos Daskalakis, Nishanth Dikkala, Gautam Kamath

Given samples from an unknown multivariate distribution $p$, is it possible to distinguish whether $p$ is the product of its marginals versus $p$ being far from every product distribution? Similarly, is it possible to distinguish whether $p$ equals a given distribution $q$ versus $p$ and $q$ being far from each other? These problems of testing independence and goodness-of-fit have received enormous attention in statistics, information theory, and theoretical computer science, with sample-optimal algorithms known in several interesting regimes of parameters. Unfortunately, it has also been understood that these problems become intractable in large dimensions, necessitating exponential sample complexity. Motivated by the exponential lower bounds for general distributions as well as the ubiquity of Markov Random Fields (MRFs) in the modeling of high-dimensional distributions, we initiate the study of distribution testing on structured multivariate distributions, and in particular the prototypical example of MRFs: the Ising Model. We demonstrate that, in this structured setting, we can avoid the curse of dimensionality, obtaining sample and time efficient testers for independence and goodness-of-fit. One of the key technical challenges we face along the way is bounding the variance of functions of the Ising model.