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Fundamental Limits of Online and Distributed Algorithms for Statistical Learning and Estimation

Ohad Shamir

arXiv:1311.3494v6cs.LGstat.ML

TL;DR

The paper addresses the lack of a general information-theoretic account of how constraints on interaction with training data affect learning performance. It develops a framework for such constraints and identifies settings where constrained algorithms cannot match unconstrained performance, while noting that its sparse-estimation hardness construction uses a tailored, spiky distribution.

  • Problem

    The paper addresses the lack of generic tools to quantify how memory, communication, or limited access to data affect learning performance independently of problem semantics.

    Source: §1 · p. 2

  • Method

    The paper develops an information-theoretic framework based on constraints on how algorithms interact with data, encompassing memory-, communication-, and partial-information settings.

    Source: §1 · p. 2

  • Results

    The paper identifies cases across memory-, communication-, and partial-information learning where constrained algorithms have inferior statistical performance to constraint-free algorithms.

    Source: §5 · p. 10

  • Takeaways & Limitations

    The results provide a first step toward understanding how information constraints affect learning ability across several settings.

    Source: §5 · p. 10

    The sparse PCA and covariance-estimation hardness result uses a tailored, spiky distribution, and similar results for natural distributions such as Gaussian remain open.

    Source: §5 · p. 11

Abstract

from arXiv · show

Many machine learning approaches are characterized by information constraints on how they interact with the training data. These include memory and sequential access constraints (e.g. fast first-order methods to solve stochastic optimization problems); communication constraints (e.g. distributed learning); partial access to the underlying data (e.g. missing features and multi-armed bandits) and more. However, currently we have little understanding how such information constraints fundamentally affect our performance, independent of the learning problem semantics. For example, are there learning problems where any algorithm which has small memory footprint (or can use any bounded number of bits from each example, or has certain communication constraints) will perform worse than what is possible without such constraints? In this paper, we describe how a single set of results implies positive answers to the above, for several different settings.

1 Introduction

The paper asks how information constraints affect learning performance independently of problem semantics, then develops a general framework and exhibits settings where constrained algorithms perform worse than unconstrained ones. Its examples span memory, communication, partial information, and sequential learning, while prior work often addressed narrower settings or did not establish statistical-learning trade-offs.

1 Introduction

  • A generic regret lower bound for partial-information online learning applies across feedback types, including coordinate feedback, projections, and restricted feedback signals.It remains applicable when the learner adaptively chooses which bits of each loss vector to retain.Source: §1 · p. 2
  • ˜Ω(d^2) memory is necessary for statistically optimal sample complexity in some sparse PCA and sparse covariance estimation problems.The paper presents this as a first formal memory/sample-complexity trade-off in statistical learning.Source: §1 · p. 3
  • ˜Ω(d^2) communication per machine can be insufficient for optimal performance in certain distributed problems, yielding a formal communication/sample-complexity trade-off for i.i.d. data.The stated regime permits communication budgets larger than the data dimension.Source: §1 · p. 3
  • Toy stochastic optimization problems exist where algorithms using memory linear in dimension, including stochastic gradient descent or mirror descent, cannot be statistically optimal.This gives examples beyond the sparse PCA and covariance-estimation memory trade-offs.Source: §1 · p. 3

Related Work

  • Prior work established some communication limits, but the cited results focused on parametric estimation with smaller communication budgets or did not cover same-distribution examples and larger budgets.The paper contrasts its detection problems, whose answers require only O(log(d)) bits, with earlier parametric-estimation results requiring O(d) bits to specify the answer.Source: Related Work · p. 3
  • Existing bandit lower bounds depend on feedback semantics, whereas this paper's results are described as generic across partial-information settings.The cited literature also leaves a gap between streaming or distributed lower bounds and statistical learning on i.i.d. data.Source: Related Work · p. 4

2 Information-Constrained Protocols

The paper models information-constrained algorithms as protocols that process batches of i.i.d. instances and pass bounded-bit messages to a final output function. This framework includes sequential memory-limited, distributed, and partial-information learning procedures.

2 Information-Constrained Protocols

  • A (b, n, m) protocol processes mn i.i.d. instances in m batches of n and limits each intermediate message to b bits before computing a final output.The message functions may depend on previous messages and may be randomized.Source: §2 · p. 4
  • For b-memory online protocols, n = 1 and each message is the bounded-size state updated after the next example.The final output depends on the last state; the passage notes that many gradient-based methods maintain state proportional to the parameter dimension.Source: §2 · p. 4
  • Partial-information online learning is a (b, 1, m) protocol in which only b bits are extracted from each d-dimensional loss vector.The passage gives most multi-armed bandit problems as examples.Source: §2 · p. 5
  • Mini-batch online algorithms update state from a b-dimensional vector extracted from each example or batch, such as a gradient or gradient average.The class includes distributed versions that parallelize mini-batch processing.Source: §2 · p. 5
  • The framework contrasts constrained protocols with constraint-free procedures that may interact with sampled instances in any manner.Message sizes may also vary across machines and be chosen in a data-dependent manner.Source: §2 · p. 5

3 Basic Results

The hide-and-seek problem asks algorithms to identify a single biased coordinate among d independent coordinates. The results contrast unconstrained detection with protocols limited in information or memory, and explain how those limits raise sample requirements.

3 Basic Results

  • Hide-and-seek uses product distributions on {−1,1}^d with one unknown coordinate whose expected value is 2ρ, and asks the learner to detect that coordinate.All other coordinates are unbiased; the sample consists of mn instances.Source: §3 · p. 5
  • Without information constraints, choosing the coordinate with the highest empirical average detects the biased coordinate using O(log(d)/ρ^2) samples.This guarantee follows from Hoeffding’s inequality and a union bound.Source: §3 · p. 5
  • With one-instance protocols, reliable detection requires m ≥ Ω((d/b)/ρ^2) samples, versus O(log(d)/ρ^2) without information constraints.The lower bound is optimal up to log-factors; a segmented online algorithm attains ˜O((d/b)/ρ^2) samples under the stated condition on ρ.Source: §3 · p. 6
  • For general (b,n,m) protocols, the results establish a sample-complexity trade-off when ρ > b log(d)/d, compared with the unconstrained O(log(d)/ρ^2) requirement.The theorem assumes ρ ≤ 1/(4n); the supplied passage states the lower bound’s threshold but does not provide its full expression.Source: §3 · pp. 5–6
  • Theorem 2’s proof also applies when n > 1, but its dependence on n is exponential.The general-protocol result is described as weaker in its dependence on the bias parameter.Source: §3 · p. 6
  • The proof bounds how much a b-bit message reveals about the unknown coordinate: its information about the relevant sample coordinate is at most b/d in expectation unless it already knows the target.The resulting contraction is O(ρ^2b/d) for n=1 and O(nρb/d) for general n; standard data processing alone gives a weaker bound.Source: §3 · p. 7

4 Applications

Across partial-feedback learning, a toy stochastic optimization task, and sparse PCA/covariance estimation, the section gives settings where information or memory constraints degrade performance. It derives regret lower bounds for partial feedback and statistical gaps for matrix estimation, with the sparse PCA construction relying on a spiky distribution.

4.1 Online Learning with Partial Information

  • With b bits retained from each d-dimensional loss vector, partial-information algorithms face a regret lower bound scaling with d/b and T, even when they choose the bits.The lower bound applies to general partial-feedback models, including settings with at most k observed features per example.Source: §4.1 · p. 8

4.2 Stochastic Optimization

  • In a toy stochastic optimization problem, protocols with b=O(d) incur higher expected error than constraint-free protocols.The construction reduces optimization to detecting a biased matrix entry; similar gaps extend to general protocols, including non-interactive distributed learning.Source: §4.2 · pp. 8–9

4.3 Sparse PCA, Sparse Covariance Estimation, and Detecting Correlations

  • For 2-sparse PCA, estimation reduces to detecting the uniquely correlated coordinate pair, and sparse-vector samples support a hide-and-seek construction despite rank-one matrices.The empirical maximum-correlation pair succeeds with probability at least 1 − d² exp(−mτ²/6).Source: §4.3 · pp. 9–10
  • For sparse PCA and covariance estimation, there are sample-size regimes where subquadratic-memory online algorithms cannot be statistically optimal.The demonstrated failure distribution is spiky; whether a similar result holds for natural distributions such as Gaussian remains open.Source: §4.3 · p. 10
  • The paper identifies the sparse PCA result as its first explicit demonstration that memory constraints can impose a statistical cost on a standard estimation problem.It distinguishes this cost from previously reported runtime constraints on sparse PCA.Source: §4.3 · p. 10

5 Discussion and Open Questions

The paper establishes performance gaps for information-constrained algorithms, while identifying open questions about the bounds, settings, and distributions to which those gaps apply.

5 Discussion and Open Questions

  • Whether Theorem 3 can be strengthened remains open; the authors conjecture a bound depending on mnρ^2b/d for any ρ ≤ O(1/√n).They state that this would allow similar gaps for (b, 1, m) and (b, n, m) protocols.Source: §5 · p. 10
  • Whether online or distributed algorithms are statistically inferior on convex stochastic optimization problems remains open; the discussed example is easily solvable but non-convex.Source: §5 · pp. 10–11
  • Extending the distributed-algorithm results from non-interactive or serial protocols to multi-round interactive protocols remains open.Source: §5 · p. 11
  • The sparse-PCA and covariance-estimation hardness result uses a tailored distribution with controlled tails that is spiky and not uniformly sub-Gaussian; similar results for natural distributions such as Gaussian remain open.Source: §5 · p. 11
  • More broadly, the paper identifies extending its results to other learning problems and information constraints as future work.Source: §5 · p. 11

A Proofs

The proofs use information-theoretic results and auxiliary lemmas, including a key bound on how constrained protocols distribute information across coordinates.

A Proofs

  • The proofs use standard quantities and results from information theory.Source: A Proofs · p. 14
  • The proofs also rely on auxiliary lemmas presented in Subsection A.1.Source: A Proofs · p. 14
  • Lemma 6 quantifies how information-constrained protocols cannot provide information on all coordinates simultaneously.Source: A Proofs · p. 14

A.1 Auxiliary Lemmas

The auxiliary lemmas provide bounds on coordinate information, probability, and concentration used in the proofs.

A.1 Auxiliary Lemmas

  • The probabilistic argument yields an index j with Pr0(˜J = j) ≤ 3/d and Prj(˜J = j) ≤ 3/d + 2B.Source: §A.1 · p. 15
  • For the balls-into-bins lemma, the derived expression is bounded by a constant less than 13.Source: §A.1 · pp. 16–17
  • If W takes at most 2^b values, the coordinate-averaged entropy bound is at most b/d.Source: §A.1 · pp. 17–18

A.2 Proof of Thm. 2

The proof handles a general (b, n, m) protocol by comparing message distributions under a uniform reference and a biased-coordinate alternative, then bounding their divergence.

A.2 Proof of Thm. 2

  • The proof establishes a more general result for any (b, n, m) protocol.Source: §A.2 · p. 18
  • The proof states that the theorem’s result follows for n = 1.Source: §A.2 · p. 18
  • The reference distribution Pr0 uses instances drawn uniformly from {−1, +1}^d, with no biased coordinate.Source: §A.2 · p. 18
  • For a message conditioned on prior messages, the proof analyzes KL divergence between its distributions under Pr0 and Prj.Source: §A.2 · p. 19
  • Lemma 6 bounds the information in each message, yielding an upper bound on the accumulated divergence across the protocol.Source: §A.2 · p. 21

A.3 Proof of Thm. 3

The proof compares the message distributions under a reference and biased-coordinate distribution, using relative entropy to bound how much the message reveals about the biased coordinate. A sparse auxiliary-variable construction yields an upper bound of 30nρb/d on the relevant divergence.

A.3 Proof of Thm. 3

  • The analysis defines Pr0 as choosing x uniformly from {−1,+1}^d, with no biased coordinate, and describes sampling under Prj using an auxiliary-vector process.The vector has independent entries, each equal to 1 with probability 4ρ; when vl=1, xj,l is 1 with probability 3/4.Source: §A.3 · p. 21, The 30nρb/d bound · p. 23
  • Conditioned on the auxiliary vector, the message’s information about coordinate xj is bounded using mutual information and the b-bit message constraint.When the auxiliary vector is zero, the reference and biased-coordinate distributions coincide, so the corresponding relative entropy is zero.Source: The 30nρb/d bound · pp. 23–24
  • The proof bounds the relative entropy between reference and biased-coordinate message distributions by 30nρb/d.The bound uses ρ ≤ 1/(4n) and an auxiliary vector whose entries equal 1 independently with probability 4ρ.Source: §A.3 · p. 21, The 30nρb/d bound · pp. 23, 25

A.4 Proof of Thm. 4

The proof turns a hypothetical small-regret guarantee into a procedure for detecting the biased coordinate. This conflicts with the detection bound from Theorem 2, yielding the stated regret lower bound.

A.4 Proof of Thm. 4

  • The construction makes coordinate j uniquely optimal, with choosing any other coordinate incurring expected loss difference ρ.The distributions are over {0,1}^d and set ρ = c2 min{1/4, √(d/(bT))}.Source: §A.4 · p. 25
  • The most commonly chosen coordinate estimates j, so a small-regret guarantee limits how often the learner can choose a coordinate other than j.Markov’s inequality converts the bound on incorrect choices into a lower bound on detection probability.Source: §A.4 · p. 25
  • A hypothetical regret guarantee below c1 min{T/4, √(dT/b)} implies detection probability above 0.874, while Theorem 2 bounds it below that threshold for some coordinate.The proof chooses c1 = 3.7 × 10^-4 and c2 = 5.9 × 10^-3, obtaining the contradiction for d > 3.Source: §A.4 · p. 25

A.5 Proof of Thm. 5

The proof reduces detecting a correlated coordinate pair in sparse instances to a hide-and-seek problem, then applies information bounds for constrained protocols. It also describes extending the construction to other sample sizes.

A.5 Proof of Thm. 5

  • The proof establishes the sparse hide-and-seek lower bound and applies it to (b,n,m) protocols and b-memory online protocols by tuning the batch size κ.A b-memory online protocol can be viewed as a protocol processing batches of κ instances, ignoring any remaining instances.Source: §A.5 · pp. 26–27
  • The constructed distributions satisfy the theorem’s moment and concentration conditions, with the target pair’s correlation exceeding that of other pairs by τ = 2ρ/(d−1).The proof verifies these properties using the distribution’s sparsity and a concentration bound on empirical coordinate products.Source: §A.5 · pp. 27–28
  • The correlated-pair detection problem reduces to hide-and-seek over d(d−1)/2 coordinates by examining the entries above the diagonal of xx⊤.Each instance is 2-sparse, and only the target coordinate pair has the specified slight correlation.Source: §A.5 · pp. 27–28
  • The construction can extend performance gaps to other sample sizes, including a regime where m is roughly linear in memory b.The passage describes adapting the support and bias of the distribution to obtain the extension, up to logarithmic factors.Source: §A.5 · p. 28

A.6 Proof of Thm. 6

The proof bounds information about the biased coordinate in a sparse-instance hide-and-seek problem using a reference distribution and the protocol’s message constraint. The resulting bound on the relevant divergence is 26mb/d.

A.6 Proof of Thm. 6

  • The proof represents each sequence of sparse instances by coordinate indices u and vectors x1,…,xd, then compares biased and reference message distributions conditioned on u.Under the reference distribution, instances are uniformly chosen from the signed coordinate vectors; under the alternative, one coordinate has bias ρ.Source: §A.6 · pp. 29, 31
  • Conditioned on u, the message’s information about xj is bounded using mutual information and the fact that each message contains at most b bits.The proof then controls the largest number of instances assigned to any coordinate using a balls-and-bins bound.Source: §A.6 · pp. 30–31
  • The proof bounds each individual term inside the expectation in Eq. (21) by 13b/d via Eq. (22), so Eq. (21) is at most 26mb/d, implying the required result.Source: §A.6 · p. 31

B Basic Results in Information Theory

This section reviews information-theoretic quantities used in the proofs of Theorems 3 and 6, focusing on discrete distributions over finite sets. It introduces entropy, mutual information, and relative entropy along with properties and inequalities used in the paper.

B Basic Results in Information Theory

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