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Dimitrios Myrisiotis

Publications and source records attributed to Dimitrios Myrisiotis.

12 recordsLinked to original sources
arXiv2026

A Witnessing Duality for ReLU Network Depth

A depth lower bound for ReLU networks asserts that every sufficiently shallow network fails on some input. What does a proof of such a bound compute? We study this question for exact computation of the maximum of $n$ real inputs, with depth measured by the number of hidden layers. For every separately fixed prime $q$ and depth bound $K$, we give a uniform $\mathsf{TC}^0$ refuter for rational networks with at most $K$ hidden layers and more than $q^K$ inputs, provided that the reduced denominators of their nonbias weights are not divisible by $q$. Biases may be arbitrary rationals. The refuter finds a rational input on which the network fails, and its correctness is provable in $\mathbf{VTC}^0$, the bounded-arithmetic theory of uniform constant-depth threshold computation. The underlying numerical depth lower bound follows from Averkov, Hojny and Merkert (ICLR 2025), extending Haase, Hertrich and Loho (ICLR 2023); our contribution is an internal proof that makes its geometric argument into a feasible refuter. On the construction side, $\mathbf{VTC}^0$ verifies locally checkable certificates for the balanced integer-weight maximum network of depth $\lceil\log_2 n\rceil$. At $q=2$, the construction and refuter meet: for each fixed $K$, the theory verifies that integer-nonbias-weight networks with arbitrary rational biases can compute the maximum with at most $K$ hidden layers exactly when $n\le 2^K$. We also give certified rational constructions of polynomial encoding length and depth $O(\log n/\log\log n)$. These results instantiate a witnessing duality: bounded proofs of certified existence and of refutation correspond, respectively, to threshold-computable constructors and refuters whose guarantees the theory proves.

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arXiv2026

The Complexity of ReLU Network Synthesis from a Parameter Multiset

We ask how hard it is to build a neural network when its parameter values are given in advance. An instance supplies a multiset of rational numbers, a target behavior, and a bound on the number of neurons, output included. The question is whether one can choose the graph of a ReLU network within the bound, and place the numbers on its edges and biases so that it meets the target. This separates the numbers from the architecture; permutation training instead fixes the hidden units and only rearranges the output weights. Matching two labeled examples exactly is already $\mathsf{NP}$-complete; we ask what changes when the target must hold on every input or on average. Agreeing with a Boolean circuit on every Boolean input is $Σ_2^{\mathsf P}$-complete, already for four neurons. Agreeing with a given ReLU network on every real input lies in $\mathsf{DP}$ for every fixed neuron bound and is $\mathsf{DP}$-complete for three neurons. Approximating a ReLU network on the unit cube within absolute error one everywhere is $Σ_2^{\mathsf P}$-complete, even with no hidden neurons and all numbers equal and positive. Deciding whether some network has expected squared loss at most a threshold is $\mathsf{NP}^{\mathsf{PP}}$-complete for uniform Boolean inputs. For uniform inputs on the unit cube and for standard Gaussian inputs with a positive threshold, the same holds for two neurons when a positive risk gap, given in binary and possibly exponentially small, separates yes from no instances. Without a gap, the cube version is complete for guessing a polynomial-length string, then comparing a sum of exponentially many rationals with a threshold, all computable in polynomial time from the input and the guess. Since the graph is chosen too, the reductions must rule out every other network built from the same numbers.

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arXiv2026

A Hand-Checkable Proof That Two Hidden ReLU Layers Compute the Maximum of Six Numbers

Exactly computing the maximum function is a standard test case for studying depth in ReLU networks. Two hidden layers are known to suffice for up to twelve inputs through computer-assisted constructions. For six real inputs, we give an explicit hexagon identity whose local structure yields a self-contained analytical proof of this depth bound. The identity was found by computer-assisted search; we prove it through explicit cancellations that can be checked entirely by hand, without executing a verification program. The identity also yields an explicit network with hidden widths $17$ and $41$, zero biases, and rational weights.

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arXiv2026

Approximating the Trace Distance Between Product Quantum States

We study the trace distance \[D_{\mathrm{tr}}(ρ,σ) =\frac12\|ρ-σ\|_1, ρ=\bigotimes_{i=1}^nρ_i,\quad σ=\bigotimes_{i=1}^nσ_i, \] when the two exponentially large states are specified by their local factors. We give a deterministic approximation within a universal constant factor for rational product inputs. Its running time is polynomial in the number of factors, the local dimension, and the input bit length. In the opposite direction, exact computation is $\#\mathsf P$-hard even for diagonal qubit states, by the corresponding hardness of total variation distance between product distributions. The proof uses local Uhlmann-optimal purifications to reduce the problem to estimating the product-fidelity defect and the trace norm of a structured first-order operator. Although this operator acts on an exponentially large space, we approximate its trace norm by a local convex surrogate that admits a polynomial-size classical conic formulation. A square-function estimate shows that the surrogate upper-bounds this trace norm. Conversely, duality and local dephasing reduce the reverse comparison to a head--tail inequality for independent centered random variables, showing that the surrogate is at most a dimension-free constant times the same norm.

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arXiv2025

Algorithms and Hardness for Estimating Statistical Similarity

We introduce and study the computational problem of determining statistical similarity between probability distributions. For distributions $P$ and $Q$ over a finite sample space, their statistical similarity is defined as $S_{\mathrm{stat}}(P, Q) := \sum_x \min(P(x), Q(x))$. Despite its fundamental nature as a measure of similarity between distributions, capturing essential concepts such as Bayes error in prediction and hypothesis testing, this computational problem has not been previously explored. Recent work on computing statistical distance has established that, somewhat surprisingly, even for the simple class of product distributions, exactly computing statistical similarity is $\#\mathsf{P}$-hard. This motivates the question of designing approximation algorithms for statistical similarity. Our first contribution is a Fully Polynomial-Time deterministic Approximation Scheme (FPTAS) for estimating statistical similarity between two product distributions. Furthermore, we also establish a complementary hardness result. In particular, we show that it is $\mathsf{NP}$-hard to estimate statistical similarity when $P$ and $Q$ are Bayes net distributions of in-degree $2$.

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arXiv2024

Computational Explorations of Total Variation Distance

We investigate some previously unexplored (or underexplored) computational aspects of total variation (TV) distance. First, we give a simple deterministic polynomial-time algorithm for checking equivalence between mixtures of product distributions, over arbitrary alphabets. This corresponds to a special case, whereby the TV distance between the two distributions is zero. Second, we prove that unless $\mathsf{NP} \subseteq \mathsf{RP}$, it is impossible to efficiently estimate the TV distance between arbitrary Ising models, even in a bounded-error randomized setting.

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arXiv2024

Efficient, Low-Regret, Online Reinforcement Learning for Linear MDPs

Reinforcement learning algorithms are usually stated without theoretical guarantees regarding their performance. Recently, Jin, Yang, Wang, and Jordan (COLT 2020) showed a polynomial-time reinforcement learning algorithm (namely, LSVI-UCB) for the setting of linear Markov decision processes, and provided theoretical guarantees regarding its running time and regret. In real-world scenarios, however, the space usage of this algorithm can be prohibitive due to a utilized linear regression step. We propose and analyze two modifications of LSVI-UCB, which alternate periods of learning and not-learning, to reduce space and time usage while maintaining sublinear regret. We show experimentally, on synthetic data and real-world benchmarks, that our algorithms achieve low space usage and running time, while not significantly sacrificing regret.

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arXiv2024

Learnability of Parameter-Bounded Bayes Nets

Bayes nets are extensively used in practice to efficiently represent joint probability distributions over a set of random variables and capture dependency relations. In a seminal paper, Chickering et al. (JMLR 2004) showed that given a distribution $\mathbb{P}$, that is defined as the marginal distribution of a Bayes net, it is $\mathsf{NP}$-hard to decide whether there is a parameter-bounded Bayes net that represents $\mathbb{P}$. They called this problem LEARN. In this work, we extend the $\mathsf{NP}$-hardness result of LEARN and prove the $\mathsf{NP}$-hardness of a promise search variant of LEARN, whereby the Bayes net in question is guaranteed to exist and one is asked to find such a Bayes net. We complement our hardness result with a positive result about the sample complexity that is sufficient to recover a parameter-bounded Bayes net that is close (in TV distance) to a given distribution $\mathbb{P}$, that is represented by some parameter-bounded Bayes net, generalizing a degree-bounded sample complexity result of Brustle et al. (EC 2020).

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arXiv2024

Total Variation Distance Meets Probabilistic Inference

In this paper, we establish a novel connection between total variation (TV) distance estimation and probabilistic inference. In particular, we present an efficient, structure-preserving reduction from relative approximation of TV distance to probabilistic inference over directed graphical models. This reduction leads to a fully polynomial randomized approximation scheme (FPRAS) for estimating TV distances between same-structure distributions over any class of Bayes nets for which there is an efficient probabilistic inference algorithm. In particular, it leads to an FPRAS for estimating TV distances between distributions that are defined over a common Bayes net of small treewidth. Prior to this work, such approximation schemes only existed for estimating TV distances between product distributions. Our approach employs a new notion of $partial$ couplings of high-dimensional distributions, which might be of independent interest.

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arXiv2023

On Approximating Total Variation Distance

Total variation distance (TV distance) is a fundamental notion of distance between probability distributions. In this work, we introduce and study the problem of computing the TV distance of two product distributions over the domain $\{0,1\}^n$. In particular, we establish the following results. 1. The problem of exactly computing the TV distance of two product distributions is $\#\mathsf{P}$-complete. This is in stark contrast with other distance measures such as KL, Chi-square, and Hellinger which tensorize over the marginals leading to efficient algorithms. 2. There is a fully polynomial-time deterministic approximation scheme (FPTAS) for computing the TV distance of two product distributions $P$ and $Q$ where $Q$ is the uniform distribution. This result is extended to the case where $Q$ has a constant number of distinct marginals. In contrast, we show that when $P$ and $Q$ are Bayes net distributions, the relative approximation of their TV distance is $\mathsf{NP}$-hard.

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arXiv2020

Algorithms and Lower Bounds for de Morgan Formulas of Low-Communication Leaf Gates

The class $FORMULA[s] \circ \mathcal{G}$ consists of Boolean functions computable by size-$s$ de Morgan formulas whose leaves are any Boolean functions from a class $\mathcal{G}$. We give lower bounds and (SAT, Learning, and PRG) algorithms for $FORMULA[n^{1.99}]\circ \mathcal{G}$, for classes $\mathcal{G}$ of functions with low communication complexity. Let $R^{(k)}(\mathcal{G})$ be the maximum $k$-party NOF randomized communication complexity of $\mathcal{G}$. We show: (1) The Generalized Inner Product function $GIP^k_n$ cannot be computed in $FORMULA[s]\circ \mathcal{G}$ on more than $1/2+\varepsilon$ fraction of inputs for $$ s = o \! \left ( \frac{n^2}{ \left(k \cdot 4^k \cdot {R}^{(k)}(\mathcal{G}) \cdot \log (n/\varepsilon) \cdot \log(1/\varepsilon) \right)^{2}} \right).$$ As a corollary, we get an average-case lower bound for $GIP^k_n$ against $FORMULA[n^{1.99}]\circ PTF^{k-1}$. (2) There is a PRG of seed length $n/2 + O\left(\sqrt{s} \cdot R^{(2)}(\mathcal{G}) \cdot\log(s/\varepsilon) \cdot \log (1/\varepsilon) \right)$ that $\varepsilon$-fools $FORMULA[s] \circ \mathcal{G}$. For $FORMULA[s] \circ LTF$, we get the better seed length $O\left(n^{1/2}\cdot s^{1/4}\cdot \log(n)\cdot \log(n/\varepsilon)\right)$. This gives the first non-trivial PRG (with seed length $o(n)$) for intersections of $n$ half-spaces in the regime where $\varepsilon \leq 1/n$. (3) There is a randomized $2^{n-t}$-time $\#$SAT algorithm for $FORMULA[s] \circ \mathcal{G}$, where $$t=Ω\left(\frac{n}{\sqrt{s}\cdot\log^2(s)\cdot R^{(2)}(\mathcal{G})}\right)^{1/2}.$$ In particular, this implies a nontrivial #SAT algorithm for $FORMULA[n^{1.99}]\circ LTF$. (4) The Minimum Circuit Size Problem is not in $FORMULA[n^{1.99}]\circ XOR$. On the algorithmic side, we show that $FORMULA[n^{1.99}] \circ XOR$ can be PAC-learned in time $2^{O(n/\log n)}$.

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