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Self-supervised Learning from a Multi-view Perspective

Yao-Hung Hubert Tsai, Yue Wu, Ruslan Salakhutdinov, Louis-Philippe Morency

arXiv:2006.05576v4cs.LGstat.ML

TL;DR

The paper addresses how self-supervised learning can succeed without downstream supervision by modeling inputs and self-supervised signals as redundant views. It develops an information-theoretical framework and composite objectives, showing that representations can retain task-relevant information, discard task-irrelevant information, and support broader objective design. The framework is also tested in visual and cross-modal settings, including cases where multi-view redundancy may fail.

  • Problem

    Self-supervised learning generalizes across downstream tasks, but the theoretical properties explaining its efficacy and the limits of the multi-view perspective remain insufficiently understood.

  • Method

    The paper develops an information-theoretical multi-view framework, connects contrastive and predictive objectives, and introduces a term for discarding task-irrelevant information.

  • Results

    Self-supervised representations can extract task-relevant information with potential loss and discard task-irrelevant information with a fixed gap, while composite contrastive-predictive objectives outperform either objective alone.

  • Takeaways & Limitations

    The framework helps characterize when self-supervised learning is likely to work and expands the space of objectives that can be composed for representation learning.

  • Takeaways & Limitations

    The analysis is constrained when inputs and self-supervised signals contain very different task-relevant information, producing a potentially large multi-view redundancy error.

Abstract

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As a subset of unsupervised representation learning, self-supervised representation learning adopts self-defined signals as supervision and uses the learned representation for downstream tasks, such as object detection and image captioning. Many proposed approaches for self-supervised learning follow naturally a multi-view perspective, where the input (e.g., original images) and the self-supervised signals (e.g., augmented images) can be seen as two redundant views of the data. Building from this multi-view perspective, this paper provides an information-theoretical framework to better understand the properties that encourage successful self-supervised learning. Specifically, we demonstrate that self-supervised learned representations can extract task-relevant information and discard task-irrelevant information. Our theoretical framework paves the way to a larger space of self-supervised learning objective design. In particular, we propose a composite objective that bridges the gap between prior contrastive and predictive learning objectives, and introduce an additional objective term to discard task-irrelevant information. To verify our analysis, we conduct controlled experiments to evaluate the impact of the composite objectives. We also explore our framework's empirical generalization beyond the multi-view perspective, where the cross-view redundancy may not be clearly observed.

1 INTRODUCTION

The paper frames self-supervised learning through a multi-view assumption in which inputs and self-supervised signals share task-relevant information. It theoretically shows that learned representations can retain task-relevant information while discarding task-irrelevant information, and uses this framework to design and evaluate composite objectives.

  • Motivation: Self-supervised learning uses proxy objectives between inputs and self-defined signals, yet its learning efficacy has received limited theoretical explanation despite broad downstream generalization.Examples include image classification, object detection, instance segmentation, and pose estimation using learned features.
  • Multi-view assumption: The multi-view assumption treats inputs and self-supervised signals as corresponding views, with either view approximately sufficient for downstream tasks.For image augmentations, this means style changes should preserve task-relevant content.
  • Theoretical contributions: Under this assumption, learned representations extract task-relevant information with potential loss and discard task-irrelevant information with a fixed gap.The framework also quantifies the smallest generalization error for classification from the retained information.
  • Objective design: The analysis connects contrastive and predictive self-supervised learning and supports composing their objectives to extract relevant information while discarding irrelevant information.The proposed combination reportedly performs better than contrastive-only or predictive-only objectives and has less over-fitting, while an additional objective targets task-irrelevant information.
  • Empirical evaluation: Controlled experiments compare objective compositions in visual settings where the assumption is likely to hold and visual-textual settings where shared task-relevant information may be limited.The second setting tests generalization when the multi-view assumption is likely to fail.

2 A MULTI-VIEW INFORMATION-THEORETICAL FRAMEWORK

The paper develops a multi-view information-theoretical framework for self-supervised learning, treating inputs and self-supervised signals as corresponding views whose shared information supports downstream tasks. It characterizes how self-supervised objectives extract task-relevant information, discard task-irrelevant information, connect contrastive and predictive learning, and bound downstream classification error.

  • 2.1 MULTI-VIEW ASSUMPTION: The framework assumes the input X and self-supervised signal S share most task-relevant information, formalized by I(X; T|S) ≤ ϵinfo; drastic augmentations or excessive masking can violate this assumption.The assumption is motivated by augmentations that alter style without changing content, but large noise or masking may remove or separate task-relevant information.
  • 2.2 LEARNING MINIMAL AND SUFFICIENT REPRESENTATIONS FOR SELF-SUPERVISION: Self-supervised representations retain nearly all task-relevant information from X, with a potential loss bounded by ϵinfo under the multi-view assumption.The theorem compares self-supervised representations with supervised representations and allows downstream tasks beyond classification, regression, or clustering.
  • 2.2 LEARNING MINIMAL AND SUFFICIENT REPRESENTATIONS FOR SELF-SUPERVISION: Minimal sufficient self-supervised representations discard task-irrelevant information up to the unavoidable compression gap I(X; S|T), the shared input-signal information unrelated to the task.This gap is larger when downstream tasks require only part of the shared information.
  • 2.3 CONNECTIONS WITH CONTRASTIVE AND PREDICTIVE LEARNING OBJECTIVES: Contrastive and forward predictive objectives extract task-relevant information by maximizing mutual information or its equivalent conditional-likelihood formulation, while inverse predictive learning discards task-irrelevant information.The inverse objective reconstructs the representation from the self-supervised signal under mutual-information maximization.
  • 2.3 CONNECTIONS WITH CONTRASTIVE AND PREDICTIVE LEARNING OBJECTIVES: The composite loss LSSL = λCLLCL + λFPLFP + λIPLIP combines contrastive, forward predictive, and inverse predictive terms to extract task-relevant and discard task-irrelevant information simultaneously.Optimizing the inverse predictive objective alone can yield degenerate constant representations, motivating composition with the other objectives.
  • 2.4 THEORETICAL ANALYSIS - BAYES ERROR RATE FOR DOWNSTREAM CLASSIFICATION: The mutual-information estimator can approximate I(ZX; S) at rate O(1/√n), linking finite-sample estimation to the downstream Bayes-error guarantees.This result is stated for a suitable neural-network estimator under the proposition’s parameter and confidence conditions.
  • 2.4 THEORETICAL ANALYSIS - BAYES ERROR RATE FOR DOWNSTREAM CLASSIFICATION: The Bayes-error analysis shows that downstream classification error improves when estimated mutual information is accurate, sample size is larger, task-irrelevant information is small, and the compression gap is small.The framework also states that minimal self-supervised representations are preferable because they contain no superfluous information conditioned on S and T.

3 CONTROLLED EXPERIMENTS

The controlled experiments evaluate SSL objectives under same-modality and cross-modality settings, showing benefits from combining objectives that preserve useful information and compress representations.

  • Experimental design: The experiments use visual data where the multi-view redundancy assumption is likely to hold and visual-textual data where it may fail.The visual experiments use Omniglot and CIFAR10, while the cross-modality experiments use MS COCO.
  • Experiment I - Visual Representation Learning: Adding inverse-predictive learning to contrastive learning boosts downstream performance, although results are sensitive to λ_IP.The authors interpret the improvement as evidence that more compressed representations can improve downstream performance.
  • Experiment I - Visual Representation Learning: Combining contrastive and forward-predictive objectives brings together contrastive learning’s stronger accuracy and faster convergence with predictive learning’s complementary behavior.Contrastive learning also suffers from overfitting with long-epoch training in the reported comparison.
  • Experiment II - Visual-Textual Representation Learning: For visual-textual learning, the authors use LCL with λ_IPL_IP and evaluate fixed visual representations on MS COCO multi-label classification.They report Micro ROC-AUC and Subset Accuracy on the validation set.
  • Experiment II - Visual-Textual Representation Learning: Cross-modal SSL remains effective: pretrained ResNet with BERT outperforms pretrained ResNet alone, while pretraining vision helps more than pretraining text.Combining LCL and LIP further improves the learned representations, suggesting that LIP can discard task-irrelevant information.
  • Optimization remarks: The inverse-predictive weight acts as a Lagrangian coefficient balancing maximized shared information against minimized conditional entropy.Its best value varies with parameterization, though empirically the LIP scale is often about one-tenth of the contrastive or forward-predictive loss scale.

4 RELATED WORK

The related-work discussion situates the framework among theoretical SSL analyses, information-bottleneck methods, and InfoMin, emphasizing both shared extraction of task-relevant information and this paper’s additional compression objective.

  • Theoretical SSL analyses: The paper connects its assumptions to prior multi-view theories, adopting I(T; X|S) ≈ 0 rather than the stronger I(X; S|T) ≈ 0 assumption criticized as impractical.This distinction defines the framework’s relationship to Arora et al., Lee et al., and Tosh et al.
  • Theoretical SSL analyses: Unlike prior analyses focused on extracting task-relevant information, this work also proposes discarding task-irrelevant information and quantifying what cannot be discarded.The comparison is made against contrastive and predictive SSL theories.
  • Information bottleneck methods: Compared with the Multi-view Information Bottleneck, the method minimizes H(Z_X|S) instead of I(Z_X; X|S), enabling optimization through inverse-predictive learning.Both approaches aim to reduce redundant information across views.
  • InfoMin: Compared with InfoMin, the method keeps standard augmentations and learns representations containing no more than the views’ shared information, rather than learning minimal augmentations first.Both approaches seek representations that avoid excessive information.
  • Empirical relationship to prior SSL: Empirically, the framework explains how contrastive and predictive objectives extract task-relevant information and shows that adding its compression objective benefits performance.The proposed objective therefore extends rather than replaces those SSL approaches.

5 CONCLUSION

The paper concludes that SSL can extract task-relevant information while discarding task-irrelevant information, and it identifies connections to minimal-sufficient representation learning.

  • 5 CONCLUSION: The framework supports both theoretical and empirical analyses of SSL, including contrastive and predictive objectives.The authors present it as a way to understand when and why SSL is likely to work.
  • 5 CONCLUSION: Self-supervised representations can retain task-relevant information with a potential loss and discard task-irrelevant information with a fixed gap.The conclusion states this as the paper’s central theoretical and empirical finding.
  • 5 CONCLUSION: The paper plans to extend the framework to SSL methods that do not fit its current analysis, including BYOL, SWAV, and Uniformity-Alignment.This identifies the current framework’s method coverage boundary.
  • A REMARKS ON LEARNING MINIMAL AND SUFFICIENT REPRESENTATIONS: Under maximized I(Z_X;T), minimizing I(Z_X;X) is equivalent to minimizing I(Z_X;X|T) and H(Z_X|T) for deterministic representations.The equivalence follows from the stated information identities and determinism.
  • A REMARKS ON LEARNING MINIMAL AND SUFFICIENT REPRESENTATIONS: The same deterministic-representation equivalence connects minimizing I(Z_X;X|S) with minimizing H(Z_X|S), while direct conditional-mutual-information optimization may be unstable.Federici et al. use a Gaussian-encoder upper bound to address that optimization issue.

B PROOFS FOR THEOREM 1 AND 2

The proofs use deterministic encoders and data-processing arguments to establish information bounds for supervised and self-supervised representations.

  • Proof setup: Deterministic encoding of X induces the Markov chain S ↔ T ↔ X → Z_X through conditional independence.This is the basis for the subsequent data-processing inequalities.
  • Theorem restatements: The restated theorem concludes that self-supervised representations retain all task-relevant information up to potential loss ϵ_info, while supervised representations retain all of it.The theorem compares the self-supervised and supervised information quantities directly.
  • Proof organization: The proof is organized into separate supervised and self-supervised cases before deriving the final inequalities.The supplied proof outline explicitly identifies these two parts.
  • Supervised Learned Representations: The supervised proof applies data processing to show that maximizing I(Z_X;T) reaches I(X;T).The result is stated for the supervised learned representations.
  • Self-supervised Learned Representations: For self-supervised representations, data processing bounds I(Z_X;S), I(Z_X;S;T), and I(Z_X;S|T) by their corresponding input quantities.The proof first expands the mutual-information expressions involving X, S, and T.

3. I(Zsslmin

Under the redundancy assumption, self-supervised representations retain nearly all task-relevant information, while minimal representations discard task-irrelevant information up to a fixed compression gap.

  • 3. I(Zsslmin: I(X; T|S) ≤ ϵinfo implies self-supervised representations retain at least I(X; T) − ϵinfo task-relevant information.The bound follows from the redundancy assumption and the information-theoretic proof.
  • 3. I(Zsslmin: The sufficient minimal self-supervised representation contains task-irrelevant information amounting to I(X; S|T), which cannot be discarded from the input.This is the fixed compression gap separating task-relevant sufficiency from minimality.
  • 3. I(Zsslmin: The proof decomposes input information into self-supervised-signal information and residual information conditioned on both the signal and task.The decomposition uses the Markov-chain data-processing inequality.
  • 3. I(Zsslmin: Minimality is characterized by vanishing conditional mutual information: I(Zsslmin X ; X|S, T) = 0 for the self-supervised minimum and I(Zsupmin X ; X|T) = 0 for the supervised minimum.

C PROOF FOR PROPOSITION 1

The proposition establishes a finite-sample mutual-information estimator using a compact neural-network family, with the proof based on approximation and uniform convergence arguments.

  • C PROOF FOR PROPOSITION 1: A compact neural-network family contains an estimator achieving the target mutual information with probability at least 1 − δ.The proposition asserts existence of a dimension d, parameter space Θ, and parameter θ* under the stated sampling setup.
  • C PROOF FOR PROPOSITION 1: The proof combines boundedness and Lipschitzness, universal neural-network approximation, and uniform convergence controlled by covering numbers.The cited prior analyses provide alternative versions of the bound, with Tsai et al. (2020) using a tighter covering-number bound than Belghazi et al. (2018).

D PROOFS FOR THEOREM 3 AND 4

This section derives Bayes-error bounds for arbitrary, supervised, and self-supervised learned representations by relating conditional entropy to task information.

  • D PROOFS FOR THEOREM 3 AND 4: The self-supervised analysis decomposes representation information into signal information, task-conditioned signal information, and residual information conditioned on signal and task.Data processing bounds the task-conditioned signal information by I(X; S|T).
  • D PROOFS FOR THEOREM 3 AND 4: Theorem 7 bounds the Bayes error of an arbitrary learned representation through its mutual information with the task and residual input information.The proof uses I(ZX; T) = H(T) − H(T|ZX) together with a lower bound involving I(ZX; S), I(X; S|T), and I(ZX; X|S, T).
  • D PROOFS FOR THEOREM 3 AND 4: The proofs use the feasible Bayes-error range 0 ≤ Pe ≤ 1 − 1/|T| to complete the lower- and upper-bound arguments.The same range is used after applying the conditional-entropy inequalities.
  • D PROOFS FOR THEOREM 3 AND 4: Theorem 8 gives corresponding Bayes-error bounds for supervised and self-supervised representations, including a gap controlled by ϵinfo.The derivation combines entropy–error inequalities with the information bounds established earlier.

E TIGHTER BOUNDS FOR THE BAYES ERROR RATES

The appendix tightens the Bayes-error bounds and outlines additional visual-learning experiments concerning architectures, objective deployment, and signal construction.

  • E TIGHTER BOUNDS FOR THE BAYES ERROR RATES: The tighter bound defines H+(Pe) as H(Pe) + Pe log(|T| − 1), where H(x) = −x log(x) − (1 − x) log(1 − x).
  • E TIGHTER BOUNDS FOR THE BAYES ERROR RATES: Tighter entropy–error inequalities improve Theorems 7 and 8 into sharper Bayes-error bounds for arbitrary and self-supervised representations.The improved bounds replace the looser entropy inequalities with H−(Pe) ≤ H(T|ZX) ≤ H+(Pe).
  • E TIGHTER BOUNDS FOR THE BAYES ERROR RATES: The refined self-supervised bound retains the supervised-versus-self-supervised comparison while incorporating the tighter entropy constraints.Theorem 10 states the corresponding result for self-supervised learned representations.
  • F MORE ON VISUAL REPRESENTATION LEARNING EXPERIMENTS: Additional visual-learning experiments are proposed around architecture design, contrastive versus forward-predictive deployment, and self-supervised signal construction.The appendix presents these as interesting future work beyond the controlled experiments in the main text.

F.1 ARCHITECTURE DESIGN

The appendix specifies the visual representation architecture and evaluates alternative objective deployments and self-supervised signal-construction strategies. Results indicate that objective formulation and signal construction materially affect test generalization, although the latter comparison incorporates label information indirectly.

  • F.1 ARCHITECTURE DESIGN: The architecture processes 28 × 28 images into 1024-dimensional L2-normalized features using convolutional, pooling, and linear layers, with a symmetric reconstruction network.The same network designs are used for mutual-information and conditional-entropy estimations, and the reconstruction network has the same parameter count as the feature network.
  • F.1 ARCHITECTURE DESIGN: Figure 5 compares compositions of self-supervised objectives, while Figure 6 compares signal-construction strategies using LCL and reports means with standard errors from five trials.Figure 6 contrasts the proposed drawing-style-plus-augmentation differences with SimCLR’s augmentation-only differences.
  • F.2 DIFFERENT DEPLOYMENTS FOR CONTRASTIVE AND PREDICTIVE LEARNING OBJECTIVES: Different contrastive and predictive objective deployments produce very different test generalization performance, making deployment selection important for a given task.The experiments vary CPC or JS for LCL and MSE, BCE, or RevBCE for the predictive reconstruction loss; RevBCE is selected in the main text after performing best among those losses.
  • F.3 DIFFERENT SELF-SUPERVISED SIGNAL CONSTRUCTION STRATEGY: The proposed signal-construction strategy generalizes better to the test set than the common augmentation-only strategy, but it implicitly uses label information and is therefore not directly comparable.The input and signal differ in drawing styles and image augmentations, whereas prior strategies differ only in augmentations; signals are sampled from the same character as the input.

G METRICS IN VISUAL-TEXTUAL REPRESENTATION LEARNING

The visual-textual evaluation uses multi-label metrics that assess exact-match correctness and ranking quality. These metrics complement one another by measuring complete prediction agreement and ROC-based discrimination.

  • G METRICS IN VISUAL-TEXTUAL REPRESENTATION LEARNING: Subset Accuracy, also called Exact Match Ratio, counts only fully correct multi-label outputs and treats partially correct outputs as incorrect.
  • G METRICS IN VISUAL-TEXTUAL REPRESENTATION LEARNING: Micro AUC ROC measures the area under a micro-averaged receiver operating characteristic curve.
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