Pseudo-Independent Algorithms

• Pseudo-Independent Algorithms are algorithmic paradigms that quantify partial, approximate, or statistical independence to enable robust computational frameworks.
• They integrate concepts from algorithmic information theory, sublinear expectations, and combinatorial optimization through techniques like k-wise independence and latent decompositions.
• Their application spans streaming, sketching, and risk management, highlighting trade-offs in achieving true independence and guiding the design of efficient algorithms.

A pseudo-independent algorithm is an algorithmic design or analysis paradigm that explicitly incorporates and quantifies partial, approximate, or statistical independence within its components, randomness, or outputs. The term draws together diverse research in algorithmic information theory, streaming, combinatorial optimization, machine learning, and sublinear expectations, providing a precise framework for understanding, harnessing, or circumventing independence constraints in computation. It has close ties to notions such as algorithmic independence, k-wise independence, pseudo-deterministic computation, and modern approaches in robust statistical inference and optimization.

1. Formal Notions of Algorithmic Independence

Algorithmic independence generalizes classical statistical independence by replacing measures of (Shannon) entropy with algorithmic (Kolmogorov) complexity—a shift that enables the analysis of independence in individual objects (such as strings or sequences) and not just random variables.

Two principle definitions are:

• Integral Independence: Sequences $x$ and $y$ satisfy, for all $n$,

$C^y(x[1..n]) \geq C(x[1..n]) - O(\log n)$

where $C^y$ is the plain Kolmogorov complexity relative to oracle $y$.

• Finitary Independence: For all $n,m$,

$$C(x[1..n]\,\|\,y[1..m]) \geq C(x[1..n]) + C(y[1..m]) - O(\log n+\log m)$$

(see (0802.0487)).

These definitions capture the idea that descriptions of finite prefixes (or joint objects) do not admit significant information compression beyond the sum of their individual complexities. Integral independence is strictly stronger than finitary independence.

A crucial measure result is that for any fixed $x$, the set of $y$ finitary-independent of $x$ has Lebesgue measure one; thus, almost every pair $(x,y)$ is finitary-independent. The existence of "nontrivial" (complex) integral-independent sequences for each $x$ remains conjectural.

2. Pseudo-Independence in Sublinear Expectations

In the nonlinear probability theory framework—motivated by risk and ambiguity in finance, insurance, and robust statistics—the concept of independence is extended via sublinear expectations. Here, the expectation operator $\mathbb{E}$ is subadditive, especially when defined as a supremum over a set $\mathcal{P}$ of probability measures (reflecting model uncertainty).

Pseudo-independence, defined in (Li, 2021), requires:

$$\mathbb{E}_P[\varphi(X_n) \mid \mathcal{F}_{n-1}] \in [\mathbb{E}_P[-\varphi(X_n)],\,\mathbb{E}_P[\varphi(X_n)]], \text{ for all } P \in \mathcal{P}$$

for bounded Lipschitz $\varphi$, and all $n$.

This generalizes Peng's (nonlinear) independence: Peng's definition implies pseudo-independence, but not vice versa. This relaxation allows for useful limit theorems when classical or full nonlinear independence is inappropriate or unachievable.

3. Structural and Combinatorial Pseudo-Independence

Pseudo-independent algorithms are often associated with algorithmic or structural properties of problem instances:

• Pseudo-independent models in probabilistic belief networks refer to settings where subsets of collectively dependent variables appear marginally independent; such models are difficult to learn via single-link structural tests, requiring multi-link or backtracking learning schemes (Hu et al., 2013).
• In combinatorial optimization and graph algorithms, "pseudo-independence" appears in settings where independence can be quantified or relaxed. For example, OBDD-based randomized graph algorithms utilize low k-wise independence (e.g., 3-wise independent random variable families) to obtain computationally tractable, approximately independent random sources (Bury, 2015). Similarly, geometric intersection graphs (e.g., pseudo-disks, k-perfectly orientable graphs) provide structures supporting approximations as if true independence held among item selections (Chan et al., 2011, Chekuri et al., 2023).
• In random matrix theory, families of pseudo-random matrices constructed via Golomb sequences exhibit asymptotic pseudo-independence: mixed centered moments vanish as the matrix size increases, emulating the behavior (but not construction) of independent random ensembles (Soloveychik et al., 2018).

4. Algorithmic Barriers and Impossibility Results

Significant impossibility results delineate what forms of independence are algorithmically constructible:

• No uniform Turing reduction can, from a given nontrivial infinite sequence, efficiently produce two new sequences that are even finitary-independent with super-logarithmic complexity (0802.0487).
• It is in general impossible to "boost" independence: there is no algorithmic counterpart to randomness extractors that can manufacture nontrivial independence between random variables or sequences solely via effective procedures on a single source with positive constructive Hausdorff dimension.
• For streaming and sketching algorithms, enforcing reproducible (pseudo-deterministic) outputs—strong forms of pseudo-independence from algorithmic randomness—comes at nearly-linear or even exponential memory cost for core tasks (counting, heavy hitters, $\ell_2$-norm estimation), while regular randomized algorithms with high output entropy require only polylogarithmic memory (Goldwasser et al., 2019, Braverman et al., 2023).

These impossibility theorems drive an emphasis on resource-bounded independence, non-uniform constructions, or accepting partial/approximate independence notions.

5. Limit Theorems and Implications for Robust Algorithms

Pseudo-independent sequences admit analogues of classical probabilistic laws when standard assumptions cannot be met. Under sublinear expectations, Marcinkiewicz-type strong and weak laws of large numbers hold for pseudo-independent random sequences with finite Choquet $r$-moments ($1 \leq r < 2$), with the scaling $n^{1/r}$ governing convergence (Fu, 30 Apr 2025). Central limit theorems also generalize: properly normalized sums of pseudo-independent variables converge in law to G-normal distributions under appropriate moment conditions (Li, 2021).

These results legitimize the use of pseudo-independent algorithmic primitives in settings where model uncertainty or weakened independence prevails, enabling robust statistical inference, learning under adversarial uncertainty, and risk management.

6. Computational and Algorithmic Frameworks

Pseudo-independent constructs underpin several practical algorithmic methods:

• Partial randomization: Using k-wise or almost-k-wise independent random functions to control space and time complexity, as in OBDD-based graph matching, where the computational cost of exact independence is provably prohibitive (Bury, 2015).
• Sample reuse in stochastic optimization: Algorithms can be adapted to reuse randomness, achieving trade-offs between full-batch and sample queries, with the influence of randomness on outputs analyzed using measures of pseudo-independence (Jin et al., 2 Sep 2025).
• Latent independence decompositions: In probabilistic model diagnosis and mixture modeling, the latent independent weight quantifies and extracts the maximal component of data attributable to an independent product measure, leading to algorithms that process the "independent core" separately from the dependent residual (Pearson et al., 2020).
• Learning under pseudo-independence: Multi-link and backtracking search strategies are formalized for belief networks when underlying dependencies are hidden by structural pseudo-independence (Hu et al., 2013).
• Streaming and sketching: Lower bounds for pseudo-deterministic computation clarify the resource cost of achieving pseudo-independent outputs, and variants such as k-pseudo-deterministic computation model partial relaxations (Goldwasser et al., 2019, Braverman et al., 2023).

7. Open Questions and Research Directions

Core unresolved issues and ongoing research areas include:

• Characterizing the space of nontrivial integral-independent sequences and establishing measure-theoretic abundance for the strong notion of algorithmic independence (0802.0487).
• Designing resource-bounded notions of independence that admit efficient, non-uniform, or instance-specific constructions.
• Developing computationally efficient algorithms for identifying, extracting, and exploiting latent independence within high-dimensional, dependent generative structures (Pearson et al., 2020).
• Tightening space and query complexity bounds for pseudo-independent (or pseudo-deterministic) streaming and sublinear algorithms, with particular attention to fundamental open tasks such as Shift Finding (Braverman et al., 2023).
• Extending limit theorems for pseudo-independent sequences under sublinear expectations to broader classes of functionals and dependence structures (Fu, 30 Apr 2025, Li, 2021).
• Quantifying and controlling the trade-off between independence guarantees and computational overhead in parallel, distributed, or memory-constrained algorithmic environments (Jin et al., 2 Sep 2025).

Summary Table: Notions and Contexts of Pseudo-Independence

Context/Area	Pseudo-Independence Manifestation	Reference
Algorithmic Information Theory	Integral/finitary algorithmic independence	(0802.0487)
Nonlinear Probability/Sublinear Exp.	Pseudo-independence via expectation bounds	(Li, 2021)
Probabilistic Modeling/Belief Networks	Marginally independent, globally dependent	(Hu et al., 2013)
Combinatorial/Geometric Algorithms	k-wise, almost k-wise random independence	(Bury, 2015)
Random Matrix Theory	Asymptotic independence via moment vanishing	(Soloveychik et al., 2018)
Statistical Testing/Mixture Models	Latent independent weight decompositions	(Pearson et al., 2020)
Streaming and Sketching Algorithms	Pseudo-deterministic/k-pseudo independence	(Goldwasser et al., 2019, Braverman et al., 2023)
Stochastic Optimization	Sample reuse/pseudo-independent outputs	(Jin et al., 2 Sep 2025)
Sublinear Law of Large Numbers	Marcinkiewicz laws for pseudo-independence	(Fu, 30 Apr 2025)

In conclusion, pseudo-independent algorithms provide a unified language and toolkit for algorithm design and analysis across domains where full independence is unavailable, is algorithmically unattainable, or is too expensive to enforce. Their application and theoretical paper span randomness extraction, learning, robust statistics, optimization, and beyond, with ongoing progress directed toward sharper resource bounds, deeper structural understanding, and broader applicability in uncertain or adversarial computational environments.