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Stability Bound for Stationary Phi-mixing and Beta-mixing Processes
Mehryar Mohri, Afshin Rostamizadeh
TL;DR
Existing stability-based generalization bounds mainly assume i.i.d. data, limiting their direct use for temporally dependent observations. This paper derives stability bounds for stationary phi-mixing and beta-mixing sequences, extending the analysis to non-i.i.d. settings and illustrating it with common learning algorithms.
Problem
Existing stability analyses and bounds assume i.i.d. samples, although many machine-learning observations have temporal dependence.
Method
The paper derives stability-based generalization bounds for stationary phi-mixing and beta-mixing sequences using mixing-aware concentration and independent-block techniques.
Results
The bounds extend i.i.d. stability bounds to non-i.i.d. scenarios and apply to stable algorithms, including SVR, KRR, and SVMs.
Takeaways & Limitations
The results provide a theoretical basis for analyzing these stable learning algorithms in the non-i.i.d. scenarios studied.
Takeaways & Limitations
Using the bounds requires knowledge of some mixing properties of the distributions.
Abstract
from arXiv · showhide
Most generalization bounds in learning theory are based on some measure of the complexity of the hypothesis class used, independently of any algorithm. In contrast, the notion of algorithmic stability can be used to derive tight generalization bounds that are tailored to specific learning algorithms by exploiting their particular properties. However, as in much of learning theory, existing stability analyses and bounds apply only in the scenario where the samples are independently and identically distributed. In many machine learning applications, however, this assumption does not hold. The observations received by the learning algorithm often have some inherent temporal dependence. This paper studies the scenario where the observations are drawn from a stationary phi-mixing or beta-mixing sequence, a widely adopted assumption in the study of non-i.i.d. processes that implies a dependence between observations weakening over time. We prove novel and distinct stability-based generalization bounds for stationary phi-mixing and beta-mixing sequences. These bounds strictly generalize the bounds given in the i.i.d. case and apply to all stable learning algorithms, thereby extending the use of stability-bounds to non-i.i.d. scenarios. We also illustrate the application of our phi-mixing generalization bounds to general classes of learning algorithms, including Support Vector Regression, Kernel Ridge Regression, and Support Vector Machines, and many other kernel regularization-based and relative entropy-based regularization algorithms. These novel bounds can thus be viewed as the first theoretical basis for the use of these algorithms in non-i.i.d. scenarios.
1. Introduction
The paper extends algorithmic-stability generalization bounds from i.i.d. samples to stationary phi-mixing and beta-mixing sequences. It derives distinct bounds for both settings and illustrates the phi-mixing results on several learning algorithms.
- Existing stability analyses generally assume i.i.d. samples, although many applications involve temporally dependent observations.Examples include system diagnosis, time-series prediction, and financial data.
- The paper proves stability-based generalization bounds for stationary phi-mixing and beta-mixing sequences.The mixing assumption models dependence that weakens over time.
- The phi-mixing analysis uses a generalized McDiarmid inequality and yields bounds with the standard exponential form.The beta-mixing analysis instead uses standard McDiarmid’s inequality.
- The bounds apply to stable algorithms and are illustrated for Support Vector Regression, Kernel Ridge Regression, and Support Vector Machines.The paper also discusses kernel regularization-based and relative entropy-based algorithms.
- The paper presents its results as stability-based extensions to non-i.i.d. learning and organizes them around definitions, phi-mixing bounds, applications, and beta-mixing bounds.
2. Preliminaries
The paper introduces standard definitions for dependent observations in mixing theory before discussing non-i.i.d. learning scenarios.
- The preliminaries establish definitions for dependent observations in mixing theory.
- The paper then briefly frames learning problems in the non-i.i.d. setting.
- These preliminaries prepare the analysis of learning with dependent data.
2.1 Non-i.i.d. Definitions
This section defines stationarity and mixing coefficients, distinguishes beta-mixing from phi-mixing, and introduces the independent-block technique used for dependent observations.
- Stationarity means that shifting a finite block of a sequence does not change its distribution, but it does not imply independence.
- A process is beta-mixing or phi-mixing when its corresponding dependence coefficient tends to zero as the time gap increases.The coefficients can decay algebraically or exponentially.
- Beta-mixing is weaker than phi-mixing and therefore covers a more general non-i.i.d. scenario.
- The paper gives stability-based bounds for both mixing cases; phi-mixing bounds are simpler and have the standard exponential form.The beta-mixing bounds use the more general setting and include an additive term depending on the mixing coefficient.
- The independent-block technique compares dependent blocks with independent blocks while preserving the distribution within each block.For beta-mixing, the comparison error depends on the smallest gap between blocks.
2.2 Learning Scenarios
The paper formulates learning with dependent stationary samples, explains why i.i.d.-based stability proofs fail, and replaces them with mixing-aware concentration and independent-block arguments. It considers both realistic dependent test points and a less general independent-test setting.
- The learning setting uses labeled samples from a stationary mixing sequence, with the goal of achieving low error on future samples.
- In the most general setting, future test samples depend on the training sample, so generalization error is measured conditionally on that sample.
- A less realistic setting assumes dependent training samples but test points independent of the training sample; earlier studies focused on this case.The paper states that previous bounds cannot be directly applied to the more general setting.
- Non-stationary sequences make generalization error time-dependent, and an averaged-over-time alternative may be difficult to compute and use.
- Stability controls how the learned hypothesis changes when one training point is replaced, enabling stability-based generalization bounds.The paper uses a stability condition introduced in earlier work and combines it with concentration inequalities.
- Because dependence breaks the replacement equality used by i.i.d. proofs, the analysis removes blocks and replaces them with independent blocks before applying mixing bounds.The construction uses blocks that may have unequal sizes and differs from earlier equal-block methods.
3.1 Lipschitz Bound
The analysis compares generalization errors when test points depend on training data and bounds their sensitivity to replacing one sample point. Stability controls hypothesis changes, while mixing arguments handle dependent sequences.
- Dependent test points: The proof relates dependent test-point errors to expectations involving a sequence with the last b points removed.The reduced sequence preserves the corresponding block distribution, while dependence contributes a mixing term.
- Stability control: ˆβ-stability bounds the effect of changing one training point on the returned hypotheses and their costs.The argument assumes a bounded cost function with upper bound M.
- Lipschitz control: The resulting Lipschitz bound enables concentration inequalities to be applied to the generalization-error function.The construction is used for samples drawn from a β-mixing stationary process.
- Lipschitz control: The proof first bounds the difference between empirical errors and then bounds the difference between true errors.Costs on agreeing points are controlled by stability, while the differing point is controlled by M.
3.2 Bound on Expectation
The expectation analysis replaces dependent block sequences with independent blocks having matching distributions, then uses stability and bounded costs to control the resulting differences.
- Expectation control: The proof bounds the expected generalization-error function by analyzing independent blocks.This supplies the expectation control required after establishing a Lipschitz bound.
- Block construction: Removing up to three blocks of at most b points from each hypothesis incurs a stability-controlled error.The construction may produce as many as four blocks before independent counterparts are introduced.
- Block comparison: Applying the mixing lemma to two bounded cost functions yields an additional error controlled by M.The proof compares sequences whose corresponding blocks are independent but distributionally matched.
- Block comparison: Stationarity allows an independent replacement point to be substituted without changing its distribution, after which stability completes the comparison.The same steps establish the opposite inequality.
3.3 ϕ-mixing Generalization Bounds
The paper combines stability with concentration inequalities to obtain generalization bounds for stationary ϕ-mixing sequences. The ϕ-mixing results have an exponential form, recover the i.i.d. bound when mixing vanishes, and support applications to stable algorithms.
- Concentration: The ϕ-mixing analysis uses a generalized McDiarmid inequality adapted to dependent sequences.The concentration result applies to Lipschitz functions under ϕ-mixing distributions.
- Concentration: The paper reports that its concentration theorem improves the prior theorem’s exponent by a factor of 4.The improvement follows from using the general form of McDiarmid’s inequality instead of Azuma’s inequality.
- General bound: Theorem 10 gives a general stability bound for hypotheses trained on stationary ϕ-mixing samples.It assumes a measurable non-negative cost bounded by M and permits a block parameter b.
- Algebraic mixing: For algebraically ϕ-mixing sequences, Theorem 11 optimizes the block parameter under ϕ(k) = ϕ0k^-r with r > 1.The theorem is obtained by optimizing Theorem 10 and substituting the resulting parameter.
- Relation to i.i.d.: When the mixing coefficient is zero, the bounds coincide with the i.i.d. stability bound.For convergence, the paper requires ˆβ = o(1/√m); it also discusses ˆβ ≤ O(1/m) for several algorithm classes.
- Applications: The resulting bounds are applied to several general algorithm classes in the algebraically mixing setting.The applications use existing stability analyses for kernel regularization, k-local rules, and relative entropy regularization.
3.4 Applications
The paper applies its stability bounds to kernel-regularized and relative-entropy-regularized algorithms under boundedness and admissibility assumptions. The applications include SVM, SVR, KRR, and convex combinations of base hypotheses.
- Kernel regularization: Kernel regularization is formulated through a regularized objective in a reproducing kernel Hilbert space.The kernel is positive definite and symmetric, and the regularization parameter controls the objective.
- Assumptions: The analysis assumes bounded outputs and labels, while differentiability simplifies presentation but is not necessary for the results.The boundedness conditions apply to hypotheses, inputs, and output labels through a common bound M.
- Kernel regularization: A bounded kernel and σ-admissible cost give kernel regularized algorithms a stability coefficient bounded as specified by Lemma 12.The result is stated under K(x, x) ≤ κ < ∞.
- Applications: The applications cover SVM with hinge loss, SVR with ϵ-insensitive loss, and Kernel Ridge Regression.The paper lists these as three specific instances of kernel regularization algorithms.
- Kernel regularization: For kernel methods, the paper provides output bounds and uses them with stability estimates to obtain the SVM, SVR, and KRR applications.The output bound depends on the regularization parameter λ and kernel bound κ².
- Kernel regularization: The paper notes that an earlier upper-bound function from related work appears incorrect and supplies a remedy.This correction concerns the later application of the bound rather than the stated stability framework.
- Relative entropy regularization: Relative-entropy regularized algorithms learn a weighting function over parameterized base hypotheses using a convex-combination hypothesis.The cost is defined through an internal cost function and the regularizer is a Kullback–Leibler divergence relative to a fixed distribution.
- Relative entropy regularization: The paper derives a high-probability corollary for the relative-entropy optimization when the internal cost is bounded by M.The result follows by applying the stability bounds to the optimization in equation 29.
3.5 Discussion
The β-mixing analysis extends stability-based generalization bounds to dependent stationary sequences by replacing dependent observations with independent blocks and balancing block parameters. The resulting bounds include an additive mixing-dependent term, recover the i.i.d. case under fast mixing, and impose convergence conditions under algebraic mixing.
- Contribution: The results provide first stability-based generalization bounds for the studied algorithm class in a non-i.i.d. scenario.The stated contribution is specifically tied to the algorithms analyzed in this part of the paper.
- β-mixing bounds: The β-mixing bound applies to stationary dependent sequences and is more general than the earlier ϕ-mixing analysis, but includes an additive term depending on the mixing coefficient.The proof requires only that the training sequence be drawn from a stationary β-mixing distribution.
- Proof strategy: The analysis partitions the sample into blocks, replaces dependent blocks with independent ones, and applies McDiarmid’s inequality to the resulting block sequence.The blocks may be non-symmetric and have unequal sizes, with parameters satisfying (a + b)µ = m.
- Parameter selection: Parameter selection balances the block size used to reduce the mixing term against the number of blocks used to control the remaining terms.The optimal choice depends on whether the mixing coefficient decays algebraically or exponentially.
- Recovery of the i.i.d. case: For fast mixing, the β-mixing result can recover an i.i.d.-form bound; when β(b) = 0, choosing a = 0, b = 1, and µ = m retrieves the Bousquet–Elisseeff i.i.d. bound.This explicitly connects the dependent-sequence result to the classical independent-sample case.
- Algebraic mixing: Under algebraic β-mixing with ˆβ ∈ O(m^-α), convergence requires α > 3r+3, while α = 1 is sufficient when r > 1.A larger mixing parameter r yields a more favorable bound, provided the convergence condition holds.
5. Conclusion
The paper extends stability-based generalization analysis to stationary non-i.i.d. sequences and identifies practical uses for analyzing and designing stable learning algorithms.
- Stability bounds for stationary ϕ-mixing and β-mixing sequences extend existing i.i.d. stability bounds to non-i.i.d. settings.They apply to broad classes of algorithms, including SVR, KRR, and SVMs.
- Because they are algorithm-specific, the bounds can often be tighter than bounds based on general hypothesis-class complexity measures.
- The bounds can analyze stable algorithms under the studied non-i.i.d. settings and support the design of novel, accurate learning algorithms.
- Using the bounds for algorithm design requires knowledge of mixing properties, although the shape of mixing coefficients may be estimable in some problems.