Learning Hidden Markov Models Using Conditional Samples

TL;DR

This paper demonstrates that access to conditional samples allows for efficient learning of Hidden Markov Models, overcoming cryptographic hardness in standard settings, and extends these methods to broader distribution classes.

Contribution

It introduces new algorithms for learning HMMs with conditional access, bypassing cryptographic hardness, and generalizes Angluin's L* algorithm to broader distribution classes.

Findings

Efficient polynomial-time algorithms for HMM learning with exact conditional probability queries.

Performance depends on a new parameter called the fidelity of the HMM.

Results extend to distributions with latent low-rank structure.

Abstract

This paper is concerned with the computational complexity of learning the Hidden Markov Model (HMM). Although HMMs are some of the most widely used tools in sequential and time series modeling, they are cryptographically hard to learn in the standard setting where one has access to i.i.d. samples of observation sequences. In this paper, we depart from this setup and consider an interactive access model, in which the algorithm can query for samples from the conditional distributions of the HMMs. We show that interactive access to the HMM enables computationally efficient learning algorithms, thereby bypassing cryptographic hardness. Specifically, we obtain efficient algorithms for learning HMMs in two settings: (a) An easier setting where we have query access to the exact conditional probabilities. Here our algorithm runs in polynomial time and makes polynomially many queries to approximate any HMM in total variation distance. (b) A harder setting where we can only obtain samples from the conditional distributions. Here the performance of the algorithm depends on a new parameter, called the fidelity of the HMM. We show that this captures cryptographically hard instances and previously known positive results. We also show that these results extend to a broader class of distributions with latent low rank structure. Our algorithms can be viewed as generalizations and robustifications of Angluin's $L^*$ algorithm for learning deterministic finite automata from membership queries.

Equations

$$\mathrm{Pr}[x_1,\dots ,x_T]:=\sum _{s_1,\dots ,s_{T+1}\in {\mathcal{S}}^{T+1}}\mu (s_1)\prod _{t=1}^T\mathbb{O}[x_t,s_t]\mathbb{T}[s_{t+1},s_t].$$

$$\mathrm{TV}(\mathrm{Pr},\widehat{\mathrm{Pr}}):=\frac{1}{2}\sum _{x_1,\dots ,x_T\in {\mathcal{O}}^T}\left|\mathrm{Pr}[x_1,\dots ,x_T]-\widehat{\mathrm{Pr}}[x_1,\dots ,x_T]\right|\le \epsilon .$$

$$\mathrm{Pr}[x_1,\dots ,x_T]=1^{\top }{\mathbb{A}}_{x_T}\dots {\mathbb{A}}_{x_1}\mu ,$$

$$\mathrm{Pr}[F_t|h]=\mathrm{Pr}[F_t|B_t]\beta (h).$$

$$\mathrm{Pr}[F_{t+1}|B_{t+1}]A_{o,t}=\mathrm{Pr}[o{F}_{t+1}|B_t],$$

$$\mathrm{Pr}[x_1,\dots ,x_T]$$

$$\dots$$

$$\forall h\in H_t,o\in \mathcal{O}:\beta (ho)=\frac{A_{o,t}\beta (h)}{\mathrm{Pr}[o|h]}.$$

$$\widehat{\mathrm{Pr}}[x_1,\dots ,x_T]={\hat{A}}_{x_T,T-1}\dots {\hat{A}}_{x_2,1}{\hat{A}}_{x_1,0},$$

$$\frac{1}{2}\sum _{x_1,\dots ,x_T\in {\mathcal{O}}^T}\left|{\hat{A}}_{x_T}\dots {\hat{A}}_{x_1}-A_{x_T}\dots A_{x_1}\right|.$$

$$\sum _{x_1,\dots ,x_T\in {\mathcal{O}}^T}\Vert {\hat{A}}_{x_T}\dots {\hat{A}}_{x_{t+2}}{\Vert }_2\cdot \Vert \left({\hat{A}}_{x_{t+1}}-A_{x_{t+1}}\right)A_{x_t}\dots A_{x_1}{\Vert }_2$$

$$\sum _{x_1,\dots ,x_{t+1}}\Vert \left({\hat{\mathbb{A}}}_{x_{t+1}}-{\mathbb{A}}_{x_{t+1}}\right){\mathbb{A}}_{x_t}\dots ,{\mathbb{A}}_{x_1}{\Vert }_1\lesssim O\epsilon \cdot \sum _{x_1,\dots ,x_t}\Vert {\mathbb{A}}_{x_t}\dots ,{\mathbb{A}}_{x_1}{\Vert }_1\le O\epsilon .$$

$$({\hat{A}}_{x_t}\dots {\hat{A}}_{x_1}-A_{x_t}\dots A_{x_1})=\sum _{h\in H_t}\beta (h){\alpha }_h+\sum _{v\in V_t^{\perp }}v{\gamma }_v,$$

$$\mathrm{Pr}[F_{t+1}|B_{t+1}{]}^{\top }{D}_{t+1}^{-1}\mathrm{Pr}[F_{t+1}|B_{t+1}]A_{o,t}=\mathrm{Pr}[F_{t+1}|B_{t+1}{]}^{\top }{D}_{t+1}^{-1}\mathrm{Pr}[o{F}_{t+1}|B_t],$$

$${\left[\mathrm{Pr}[F_{t+1}|B_{t+1}{]}^{\top }{D}_{t+1}^{-1}\mathrm{Pr}[F_{t+1}|B_{t+1}]\right]}_{i,j}=\sum _{f\in F_{t+1}}{d}_{t+1}(f)\left[\frac{\mathrm{Pr}[f|b_i]\mathrm{Pr}[f|b_j]}{d_{t+1}(f{)}^2}\right],$$

$$\forall t\in [T]:$$

$$\mathrm{Pr}[{\Lambda }_t|B_t]A_{o,t}=\mathrm{Pr}[o{\Lambda }_t|B_t]$$

$$\widehat{\mathrm{Pr}}[x_1,\dots ,x_t,{\Lambda }_t]\stackrel{?}{=}\mathrm{Pr}[x_1,\dots ,x_t,{\Lambda }_t].$$

$$\mathrm{Pr}[F_t|x]=\mathrm{Pr}[F_t|B_t]\beta (x).$$

$$\mathrm{Pr}[x_1\dots x_T]=A_{x_T,T-1}{A}_{x_{T-1},T-2}\dots A_{x_1,0}$$

$$\mathrm{Pr}[F_{t+1}|B_{t+1}]A_{o,t}=\mathrm{Pr}[o{F}_{t+1}|B_t].$$

$$A_{o,t}=\left[\begin{array}{cccc}\beta (b_{1}o)& \beta (b_{2}o)& \cdots & \beta (b_{n}o)\end{array}\right]\left[\begin{array}{cccc}\mathrm{Pr}[o|b_1]& 0& \cdots & 0\\ 0& \mathrm{Pr}[o|b_2]& \cdots & 0\\ ⋮& ⋮& \ddots & 0\\ 0& \cdots & 0& \mathrm{Pr}[o|b_n]\end{array}\right]$$

$$\mathrm{Pr}[F_{t+1}|B_{t+1}]A_{o,t}$$

$$=$$

$$\mathrm{Pr}[F_T|B_T]A_{x_T,T-1}{A}_{x_{T-1},T-2}\dots A_{x_1,0}=\mathrm{Pr}[x_T{F}_T\mid B_{T-1}]A_{x_{T-1},T-2}\dots A_{x_1,0}=\mathrm{Pr}[x_1{x}_2\dots x_T{F}_T|B_0]$$

$$\mathrm{Pr}[{\Lambda }_{t+1}|B_{t+1}]{\hat{A}}_{o,t}=\mathrm{Pr}[o{\Lambda }_{t+1}|B_t]$$

$$\overline{\mathrm{Pr}}[x_1,\dots ,x_t,{\Lambda }_t]=\mathrm{Pr}[{\Lambda }_t|B_t]{\hat{A}}_{x_t,t-1}\dots {\hat{A}}_{x_1,0}$$

$$\overline{\mathrm{Pr}}[x_1,\dots ,x_t,{\Lambda }_t]\ne \mathrm{Pr}[x_1,\dots ,x_t,{\Lambda }_t]$$

$$\mathrm{Pr}[x_1\dots x_{\tau }{\Lambda }_{\tau }]$$

Full text

Specifically, we obtain efficient algorithms for learning HMMs in two settings:

An easier setting where we have query access to the exact conditional probabilities. Here our algorithm runs in polynomial time and makes polynomially many queries to approximate any HMM in total variation distance. 2. 2.

A harder setting where we can only obtain samples from the conditional distributions. Here the performance of the algorithm depends on a new parameter, called the fidelity of the HMM. We show that this captures cryptographically hard instances and previously known positive results.

We also show that these results extend to a broader class of distributions with latent low rank structure. Our algorithms can be viewed as generalizations and robustifications of Angluin’s $L^{*}$ algorithm for learning deterministic finite automata from membership queries.

1. 1 Introduction

2. 1.1 Preliminaries

3. 1.1.1 Hidden Markov Models and low rank distributions

4. 1.1.2 Learning models

5. 1.2 Our results

6. 2 Technical overview

7. 2.1 Background: Observable operators and hard instances

8. 2.2 Efficient representation

9. 2.3 Error propagation

10. 2.4 Estimating operators

11. 2.5 Finding the basis

12. 3 Learning with conditional probabilities (Theorem 1)

13. 3.1 Algorithm

14. 3.2 Analysis

15. 4 Learning with conditional samples (Theorem 2)

16. 4.1 Algorithm

17. 4.2 Analysis

18. 5 Discussion

19. A Proofs for Section 3

20. B Proofs for Section 4

21. B.1 Finding robust basis

22. B.2 Learning operators

23. B.3 Perturbation analysis: Error in coefficients

24. B.4 Proof of Theorem 2

25. B.5 Properties of operator $A_{o}$

26. B.6 Estimating covariance matrix in Frobenius norm

27. C Examples

28. C.1 HMMs induce low rank distributions

29. C.2 Parity with noise

30. C.3 Full rank HMMs

31. C.4 Overcomplete HMMs

32. D General algorithm for finding approximate basis

33. D.1 Learning coefficients

34. D.2 Algorithm

35. E Helper propositions

1 Introduction

Hidden Markov Models (HMMs) are among the most fundamental tools for modeling temporal and sequential phenomena. These probabilistic models specify a joint distribution over a sequence of observations generated via a Markov chain of latent states. This structure enjoys the simultaneous benefits of low description complexity, sufficient expressivity to capture long-range dependencies, and efficient inference algorithms. For these reasons, HMMs have become ubiquitous building blocks for sequence modeling in varied fields, ranging from bioinformatics to natural language processing to finance. A long-standing challenge, in both theory and practice, is the computational difficulty of learning an unknown HMM from samples. In this paper, we are interested in the computational complexity of this estimation/learning task.

Although one can consider several notions of learnability, we focus on distribution learning, in total variation (TV) distance. In the standard realizable formulation, we are given observation sequences generated by an underlying HMM and are asked to efficiently compute a distribution that is close to the HMM in TV distance. Maximum likelihood estimation is known to be statistically efficient, but no computationally efficient implementations of this approach are known. More generally, HMMs can encode the parity with noise problem [MR05], which is widely believed to be computationally hard [Blu+94, Kea+94, Ale03], and so we do not expect to find efficient algorithms for general HMMs. Recent works have therefore focused on obtaining computationally efficient algorithms under structural assumptions which evade these hard instances [CGG01, HKZ12, Sha+17].

This work takes a different perspective. We ask: can we evade computational hardness by allowing the learner to access the HMM interactively? Specifically we consider a conditional sampling oracle: we allow the learner to sample a “future sequence” from the HMM conditioned on a “past sequence” or history. This approach is closely related to recent work in distribution testing [CRS15, BC18, Che+21, e.g.,], which demonstrates improvements in various property testing tasks via conditional sampling. One conceptual difference is that we use conditional sampling to evade computational hardness, rather than obtaining (sometimes exponential) statistical improvements.

From a practical perspective, we are motivated by potential applications of interactive learning to training language models or world models more generally. Indeed, it is quite natural to fine-tune a language model by asking annotators to complete prompts generated by the model; this precisely corresponds to conditional sampling if we view the annotators as representative of the population [ZSH22]. When training world models for decision making, it may be possible to request expert demonstrations starting from a particular state, which again is effectively conditional sampling. This latter approach is closely related to interactive imitation learning [RGB11].

We are further motivated by two theoretical considerations. First, it is not hard to show that parity with noise can be efficiently learned in this model, as we can sample the label conditioned on each history with a single observation set to $1$ and naively denoise these samples (we describe this in detail in Appendix C). However, this approach is quite tailored to noisy parity, and so it is natural to ask if it can be generalized to arbitrary HMMs. Second, learning HMMs with conditional samples can be seen as a statistical generalization of learning deterministic finite automata (DFAs) with membership queries, for which Angluin’s seminal $L^{*}$ algorithm provides a strong computational separation between interactive and non-interactive PAC learning [Ang87]. We believe it is natural to ask if $L^{*}$ can be extended to HMMs and be made robust to sampling, thereby providing further evidence for the computational benefits of interactive learning.

Contributions.

In this paper, we develop new algorithms and techniques for learning Hidden Markov models when provided with interactive access. As our first result, we show how a generalization of Angluin’s $L^{*}$ algorithm can efficiently learn any HMM in the stronger access model where the learner can query for exact conditional probabilities. As our main result, we consider the more natural conditional sampling access model and obtain an algorithm that is efficient for all HMMs with “high fidelity,” a new property we introduce. We show that this property captures the cryptographically hard instances and prior positive results, but we leave open the question of efficiently learning all HMMs via conditional sampling. Our results require a number of new algorithmic ideas and analysis techniques, most notably: an efficient representation for distributions over exponentially large domains and a new perturbation argument for mitigating error amplification over long sequences. We hope these techniques find application in other settings.

1.1 Preliminaries

Notation.

Let $\mathcal{O}:=\{1,\ldots,O\}$ denote a finite observation space and let $\mathcal{O}^{*}$ denote observation sequences of arbitrary length. We consider a distribution $\Pr[\cdot]$ over $T$ random variables $\textbf{x}_{1},\ldots,\textbf{x}_{T}$ with a sequential ordering, and we use $x_{t}\in\mathcal{O}$ to denote the value taken by the $t^{\mathrm{th}}$ random variable. For convenience, we often simply write $\Pr[x_{1},x_{2},\ldots,x_{T}]$ in lieu of $\Pr[\textbf{x}_{1}{=}x_{1},\ldots,\textbf{x}_{T}{=}x_{T}]$, omitting explicit reference to the random variables themselves.

When considering conditionals of this distribution, we always condition on assignment to a prefix of the random variables and marginalize out a suffix. For example, we consider conditionals of the form $\Pr[\textbf{x}_{t+1}{=}x_{t+1},\ldots,\textbf{x}_{t+k}{=}x_{t+k}|\textbf{x}_{1}{=}x_{1},\ldots,\textbf{x}_{t}{=}x_{t}]$, and we write this as $\Pr[x_{t+1},\ldots,x_{t+k}|x_{1},\ldots,x_{t}]$. Similarly, when considering tuples $f:=(x^{\prime}_{1},\ldots,x^{\prime}_{k})\in\mathcal{O}^{k}$ and $h:=(x_{1},\ldots,x_{t})\in\mathcal{O}^{t}$, we write $\Pr[\textbf{x}_{t+1}{=}x^{\prime}_{1},\ldots,\textbf{x}_{t+k}{=}x^{\prime}_{k}|\textbf{x}_{1}{=}x_{1},\ldots,\textbf{x}_{t}{=}x_{t}]$ as $\Pr[f|h]$, noting that the random variables assigned to $f$ are determined by the length of $h$.

We lift this conditioning notation to sets of observation sequences in the following manner. If $F:=\{f_{1},f_{2},\ldots\}$ and $H:=\{h_{1},h_{2},\ldots\}$ where each $f_{i},h_{j}\in\mathcal{O}^{*}$, we write $\Pr[F|H]$ to denote the $|F|\times|H|$ matrix whose $(i,j)^{\mathrm{th}}$ entry is $\Pr[f_{i}|h_{j}]$. We allow the sequences in $F$ and $H$ to have different lengths, but always ensure that $\mathrm{len}(f_{i})+\mathrm{len}(h_{j})\leq T$ so that this matrix is well-defined. We refer to rows and columns of this matrix as $\Pr[f|H]$ and $\Pr[F|h]$ respectively.111We always refer to rows, columns, and entries of these matrices in this manner, so no confusion arises when constructing these matrices from (unordered) sets of sequences.

Lastly, for $h=(x_{1},\ldots,x_{t})$ we use $ho=(x_{1},\ldots,x_{t},o)$ to denote concatenation, and we lift this notation to sequences and sets. For instance, if $H=\{h_{1},h_{2},\ldots\}$ then $Ho=\{h_{1}o,h_{2}o,\ldots\}$.

1.1.1 Hidden Markov Models and low rank distributions

Hidden Markov Models provide a low-complexity parametrization for distributions over observation sequences. These models are defined formally as follows.

Definition 1.1** (Hidden Markov Models).**

Let $\mathcal{S}:=\{1,\ldots,S\}$. An HMM with $S\in\mathbb{N}$ hidden states is specified by (1) an initial distribution $\mu\in\Delta(\mathcal{S})$, (2) an emission matrix $\mathbb{O}\in\mathbb{R}^{O\times S}$, and (3) a state transition matrix $\mathbb{T}\in\mathbb{R}^{S\times S}$, and defines a distribution over sequences of length $T$ via:

[TABLE]

Here $M[i,j]$ represents the $(i,j)^{\mathrm{th}}$ entry of a matrix $M$.

As the name suggests, HMMs parameterize the distribution with a Markov chain over a hidden state sequence along with an emission function that generates observations. While this specific model is particularly natural, our analysis only leverages a certain low rank structure present in HMMs. To highlight the importance of this structure, we define the rank of a distribution.

Definition 1.2** (Rank of a distribution).**

We say distribution $\Pr[\cdot]$ over observation sequences of length $T$ has rank $r$ if, for each $t\in[T]$, the conditional probability matrix $\Pr[\mathcal{O}^{\leq T-t}|\mathcal{O}^{t}]$ has rank at most $r$.222When some histories occur with zero probability, there might be multiple consistent conditional probability functions associated to a distribution, in which case the rank is not uniquely defined. We address this by defining the distribution via its conditionals (which determine the rank); see Remark C.1.

An HMM with $S$ hidden states has rank at most $S$, which can be verified using the fact that the hidden states form a Markov chain (we give a proof in Section C.1).333In fact the rank of the HMM can be much smaller, since the decomposition alluded to above realizes the non-negative rank of the matrix, which can be exponentially larger than the rank. More generally, the rank identifies a low dimensional structure in the distribution: we have exponentially many vectors $\Pr[\mathcal{O}^{\leq T-t}|h]$, one for each history $h$, in an $r$-dimensional subspace of an exponentially larger ambient space. Thus, we are interested in algorithms that exploit the low dimensional structure and admit statistical and computational guarantees scaling polynomially with the rank.

1.1.2 Learning models

To circumvent computational hardness, we allow the learner to access conditional distributions of the underlying distribution $\Pr[\cdot]$. We specifically consider two access models formalized with the following oracles: 444Both oracles require committing to a consistent choice of conditional probability distribution when conditioning on zero probability events. See Remark C.1.

Definition 1.3** (Exact conditional probability oracle).**

The exact conditional probability oracle is given as input: observation sequences $h$ and $f$ of length $t\leq T$ and $T-t$ respectively, chosen by the algorithm, and returns the scalar $\Pr[f|h]$.

Definition 1.4** (Conditional sampling oracle).**

The conditional sampling oracle is given as input: an observation sequence $h$ of length $t\leq T$, chosen by the algorithm, and returns an observation sequence $f$ of length $T-t$ such that the probability that $f$ is returned is $\Pr[f|h]$, independently of all other randomness.

When considering the exact probability oracle, we also allow the learner to obtain independent samples from the joint distribution $\Pr[\cdot]$. Note that this oracle equivalently provides access to exact (unconditional) probabilities of length $T$ sequences. We view this as a noiseless analog of the conditional sampling oracle, which is the main model of interest.

As a learning goal, we consider distribution learning in total variation distance as studied in prior works [Kea+94, MR05, HKZ12, Ana+14]. Given access to a target distribution $\Pr[\cdot]$ we want to efficiently compute an estimate $\widehat{\Pr}[\cdot]$ that is close in total variation distance to $\Pr[\cdot]$. Formally, we want an algorithm that, when given parameters $\varepsilon,\delta>0$, computes an estimate $\widehat{\Pr}[\cdot]$ such that with probability at least $1-\delta$:

[TABLE]

The algorithm is efficient if its computational complexity (and hence number of oracle calls) scale polynomially in $r,T,O,1/\varepsilon$ and $\log(1/\delta)$.

Remark 1.5**.**

Note that, as the support of $\Pr[\cdot]$ is exponentially large in $T$, it is not possible to write down all $\mathcal{O}^{T}$ values of $\widehat{\Pr}$ efficiently. Instead, the goal is to return an efficient representation from which we can evaluate $\widehat{\Pr}[x_{1},\ldots,x_{T}]$ for any sequence $x_{1},\ldots,x_{T}$ efficiently. It will become clear what constitutes an efficient representation for low rank distributions in the sequel; indeed the fact that one even exists is one of our central structural results. for HMMs, for example, the tuple of initial distribution $\mu$, observation operator $\mathbb{O}$, and transition operator $\mathbb{T}$ form an efficient representation.

1.2 Our results

Our first result studies the computational power provided by the exact probability oracle (Definition 1.3). We show how a generalization of Angluin’s $L^{*}$ algorithm can efficiently learn any HMM given access to this oracle. The result is summarized in the following theorem:555As this result is a warmup for our main result, we focus on the setting where $\mathcal{O}=\{0,1\}$ for simplicity.

Theorem 1** (Learning with exact conditional probabilities).**

*Assume $\mathcal{O}=\{0,1\}$. Let $\Pr[\cdot]$ be any rank $r$ distribution over observation sequences of length $T$. Pick any $0<\varepsilon,\delta<1$. Then Algorithm 1 with access to an exact probability oracle and samples from $\Pr[\cdot]$, runs in $\operatorname{poly}(r,T,1/\varepsilon,\log(1/\delta))$ time and returns an efficiently represented approximation $\widehat{\Pr}[\cdot]$ satisfying $\mathrm{TV}(\Pr,\widehat{\Pr})\leq\varepsilon$ with probability at least $1-\delta$. *

The main technical challenge is finding a succinct and observable parametrization/representation of the distribution, so that we can infer all conditional distributions using polynomially many queries. This observable parameterization plays a central role in our main result, and in this sense Theorem 1 can be seen as an insightful warmup.

Our main contribution is in extending this result to the more natural interactive setting where the learner only accesses conditional samples via the oracle in Definition 1.4. Our algorithm here can be viewed as a robust version of $L^{*}$, and we obtain the following guarantee:

Theorem 2** (Learning with conditional samples).**

Let $\Pr[\cdot]$ be any rank $r$ distribution over observation sequences of length $T$. Assume distribution $\Pr[\cdot]$ has fidelity $\Delta^{*}$. Pick any $0<\varepsilon,\delta<1$. Then Algorithm 2 with access to a conditional sampling oracle runs in $\operatorname{poly}(r,T,O,1/\Delta^{*},1/\varepsilon,\log(1/\delta))$ time and returns an efficiently represented approximation $\widehat{\Pr}[\cdot]$ satisfying $\mathrm{TV}(\Pr,\widehat{\Pr})\leq\varepsilon$ with probability at least $1-\delta$.

The theorem provides a robust analog to Theorem 1 in the much weaker conditional sampling access model. The caveat is that the guarantee depends on a spectral property of a distribution, which we call the fidelity. The definition of fidelity (Definition 2.2) requires further development of the algebraic structure in $\Pr[\cdot]$ and is deferred to Section 2. Nevertheless, we can show that the cryptographically hard examples of HMMs and positive results from prior work on learning HMMs have fidelity that is lower bounded by a (small) polynomial of the other parameters and thus are efficiently learnable by our algorithm (see Appendix C). On the other hand, there are HMMs with exponentially small fidelity, and we have no evidence that these instances are computationally intractable when provided with conditional samples. This leads to the main open question stemming from our work.

Open Problem 1.6**.**

Is there a computationally efficient algorithm for learning any low rank distribution given access to a conditional sampling oracle?

Paper organization.

In Section 2, we present an overview of our techniques, explaining the challenges and how we address them. Then we turn to the more formal presentation of the proofs, with Section 3 devoted to Theorem 1 and Section 4 devoted to Theorem 2. These sections present our algorithms and the main ingredients for their analysis, with some details deferred to the appendices. We close the main body of the paper in Section 5, with some further discussion regarding 1.6.

2 Technical overview

To explain the central challenges with learning low rank distributions and how we overcome them, let us introduce the following notation: let $H_{t}:=\mathcal{O}^{t}$ and $F_{t}:=\mathcal{O}^{T-t}$ denote the observation sequences of length $t$ and $T-t$ respectively. Then the matrix $\Pr[F_{t}|H_{t}]$ is a submatrix of $\Pr[\mathcal{O}^{\leq T-t}|\mathcal{O}^{t}]$ and hence is rank at most $r$ by assumption. If we define these matrices for each length $t\in[T]$, then clearly we have encoded the entire distribution. Hence, estimating these matrices in an appropriate sense would suffice for distribution learning. Although the matrices all have rank at most $r$, they are exponentially large, so the low rank property does not immediately yield an efficient representation of the distribution. Indeed, we must leverage further structure to obtain efficient algorithms.

2.1 Background: Observable operators and hard instances

For HMMs, we can hope to leverage the explicit formula for the probability of a sequence (Equation 1) to obtain an efficient algorithm. Indeed, this is the approach adopted by Hsu, Kakade, and Zhang [HKZ12]. Specifically, they use the observable operator representation [Jae00]: if we define $S\times S$ matrices $\{\mathbb{A}_{o}\}_{o\in\mathcal{O}}$ as $\mathbb{A}_{o}:=\mathbb{T}\mathrm{diag}(\mathbb{O}[o,\cdot])$ then we can write the probability of any observation sequence as

[TABLE]

where ${\bf 1}$ is the all-ones vector and recall that $\mu$ is the initial state distribution. Hsu, Kakade and Zhang show that these operators can be estimated, up to a linear transformation, whenever $\mathbb{T}$ and $\mathbb{O}$ have full column rank. In fact, under their assumptions, these operators can be recovered from $\Pr[\textbf{x}_{1}{=}\cdot,\textbf{x}_{2}{=}\cdot,\textbf{x}_{3}{=}\cdot]$ alone; no higher order moments of the distribution are required.

Unfortunately, this approach fails if either $\mathbb{T}$ or $\mathbb{O}$ are (column) rank deficient, and it is conjectured that the rank deficient HMMs are precisely the hard instances [MR05]. On the other hand, many interesting HMMs are rank deficient. For example, any overcomplete HMM—one with fewer observations than states—cannot have a full column rank $\mathbb{O}$ matrix. This captures all deterministic finite automata where the alphabet size is smaller than the number of states as well as the parity with noise problem.

Learning parity with noise is a particularly interesting case. The standard formulation is that we obtain samples of the form $(\textbf{z},\textbf{y})\in\{0,1\}^{T-1}\times\{0,1\}$ where z is uniformly distributed on the hypercube and $\textbf{y}=\bigoplus_{i\in I}\textbf{z}_{i}$ with probability $1-\alpha$ and $\textbf{y}=1-\bigoplus_{i\in I}\textbf{z}_{i}$ with the remaining probability. Here $\bigoplus$ denotes the parity operation, $I$ is a secret subset of indices $I\subseteq[T-1]$, and $\alpha\in(0,1/2)$ is a noise parameter. We want to learn the subset $I$, given samples from this process. This problem is widely believed to be computationally hard and can be encoded as an HMM with $\mathcal{O}=\{0,1\}$ and $4T$ states (see Appendix C). Considering this problem, it is quite apparent that low degree moments, like those used by Hsu, Kakade, and Zhang, reveal no information about the subset $I$. In particular, the observable operators $\mathbb{A}_{o}$ are not identifiable from low degree moments. One must use higher order information, i.e., statistics about long sequences, to solve this problem.

2.2 Efficient representation

For rank deficient HMMs, it is not clear how to identify the observable operators and it is not even clear that such operators exist for the more general case of low rank distributions. So, we must return to the question of how to efficiently represent the distribution. Here, our first observation is that any submatrix of $\Pr[F_{t}|H_{t}]$ that has the same rank as the entire matrix can be used to build an efficient representation. To see why, suppose we have such a submatrix, and let us index the columns/histories of the submatrix by $B_{t}$, which we refer to as the basis. It follows that $\Pr[F_{t}|B_{t}]$ spans the column space of $\Pr[F_{t}|H_{t}]$, which implies that for any history $h\in H_{t}$ there exists coefficients $\beta(h)\in\mathbb{R}^{|B_{t}|}$ such that

[TABLE]

The main observation toward obtaining an efficient representation is to exploit a certain circulant structure in the matrices $\{\Pr[F_{t}|H_{t}]\}_{t\leq T}$ to model the evolution of the coefficients (visualized in Figure 1). The circulant structure is simply that for basis $B_{t}$, observation $o$, and future $f\in F_{t+1}$ (i.e., of length $T-t-1$) the vector $\Pr[B_{t}of]$ appears in two of the matrices (albeit with different scaling). It appears in the matrix $\Pr[F_{t}|H_{t}]$ in row $of$ and columns $B_{t}$, and it appears in the matrix $\Pr[F_{t+1}|H_{t+1}]$ in row $f$ and columns $B_{t}o$. Thus, if we learn how to represent the columns $\Pr[F_{t+1}|B_{t}o]$ in terms of the columns $\Pr[F_{t+1}|B_{t+1}]$—which we can do via the coefficients—the circulant property provides a connection between the matrices $\Pr[F_{t+1}|H_{t+1}]$ and $\Pr[F_{t}|H_{t}]$.

Formally, we can define operators $\{A_{o,t}\}$ for each observation $o\in\mathcal{O}$ and sequence length $t\in[T]$ satisfying

[TABLE]

which can then be used to express sequence probabilities by iterated application. Indeed, we have

[TABLE]

where by an explicit choice of $B_{0}$, $B_{T}$ and $F_{T}$, the matrices $A_{x_{1},0}$ and $A_{x_{T},T-1}$ are column and row vectors respectively, and so the right-hand side is a scalar (see Proposition 3.2 for details)666We define $B_{0}$, $B_{T}$ and $F_{T}$ to be singleton sets. $B_{0}$ and $F_{T}$ contain the empty string $\varphi$ and $B_{T}$ contains any length $T$ observation sequence. These new definitions, in conjunction with Proposition 3.2 imply: $A_{x_{T},T-1}=\Pr[x_{T}|B_{T-1}]$ and therefore will be a row vector. Similarly, $A_{x_{1},0}$ is a solution of $\Pr[F_{1}|B_{1}]A_{x_{1},0}=\Pr[x_{1}F_{1}|\varphi]$ and is therefore a column vector.. More importantly, these operators can also be viewed as evolving the coefficients via the identity:

[TABLE]

This identity is proved in Proposition 3.2 and Proposition B.11. We highlight the scaling, which results in a nonlinear update equation and appears because the coefficients express conditional rather than joint probabilities. This viewpoint of operators evolving coefficients will play a central role in our error analysis.

Thus, it remains to find the bases $\{B_{t}\}_{t\leq T}$, estimate the operators $\{A_{o,t}\}_{o\in\mathcal{O},t\leq T}$, and control the error amplification from iteratively multiplying these estimates. We turn to these issues next.

Remark 2.1**.**

The approach of Hsu, Kakade, and Zhang can also be viewed as estimating operators via Equation 2 with the particular choice of basis. They show that conditional distribution of futures given any history can be written in the span of the conditional distributions of the single observation histories, so that $\mathcal{O}$ itself forms a basis. This is implied by their assumptions and it permits using only second and third degree moments to estimate the operators. However, in general we will need to use long sequences in our bases and interactive access will be crucial for estimation. Additionally, under their choice of bases and their assumptions they show that the solution of Equation 2 is related to the observable operators [Jae00], explicitly given by $\mathbb{T}$ and $\mathbb{O}$, by an invertible and bounded transformation, which is instrumental in their error analysis. When considering general bases $B$, we do not have such a connection and will require a novel error propagation argument.

2.3 Error propagation

Although finding the bases $B_{t}$ and estimating corresponding operators $A_{o,t}$ is nontrivial, even if we have estimated these operators accurately, we must address the error amplification that can arise from repeated application of the learned operators. This challenge makes up the majority of our technical analysis. We discuss estimating operators $A_{o,t}$ in Section 2.4 and how to find the basis in Section 2.5.

To explain the error amplification challenge, suppose for now that we are given bases $\{B_{t}\}_{t\leq T}$ and subsequently estimate the operators $A_{o,t}$ in $\ell_{2}$ norm, i.e., we have estimate $\widehat{A}_{o,t}$ satisfying $\|\widehat{A}_{o,t}-A_{o,t}\|_{2}\leq\varepsilon$. We first define our estimated model $\widehat{\Pr}$ in terms of the estimated operators $\widehat{A}_{o,t}$. Considering Equation 3, the natural estimator is

[TABLE]

where, as before, the matrices $\widehat{A}_{x_{1},0}$ and $\widehat{A}_{x_{T},T-1}$ are column and row vectors respectively, so the right hand side is a scalar. To simplify notation for this section, we omit the time indexing on the operators.

Given this estimate, the total variation distance is

[TABLE]

Let us first discuss two strategies for bounding this expression that can work in some cases, but do not seem to work in our setting. One idea is to pass to the $\ell_{2}$ norm and use a telescoping argument to obtain several terms of the form

[TABLE]

These terms are convenient because the matrix products only disagree in the $t^{\mathrm{th}}$ operator. However, both the “incoming” product $A_{x_{t}}\ldots A_{x_{1}}$ that pre-multiplies this difference and the “outgoing” product $\widehat{A}_{x_{T}}\ldots\widehat{A}_{x_{t+2}}$ whose norm we must bound can be rather poorly behaved. For example, the product $A_{x_{t}}\ldots A_{x_{1}}$ can have $\ell_{2}$ norm that grows exponentially with $t$, since the $\ell_{2}$ norm of the individual matrices can be much larger than $1$. An even worse problem is that we have exponentially many terms in the sum, so that even bounding each term by $\varepsilon$ (which would be possible if the incoming and outgoing products were well behaved) is grossly insufficient.

The other approach is the strategy adopted by Hsu, Kakade, and Zhang [HKZ12], which uses the definition of the observable operators [Jae00], $\mathbb{A}_{x}=\mathbb{T}{\rm diag}(\mathbb{O}[x,\cdot])$, explicitly. This allows them to control the incoming and outgoing products in a decomposition analogous to the one above, but in the $\ell_{1}$ norm. Their decomposition involves several terms, but to convey the main idea, observe that we can bound

[TABLE]

The idea is that each term in the final sum can be seen as a joint probability of the history $x_{1},\ldots,x_{t}$ and the hidden state $s_{t+1}$, so we can sum over all histories with no error amplification. Unfortunately, there is no hidden state in the more general setting (and for the rank deficient case, the observable operators can not be learned accurately as discussed in Section 2.1), so we cannot appeal to an argument of this form. Indeed, our main technical contribution is a new perturbation analysis that relies on no structural assumptions.

At a more technical level, the issue with both of these arguments is that passing to any norm, seems to be too coarse to adequately control the error amplification. Instead, our argument carefully tracks the error in the space of the coefficients. Precisely, given estimates $\widehat{A}_{o,t}$ that satisfy $\|\widehat{A}_{o,t}-A_{o,t}\|_{2}\leq\varepsilon$, we can show, via an inductive argument, that for any $x_{1},\ldots,x_{t}$

[TABLE]

where $V_{t}^{\perp}$ is an orthonormal basis for the kernel of $\Pr[F_{t}\mid B_{t}]$ and $\alpha_{h},\gamma_{v}$ are scalars. Moreover, the TV distance between $\Pr[\cdot]$ and $\widehat{\Pr}[\cdot]$ is exactly equal to the sum of these scalars over all sequences $x_{1},\ldots,x_{T}$. Even though there could be exponentially many terms in this sum, we show that this sum is small via an inductive argument. This makes up the most technical component of our proof, and we give a more detailed overview in Section 4 with the formal proofs in Section B.3.

2.4 Estimating operators

We next discuss estimating the operators $\{A_{o,t}\}_{o\in\mathcal{O},t\leq T}$ using the conditional sampling oracle. A natural idea is to use samples to estimate both sides of the system in Equation 2 and solve the noisy version via linear regression. Unfortunately, this system may have exponentially small (in $T-t$) singular values, making it highly sensitive to perturbation. There is also a cosmetic issue when working with $\Pr[F_{t+1}|B_{t+1}]$, namely this matrix is exponentially large.

To address these challenges, we introduce a particular preconditioner that stabilizes the system. Specifically, we instead estimate and solve

[TABLE]

where $D_{t+1}$ is a diagonal matrix with entries $d_{t+1}(f):=\frac{1}{|B_{t+1}|}\sum_{b\in B_{t+1}}\Pr[f|b]$ on the diagonal.777This choice of $D_{t+1}$ ensures there is no division-by-zero issue, see Remark B.8. The benefit of this preconditioner is that the new matrices are of size $|B_{t+1}|\times|B_{t+1}|$ rather than exponentially large, and yet they can still be estimated efficiently using the conditional sampling oracle. To see why the latter holds, observe that the $(i,j)^{\mathrm{th}}$ entry of the matrix on the LHS is

[TABLE]

where $B_{t+1}=\{b_{1},b_{2},\ldots,\}$. Intuitively, we can estimate this entry by sampling futures $f$ from $\Pr[\cdot|b]$ to approximate any term in the sum and sampling futures from $d_{t+1}(\cdot)$ to approximate the sum itself. While this is true, there is one technical issue to overcome: to estimate the ratio to additive accuracy, we must estimate the individual probabilities $\Pr[f\mid b_{i}]$, $\Pr[f\mid b_{j}]$ and $d_{t+1}(f)$ to relative accuracy. We can obtain $(1\pm\zeta)$ relative error estimates using conditional samples as long as the one-step probabilities are at least $\Omega(\zeta/T)$, but this is challenging when even a single one-step probability is small. To address this issue, we show that such futures actually contribute very little to the overall sum, and we design a test to safely ignore them. See Section B.6 for details.

While the ability to estimate the entries is clearly important, the hope with preconditioning is that it dramatically amplifies the singular values of the matrix on the left hand side. In particular, we want that the matrix $\Pr[F_{t+1}|B_{t+1}]^{\top}D_{t+1}^{-1}\Pr[F_{t+1}|B_{t+1}]$ has large (non-zero) singular values, as this will allow us to estimate the operators $A_{o,t}$ in the $\ell_{2}$ norm. Our choice of preconditioner does achieve this in the important example of parity with noise: we can show that $\Pr[F_{t+1}|B_{t+1}]$ has exponentially small (in $T-t$) singular values for every choice of $B_{t+1}$, while there exists a basis $B_{t+1}$ for which the non-zero singular values of the preconditioned matrix are $\Omega(1)$ (see Appendix C). Unfortunately, in general, a basis which ensures the preconditioner has large singular values might not exist, and we address this by introducing the notion of fidelity.

Definition 2.2** (Fidelity).**

We say that distribution $\Pr[\cdot]$ has fidelity $\Delta^{*}$ if there exists some bases $\{B_{t}\}_{t\in[T]}$, such that $\max_{t}|B_{t}|\leq 1/\Delta^{*}$ and

[TABLE]

where $\sigma_{+}(M)$ denotes the magnitude of the smallest non-zero eigenvalue of $M$, $D_{t}$ is a diagonal matrix of size $|F_{t}|\times|F_{t}|$ with entries $d_{t}(f):=\frac{1}{|B_{t}|}\sum_{b\in B_{t}}\Pr[f|b]$, and $S_{t}$ is a diagonal matrix of size $|H_{t}|\times|H_{t}|$ with entries $s_{t}(h):=\Pr[h]$.

Importantly, we only assume the existence of bases with this property, not that it is given to us or otherwise known in advance. Note that, although the matrix with large eigenvalues according to the fidelity definition is not the same as the preconditioned matrix we care about for learning operators, nevertheless when the distribution has high fidelity (i.e., $\Delta^{*}$ is large), we can find a basis for which $\Pr[F_{t+1}|B_{t+1}]^{\top}D_{t+1}^{-1}\Pr[F_{t+1}|B_{t+1}]$ has large eigenvalues. This, combined with our approach for estimating entries of the preconditioned matrix, allow us to learn operators $A_{o,t}$ in the $\ell_{2}$ norm. We provide details in Section B.2.

Remark 2.3**.**

Although our approach seems to require large fidelity, the parity with noise example suggests that this definition of fidelity, which can lead to a favorable preconditioned system, is more appropriate than directly assuming $\Pr[F_{t+1}|B_{t+1}]$ has large singular values. Indeed, we can also show that fidelity captures all previously studied positive results for learning HMMs. We also believe our approach can be extended to learn HMMs with small fidelity as described in Section 5.

2.5 Finding the basis

The only remaining challenge is to find the bases $\{B_{t}\}_{t\in[T]}$. Recall that, when considering the conditional sampling oracle, we want bases for which the preconditioned matrices have large eigenvalues. It turns out that when the distribution has high fidelity a random sample of polynomially many histories will form a basis with this property with high probability. Given that the other aspects of our analysis seem to require high fidelity, this random sampling approach thus suffices to prove Theorem 2.

On the other hand, for low fidelity distributions, random sampling will fail to cover the directions with small singular value, and so basis finding becomes an intriguing aspect of learning with the conditional sampling oracle. Basis finding is also the final issue to address for Theorem 1, using the exact oracle. In both cases, we provide adaptations of Angluin’s $L^{*}$ algorithm that finds bases for any low rank distribution. We defer discussion of the conditional sampling version to Appendix D and hope that it serves as a starting point toward resolving 1.6.

Adapting $L^{*}$ for basis finding with the exact oracle.

We close this section by explaining how to find a basis when provided with the exact probability oracle. As a first observation, note that we need not construct the entire system in Equation 2 to identify operators $A_{o,t}$. It suffices to find a set of futures $\Lambda_{t}\subset F_{t}$ such that $\Pr[\Lambda_{t}\mid H_{t}]$ spans the row space of $\Pr[F_{t}\mid H_{t}]$. In other words, we just need $B_{t}$ and $\Lambda_{t}$ for which $\Pr[\Lambda_{t}\mid B_{t}]$ has the same rank as $\Pr[F_{t}\mid H_{t}]$.

The difficulty is that there is no universal choice of $B_{t},\Lambda_{t}$ for general low rank distributions, and finding these sets poses a challenge search problem in an exponentially large space. We address this challenge using the exact probability oracle and an adaptation of Angluin’s $L^{*}$ algorithm for learning DFAs. The basic idea is as follows: given sets $B_{t},\Lambda_{t}$ whose submatrix is not of the required rank, we can still solve the underdetermined system

[TABLE]

and obtain an estimate $\widehat{\Pr}[\cdot]$ via Equation 5. Then, we can sample sequences $x_{1},\ldots,x_{t}\sim\Pr[\cdot]$ and check if our estimate makes the correct predictions on these sequences. In particular, we check

[TABLE]

If the predictions are accurate (i.e., these equalities hold) for each $t$ and for polynomially many random sequences, then we can show that $\widehat{\Pr}[\cdot]$ is close $\Pr[\cdot]$ in total variation distance.

On the other hand, if these equalities do not hold for some sample $x_{1},\ldots,x_{t}$, then we can use it as a counterexample to improve our basis. We provide all the details in Section 3.

3 Learning with conditional probabilities (Theorem 1)

In this section we prove Theorem 1.

See 1

We first introduce some notation, which differs from Section 2 slightly. We define $H_{t}:=\mathcal{O}^{t}$ to be the set of histories of length $t$. Similarly, we define $F_{t}:=\mathcal{O}^{\leq T-t}$ to be the set of futures of length $\leq T-t$, coinciding with our rank definition. Notice that unlike in Section 2, we take $F_{t}$ to be all futures of length up to $T-t$, so that one may append elements from the futures $F_{t}$ to elements from the histories $H_{t}$ to obtain a valid observation sequence of length at most $T$. To simplify the technical notation, let $\varphi$ be the empty string and define probabilities associated to empty string as: $\Pr[x_{1}\ldots x_{T}|\varphi]=\Pr[x_{1}\ldots x_{T}]$ and $\Pr[\varphi|x_{1}\ldots x_{T}]=1$ for any $T$-length sequence $x_{1},\ldots,x_{T}$.

We now formally define the notion of bases for distribution $\Pr[\cdot]$.

Definition 3.1** (Basis).**

Let $\Pr[\cdot]$ be any distribution over observation sequences of length $T$. A set $\{B_{t}\}_{t\in[T]}$, where each $B_{t}\subset H_{t}$, forms bases for $\Pr[\cdot]$, if for each $t\in[T]$ and all $x\in\mathcal{O}^{t}$, there exists coefficients $\beta(x)$ such that:

[TABLE]

We call each $B_{t}$ a basis for $\Pr[\cdot]$ at sequence length $t$.

In other words, a set $B_{t}\subset H_{t}$ forms a basis for distribution $\Pr[\cdot]$ if the column vectors $\Pr[F_{t}|B_{t}]$ span the column space of $\Pr[F_{t}|H_{t}]$. For now, when choosing $B_{t}$, we impose no constraint on the size of these coefficients, and we also do not require the columns $\Pr[F_{t}\mid B_{t}]$ to be linearly independent. The low rank property of $\Pr[\cdot]$ directly implies that for each $t$, there exists a basis $B_{t}$ with $|B_{t}|\leq r$. However, as discussed in Section 2.2, there are exponentially many histories in $H_{t}$, so even if we had such a small basis $B_{t}$, simply learning the coefficients for each history will not suffice for an efficient algorithm. We address this issue with the following structural result: because of the circulant structure of the conditional probability matrix, we can generate all the coefficients using $OT$ matrices each of size at most $r\times r$.

Proposition 3.2** (Existence of efficient representation).**

Let $B_{0}=F_{T}=\{\varphi\}$ and $B_{T}=\{h\}$ for any observation sequence $h\in H_{T}$.888We set $B_{T}$ to be a singleton set for notational clarity, as otherwise we would have to pre-multiply our probability estimate with the all ones row vector. Note that any singleton set forms a basis because $\Pr[F_{T}|H_{T}]$ is the all ones matrix. For $t\in\{1,\ldots,T-1\}$, let $B_{t}\subset H_{t}$ be any basis for distribution $\Pr[\cdot]$ at sequence length $t$. Then, the probability distribution $\Pr[\cdot]$ can be written as999Here by choice of basis $B_{0}$ and $B_{T}$, $A_{x_{T},T-1}=\Pr[x_{T}|B_{T-1}]$ by definition and is therefore a row vector. Similarly, $A_{x_{1},0}$ is a solution of $\Pr[F_{1}|B_{1}]A_{x_{1},0}=\Pr[x_{1}F_{1}|\varphi]$ and is therefore a column vector.:

[TABLE]

where matrices $A_{o,t}$ for every $o\in\mathcal{O}$ and $t\in\{0,\ldots,T-1\}$ satisfy

[TABLE]

Moreover, this equation always has a solution.

Proof.

We first show there exists a solution $A_{o,t}$ for Equation 7. For basis $B_{t}=\{b_{1},\ldots,b_{n}\}$ and $B_{t+1}$, we claim the following $A_{o,t}$ is a solution:

[TABLE]

Here $\beta(x)$ and $\beta(xo)$ are the coefficients associated to history $x$ of length $t$ under $B_{t}$ and history $xo$ of length $t+1$ under $B_{t+1}$ respectively. Recall that, in particular, these coefficients are such that $\Pr[F_{t+1}\mid B_{t+1}]\beta(xo)=\Pr[F_{t+1}\mid xo]$. By definition of $A_{o,t}$,

[TABLE]

Since $oF_{t+1}$ is a subset of $F_{t}$, by repeatedly applying this equation, we get

[TABLE]

Noting $\Pr[F_{T}|B_{T}]=1$ and $\Pr[x_{1}x_{2}\ldots x_{T}F_{T}|B_{0}]=\Pr[x_{1}x_{2}\ldots x_{T}\varphi|\varphi]=\Pr[x_{1}x_{2}\ldots x_{T}]$ as $F_{T}=B_{0}=\{\varphi\}$ completes the proof. ∎

3.1 Algorithm

We now present our algorithm (Algorithm 1). The user furnishes $\varepsilon$, the accuracy with which the distribution is to be learned; and $\delta$, a confidence parameter. The parameter $n$ depends on the input and is detailed in the proof of Theorem 1.

As discussed in Section 2, the algorithm iteratively builds a set of histories $B_{t}\subset H_{t}$ and a set of futures $\Lambda_{t}\subset F_{t}$ (for each $t$), to span the column/row space of $\Pr[F_{t}\mid H_{t}]$, respectively. Via Proposition 3.2, if we can find such sets, they would provide an efficient representation of the distribution. We refer to $B_{t}$ and $\Lambda_{t}$ as representative histories and test futures, respectively, and as we grow these sets, we maintain the invariant that the matrix $\Pr[\Lambda_{t}\mid B_{t}]$ is square and invertible.

We start with $B_{t},\Lambda_{t}$ of size $1$. Then, we repeat the following: motivated by the evolving equation in Proposition 3.2, we use Equation 9 to compute estimates $\widehat{A}_{o,t}$ using our current representative histories and test futures. This may be an under-determined linear system, but Proposition 3.2 guarantees that it has a solution, and we take $\widehat{A}_{o,t}$ to be any such solution. We use these operators to define our estimate for the distribution, given in Equation 10, via iterated multiplication of the operators. Then, we sample several sequences from the distribution and check if any of them certify that our estimate is incorrect, i.e., serve as a counterexample. If we do find a counterexample, then the algorithm finds a time step $\tau$, a new history $b^{\prime}\in H_{\tau}$ and a new future $\lambda^{\prime}\in F_{\tau}$ that increases the rank of $\Pr[F_{\tau}\mid B_{\tau}]$ (this step is described in Proposition 3.3 below). This can only happen $rT$ times if the distribution has rank $r$. On the other hand, if we do not find a counterexample, then we simply output our current estimate.

3.2 Analysis

We first show how to use a counterexample to improve our set of representative histories and test futures.

Proposition 3.3** (Finding representative histories and test futures).**

If $x_{1}\ldots x_{t}$ is a counterexample, that is, it satisfies the following:

[TABLE]

then we can find a new test future $\lambda^{\prime}\in F_{\tau}$ and representative history $b^{\prime}\in H_{\tau}$ for $\tau\in[t]$ in at most $\operatorname{poly}(r,T)$ time such that $\operatorname{rank}(\Pr[\Lambda_{\tau}\cup\{\lambda^{\prime}\}|B_{\tau}\cup\{b^{\prime}\}])=\operatorname{rank}(\Pr[\Lambda_{\tau}|B_{\tau}])+1$.

Proof.

For clarity, in the poof, we abuse notation and do not explicitly mention the sequence length when writing the operator $A_{o,t}$, i.e., we use $A_{x_{t}}$ instead of $A_{x_{t},t-1}$. First, we find a time $\tau\in[t]$ where the following equations hold:

[TABLE]

Such a $\tau$ must exist because (a) the first equation is true for $\tau=0$ by definition, and (b) the second equation is true for $\tau=t-1$ because of the counterexample property (Equation 11). Now, we can simplify the equations above by substituting the vector $\mathbf{v}:=(\Pr[x_{1}\ldots x_{\tau}])^{-1}\widehat{A}_{x_{\tau}}\ldots\widehat{A}_{x_{1}}$ which gives

[TABLE]

where the last step holds by definition of $\widehat{A}_{x_{\tau+1}}$ (Equation 9). Let $x_{\tau+1}\lambda_{\tau+1}$ index the row of Equation 13 where equality does not hold. Define $\lambda^{\prime}=x_{\tau+1}\lambda_{\tau+1}$ and $b^{\prime}=x_{1}\ldots x_{\tau}$. We show that the equations above imply that the row vector $\Pr[\lambda^{\prime}|B^{\prime}_{\tau}]:=\Pr[x_{\tau+1}\lambda_{\tau+1}|B^{\prime}_{\tau}]$ is linearly independent of the rows of $\Pr[\Lambda_{\tau}|B^{\prime}_{\tau}]$. This is enough to prove our claim that $\operatorname{rank}(\Pr[\Lambda^{\prime}_{\tau}|B^{\prime}_{\tau}])=\operatorname{rank}(\Pr[\Lambda_{\tau}|B_{\tau}])+1$.

We establish linear independence by contradiction. Assume that $\Pr[\lambda^{\prime}\mid B^{\prime}_{\tau}]$ is in the span of the rows of $\Pr[\Lambda_{\tau}\mid B^{\prime}_{\tau}]$. Then, there exists a vector w such that:

[TABLE]

Then, we reach a contradiction as

[TABLE]

where the first and third equality follows from linear dependence (Equation 14), the second equality follows from Equation 12, and the last inequality follows from Equation 13. ∎

Finally, we need a technical lemma which allows us to estimate the TV distance using conditional samples. This lemma implies that if our algorithm does not find a violation, then with high probability our estimate is close to the true distribution in TV distance.

Proposition 3.4** (Substitute for TV oracle).**

Let $\Pr[\cdot]$ and $\widehat{\Pr}[\cdot]$ be two probability distributions over observation sequences of length $T$. Suppose that for all $t\in\{0,\ldots,T\}$ and observations $o\in\mathcal{O}$

[TABLE]

Then

[TABLE]

Since, $\overline{\Pr}[\cdot]$ might not be a probability distribution, we need to apply this proposition to a probability distribution $\widehat{\Pr}[\cdot]$ that is close to $\overline{\Pr}[\cdot]$, which can be obtained by a simple construction. These details and the proof of Proposition 3.4 are relatively straightforward and deferred to Appendix A.

4 Learning with conditional samples (Theorem 2)

In this section, we prove Theorem 2

See 2

Throughout this section, we use the same notation as Section 2, the set of futures $F_{t}:=\mathcal{O}^{T-t}$.

4.1 Algorithm

Algorithm pseudocode is displayed in Algorithm 2. The user furnishes $\varepsilon$, the accuracy with which the distribution is to be learned; $\delta$, a confidence parameter; $\Delta^{*}$, the fidelity of the distribution and $r$, the rank of the distribution. The parameters $\Delta,\lambda,n$ and $m$ are detailed in the proof of Theorem 2 in Section B.4.

As with the previous algorithm, Algorithm 2 relies on the efficient representation provided by Proposition 3.2. First, the algorithm finds basis histories $B_{t}$ for each $t\in[T]$. As discussed in Section 2, under the fidelity assumption, this is not particularly challenging and can be done by sampling from the distribution. The remaining steps in the algorithm constitute a specialized technique for estimating the operators $A_{o,t-1}$ specified in Proposition 3.2.

Our estimate $\widehat{A}_{o,t-1}$ is based on the formula for $A_{o,t-1}$ given in Equation 8 and involves three components: (a) projection onto (an estimate of) the row space of $\Pr[F_{t}\mid B_{t}]$, (b) estimates of coefficients $\beta(bo)$ and (c) estimates of probabilities $\Pr[o\mid b]$, where the latter two are for $b_{i}\in B_{t-1}$. Item (c) is straightforward using conditional samples. For item (a), we define the “preconditioned matrix”

[TABLE]

where $D_{t}$ is a $|F_{t}|\times|F_{t}|$ diagonal matrix with $d_{t}(f)=\frac{1}{|B_{t}|}\sum_{b\in B_{t}}\Pr[f\mid b]$ on the diagonal. We show in Section B.6, how this matrix can be estimated using conditional samples. We project onto the principal subspace of the estimated matrix, i.e., onto the span of the eigenvectors with eigenvalue larger than $\Delta/2$. These projections help with error propagation, as it eliminates errors that leave the principal subspace.

For item (b), we estimate the coefficients $\beta(bo)$ for $b\in B_{t-1}$ via linear regression. Using our preconditioner and the definition of the coefficients, we can see that the coefficients satisfy:

[TABLE]

As with $\Sigma_{B_{t}}$, $q(bo)$ can also be estimated using conditional samples (via the approach in Section B.6). Moreover, our basis $B_{t}$ will ensure that $\|\beta(bo)\|_{2}$ is bounded by a universal constant, so we can use ridge regression to find estimates $\widehat{\beta}(bo)$. Then we can plug these into Equation 15 to obtain estimates $\widehat{A}_{o,t-1}$. We return these matrices as the representation of our estimated distribution.

4.2 Analysis

In the previous setting, when we had access to exact conditional probability oracle, the main challenge was finding the bases. In contrast, now that we can only obtain samples, even if we know the bases, we can only learn operators $A_{o,t}$ approximately. As discussed in Section 2, controlling estimation errors will require the notion of robust bases, which we define next.

Definition 4.1** (Robust bases).**

Bases $\{B_{t}\}_{t\in[T]}$ for distribution $\Pr[\cdot]$ are $\Delta$-robust if for every $t\in[T]$:

[TABLE]

where $\sigma_{+}(M)$ denotes the minimum non-zero eigenvalue of $M$ and $D_{t}$ is a diagonal matrix of size $|F_{t}|\times|F_{t}|$ with entries $d_{t}(f):=\frac{1}{|B_{t}|}\sum_{b\in B_{t}}\Pr[f|b]$ on the diagonal.

A priori, it is unclear if such bases exists for arbitrary low rank distributions. Moreover, even if robust bases exist, how do we find them? Our first lemma show how to find robust bases for high fidelity distributions (Definition 2.2).

Lemma 4.2** (Finding robust bases).**

Assume distribution $\Pr[\cdot]$ has rank $r$ and fidelity $\Delta^{*}$. Pick $0<\delta<1$. Let $n=O(\Delta^{*-8}\log(r/\delta T))$ and $\Delta=\Omega(\Delta^{*-11/2}\log(r/\delta T))$. For each $t\in[T]$, let $S_{t}$ be a random sample of size $n$ of observation sequences of length $t$ from distribution $\Pr[\cdot]$. Then, with probability $1-\delta$, $\{S_{t}\}_{t\in[T]}$ form $\Delta$-robust bases for $\Pr[\cdot]$.

We provide a proof in Section B.1. According to this lemma, a random sample from a high fidelity distribution forms a robust basis for each $t$. With access to a $\Delta$-robust basis $B_{t}$, we turn to the issues of estimation and error analysis. First we study estimation of the preconditioned quantities $q(bo)$ and $\Sigma_{B_{t}}$ used by the algorithm. Note that all entries of these vectors and matrices are of the following form, where $b^{*}\in B_{t}$ and $x$ is a history of length $t$:

[TABLE]

We show such quantities can be estimated efficiently using conditional samples.

Lemma 4.3** (Estimating preconditioned quantities).**

Let $\{B_{t}\}_{t\in[T]}$ be bases for distribution $\Pr[\cdot]$ where $\max_{0\leq t\leq T}|B_{t}|\leq n$. Pick any $0<\varepsilon,\delta<1$. Fix $b^{*}\in B_{t}$ and $x\in H_{t}$. Then we can build estimate $\widehat{s}(b^{*},x)$ in $\operatorname{poly}(n,|\mathcal{O}|,T,1/\varepsilon,\log(1/\delta))$ time such that with probability $1-\delta$,

[TABLE]

We provide the estimation algorithm and a proof in Section B.6. Using this lemma, we can estimate the operators $A_{o,t}$ via Equation 15. The next lemma provides a precise characterization of the estimation error for these operators.

Lemma 4.4** (Estimating operators).**

Assume the distribution $\Pr[\cdot]$ has rank $r$ and that $\{B_{t}\}_{t\in[T]}$ are $\Delta$-robust bases. Pick $0<\varepsilon,\delta<1$. Then, we can learn approximations $\widehat{A}_{o,t}$ for all observations $o\in\mathcal{O}$ and $t\in[T]$ in $\operatorname{poly}(r$, $|\mathcal{O}|$, $T$, $1/\varepsilon$, $1/\Delta$, $\log(1/\delta))$ time such that with probability $1-\delta$, for any unit vector $v$

[TABLE]

where $\beta(B_{t+1})$ is a matrix with columns $\beta(b)$ for $b\in B_{t+1}$, $V_{t+1}^{\perp}$ is a matrix whose columns form an orthonormal basis for the kernel of $\Pr[F_{t+1}|B_{t+1}]^{\top}D_{t+1}^{-1}\Pr[F_{t+1}|B_{t+1}]$, and the vectors $\alpha(o,v)$ and $\alpha^{\perp}(o,v)$ are $\ell_{1}$ bounded, i.e.,

[TABLE]

We provide the proof in Section B.2. As noted in Section 2, the main technical challenge is in analyzing how the estimation error propagates to errors in induced distributions. Using this structured error, we can show how to bound the TV distance between the induced distributions.

Lemma 4.5** (Perturbation argument).**

Assume for each sequence length $t\in[T]$ and observation $o\in\mathcal{O}$, we have an operator $\widehat{A}_{o,t}$ which is close to $A_{o,t}$ as defined above in Lemma 4.4. Let $\widehat{\Pr}[\cdot]$ be a function over observation sequences of length $T$ given by

[TABLE]

Then, the functions $\Pr[\cdot]$ and $\widehat{\Pr}[\cdot]$ are close in TV distance:

[TABLE]

This makes up the most technical component of our proof, and we give the formal proofs in Section B.3. Together with previous lemmas, this proves our main theorem, Theorem 2. We provide the formal proof in Section B.4.

5 Discussion

In this paper we show how interactive access to hidden Markov models (and more generally low rank distributions) can circumvent computational barriers to efficient learning. In particular, we show that all low rank distributions with a certain fidelity property can be efficiently learned assuming access to a conditional sampling oracle. In Appendix C, we show that fidelity captures the assumptions considered in prior work on (non-interactive) learning of HMMs, specifically:

• •

Parity with noise admits bases $B_{t}$ each of cardinality $2$ with fidelity $(1-2\alpha^{2})/2$, where $\alpha$ is the noise parameter.

• •

Full rank HMMs, where $\mathbb{T}$ and $\mathbb{O}$ are full column rank, admit bases of size $O$ with fidelity bounded by the minimum singular value of the second moment matrix $\Pr[\textbf{x}_{2}=\cdot,\textbf{x}_{1}=\cdot]$. This parameter also appears polynomially in the analysis of [HKZ12].

• •

The overcomplete setting of [Sha+17], where sequences of length $\log S$ are used for estimation, admits bases of size $S$ with fidelity $1/\operatorname{poly}(S)$, matching their parameters.

Despite this, the reliance on the fidelity parameter is the main limitation of our results. We believe this dependence is not necessary, which leads to the main open problem, 1.6. We close the paper with some final remarks regarding this open problem.

As we have mentioned previously, although fidelity greatly simplifies the basis finding aspect of our algorithm, it is not necessary for this part and refer the reader to Appendix D where we give a general algorithm for basis finding. Indeed the only place where fidelity is required is in our error propagation analysis, where our techniques require that operators $\widehat{A}_{o,t}$ are estimated in $\ell_{2}$ norm. In the general case, we will only be able to learn operators in the directions for which the preconditioned matrix has large eigenvalues, and ideally we should be able to ignore the directions with small eigenvalues. This strategy would work if we can show that ignoring the small directions preserves the low rank property, which is the linear-algebraic analog of approximating an HMM by one with fewer states. Unfortunately, we do not know if the latter holds, and we believe this is the key challenge to resolving 1.6. We look forward to further progress on this problem.

Acknowledgement

SK acknowledges funding from the Office of Naval Research under award N00014-22-1-2377 and the National Science Foundation Grant under award #CCF-2212841. This work has been made possible in part by a gift from the Chan Zuckerberg Initiative Foundation to establish the Kempner Institute for the Study of Natural and Artificial Intelligence. We thank Weihao Kong, Vatsal Sharan, Geelon So and Shachar Lovett for insightful discussions.

Appendix A Proofs for Section 3

See 1

Proof.

First, the total number of rounds are at most $rT$. This is because by Proposition 3.3, we increase $\operatorname{rank}(\Pr[\Lambda_{\tau}|B_{\tau}])$ by $1$ in every round for some $\tau\in[T]$, and $\operatorname{rank}(\Pr[\Lambda_{\tau}|B_{\tau}])$ can be at most $r$ by our low rank assumption. So we only need to show that when the algorithm ends, we have found a good estimate. Note that this happens when we can not find a counterexample in 7.

Let $z_{t}$ be the initial bit choice of $\Lambda_{t}$ i.e. $z_{t}=0$ if initial choice of $\Lambda_{t}=\{0\}$ and $z_{t}=1$ if initial choice of $\Lambda_{t}=\{1\}$. By Hoeffding’s inequality, for $n=O(\log(Tr/\delta)/\varepsilon^{2})$, we get with probability $1-\delta/Tr$ for all $t\in\{0,\ldots,T-1\}$:

[TABLE]

as $\Lambda_{t}$ contains $z_{t}$ for all such $t$. Moreover, define a probability distribution $\overline{\Pr}$ over sequences of length up to $T$ using $\widehat{\Pr}$ as follows: for any $t\in\{0,\ldots,T-1\}$,

[TABLE]

where $\Pi_{[0,1]}$ projects onto interval $[0,1]$. Note that for a sequence $(x_{1}\ldots x_{T})$, if for all $t\in\{0,\ldots,T-1\}$, $\Pr[x_{1}\ldots x_{t}\Lambda_{t}]=\overline{\Pr}[x_{1}\ldots x_{t}\Lambda_{t}]$, then $\Pr[x_{1}\ldots x_{T}]=\widehat{\Pr}[x_{1}\ldots x_{T}]$. Therefore, together with Equation 18, we get for each $t\in\{0,\ldots,T-1\}$ and $o\in\mathcal{O}$,

[TABLE]

which implies

[TABLE]

Using Proposition 3.4, we get for distribution $\widehat{\Pr}[\cdot]$

[TABLE]

Re-substituting the value of $\varepsilon$, we get $TV(\widehat{\Pr},\Pr)\leq\varepsilon$ using at most $O(rT^{5}\log(Tr/\delta)/\varepsilon^{2})$ samples from $\Pr[\cdot]$ and queries to the exact conditional probability oracle. ∎

See 3.4

Proof.

We prove this by induction. Assume

[TABLE]

Then,

[TABLE]

We handle the two terms separately. The first term

[TABLE]

where the last step follows from our assumption. The second term

[TABLE]

∎

Appendix B Proofs for Section 4

In this section, we will always restrict attention to basis $B_{t}$, for which the norm of the coefficient $||\beta(x)||_{2}\leq c$ for some universal constant $c\leq 1$ and every history $x\in H_{t}$. Note that by repeating elements in the basis, we can always make the norm smaller than $1$. So, this norm bound is without loss of generality. Finding such basis is more involved, and we show how to find them later on.

B.1 Finding robust basis

We first show how to find a robust basis. Let us define some notation: the covariance matrix for a basis is defined as

[TABLE]

where $D_{t}$ is a diagonal matrix of size $|F_{t}|\times|F_{t}|$ with entries $d_{t}(f):=\mathbb{E}_{b\in B_{t}}\Pr[f|b]$ on the diagonal. We further define the inner covariance as

[TABLE]

Note that the matrices $\Sigma_{B}$ and $\bar{\Sigma}_{B}$ share their non-zero eigenvalues (with extra eigenvalues all [math]). As we will see, random sampling from a high fidelity distribution gives a robust basis. We show in Appendix D more efficient ways of building a basis. See 4.2

Proof.

Let $S_{t}$ be a random sample of size $n$ of observation sequences of length $t$ from distribution $\Pr[\cdot]$. Let $B_{t}^{*}$ be the unknown basis of length $t$ sequences under which distribution $\Pr[\cdot]$ has high fidelity and $|B_{t}^{*}|=n^{*}\leq 1/\Delta^{*}$. Define a distribution $d_{t}^{*}$ over futures given by $d_{t}^{*}(f)=\mathbb{E}_{b\in B_{t}^{*}}[\Pr[f|b]]$. Define $D^{*}_{t}$ to be a diagonal matrix with diagonal entries given by $d_{t}^{*}(f)$.

We ignore the $t$ dependence in notation in this proof for clarity. Before, we prove that $S$ is a robust basis at length $t$, we first show some properties of $d^{*}$ that will come in handy. First, the norm of $\Pr[F|x]$ under $d$ is upper bounded:

[TABLE]

since $\Pr[f|b^{*}_{i}]/d^{*}(f)\leq n^{*}$ by definition and $||\alpha(x)||_{1}\leq\sqrt{n^{*}}||\alpha(x)||_{2}\leq\sqrt{n^{*}}$ as $\alpha(x)$ are coefficients under basis $B^{*}$.

Our next step is to show that the eigenvalues under $S$ and $B^{*}$ are not so different. For clarity, let

[TABLE]

Then, by Matrix Bernstein inequality ([Tro+15, Theorem 1.6.2]) and Equation 19, for $|S|=n=O(n^{*6}(\Delta^{*})^{-2}\log(2r/\delta T))$ and $\Delta^{*}<1$, we get the following bound on $H$ with probability $1-\delta$:

[TABLE]

And therefore by Weyl’s inequality, the top $r_{t}$ eigenvalues of

[TABLE]

are $>\Delta^{*}/2$ and all other eigenvalues are [math]. We now show that $S$ forms a basis. For clarity, let $\Pr[F|S]$ be a matrix with $\Pr[F|s_{i}]$ as columns. Then, $\Pr[F|S]^{\top}D^{*-1}\Pr[F|S]$ has $r_{t}$ eigenvalues $>n\Delta^{*}/2$. Let $V_{t}$ be the eigenvectors corresponding to non-zero eigenvalues of the latter matrix. Next we note that $\operatorname{span}(\{\Pr[F|s_{1}],\ldots,\Pr[F|s_{n}]\})$ has dimension $r_{t}$ and therefore there exists coefficients $\beta(x)\in\operatorname{span}(V_{t})$ such that

[TABLE]

Multiplying both sides by $D^{*\frac{-1}{2}}$, we get

[TABLE]

where the first inequality follows from Equation 19 and the last inequality follows from Equation 20. Simplifying this shows $S$ forms an basis with the following upper bound on the coefficients

[TABLE]

The only remaining part is to show that the two distributions $d$ and $d^{*}$ are only a small factor apart. For this, we first note from Equation 19

[TABLE]

and therefore non-zero eigenvalues of $D^{\frac{-1}{2}}D^{*\frac{1}{2}}$ are $>1/n^{*3/4}$. Since,

[TABLE]

it follows from discussion above and Equation 20 that $\Sigma_{S}$ has $r_{t}$ eigenvalues greater than

[TABLE]

Substituting $n^{*}\leq 1/\Delta^{*}$ proves the claim. ∎

B.2 Learning operators

With access to a $\Delta$-robust basis $\{B_{t}\}_{t\in[T]}$, we now show how to learn an approximation of operators $A_{o}$. For this, we need to learn approximations of projections $P_{V_{t}}$ and coefficients $\beta(b_{i}o)$ for the one-step extensions of basis $B_{t}=\{b_{1},\ldots,b_{n}\}$. Towards this end, using the techniques in Section B.6, we first estimate the following:

[TABLE]

Note $\Sigma_{B_{t}}$ is a covariance matrix with $r_{t}<r$ non-zero eigenvalues. Let $\widehat{\Sigma}_{B_{t}}$ be an approximation for covariance matrix $\Sigma_{B_{t}}$ from Corollary B.15. Compute SVD of $\widehat{\Sigma}_{B_{t}}$, and let $\widehat{V}_{t}$ be the matrix of top $r_{t}$ eigenvectors (or equivalently corresponding to eigenvalues $>\Delta/2$ according to the proof of proposition below). Then, we claim $\widehat{V}_{t}$ is close to $V_{t}$:

Proposition B.1**.**

Let $\Pr[\cdot]$ be any rank $r$ distribution over observation sequences of length $T$. Assume knowledge of a $\Delta$-robust basis $\{B_{t}\}_{t\in[T]}$ for distribution $\Pr[\cdot]$. Let $\Sigma_{B_{t}},\widehat{\Sigma}_{B_{t}},V_{t}$ and $\widehat{V}_{t}$ be as defined above. Then, the approximate projection $P_{\widehat{V}_{t}}$ satisfies

[TABLE]

*for $||\widehat{\Sigma}_{B_{t}}-\Sigma_{B_{t}}||_{2}\leq\Delta/2$. *

Proof.

Let estimate $\Sigma_{B_{t}}$ be such that

[TABLE]

The claim now follows from Davis-Kahan $\sin(\theta)$ theorem. From our assumptions, eigenvalues of $V_{t}$ are $>\Delta$ and rest are all [math]. Moreover, from Equation 23 and Weyl’s inequality, all the eigenvalues associated to $\widehat{V}^{\perp}_{t}$ are $<\alpha$. Then, for $\alpha<\Delta/2$, we get using Corollary E.3,

[TABLE]

∎

To compute an approximation for operators $A_{o,t}$, we also need to get the coefficients for one-step extensions of elements in the basis.

Proposition B.2**.**

Let $\Pr[\cdot]$ be any rank $r$ distribution over observation sequences of length $T$. Assume knowledge of a $\Delta$-robust basis $\{B_{t}\}_{t\in[T]}$ for distribution $\Pr[\cdot]$. Also, assume the coefficients under basis $\{B_{t}\}_{t\in[T]}$ are bounded i.e. $||\beta(h)||_{2}\leq c$ for all histories $h\in\mathcal{O}^{\leq T}$ and some constant $c\leq 1$. Suppose we have estimates $\widehat{q}(b_{i}o)$ and $\widehat{\Sigma}_{B_{t}}$ such that

[TABLE]

Define $\widehat{\beta}(b_{i}o)$, approximation of $\beta(b_{i}o)$, for $\lambda=4\alpha^{2}/c^{2}$:

[TABLE]

Then, the following are true:

the norm of the approximation $\widehat{\beta}(b_{i}o)$ is bounded:

[TABLE] 2. 2.

the approximation $\widehat{\beta}(b_{i}o)$ is accurate in the $\operatorname{span}(V_{t+1})$:

[TABLE]

Proof.

Let

[TABLE]

We know that $\beta(b_{i}o)$ satisfies

[TABLE]

and approximate it by the following program (for $\lambda=4\alpha^{2}$):

[TABLE]

First, we can see that the above error is small as

[TABLE]

as $||\beta(b_{i}o)||_{2}\leq c$ and

[TABLE]

Note that this also means

[TABLE]

Using these, we get

[TABLE]

for $c<1$. Now, using that our assumption on $\Sigma_{B_{t}}$,

[TABLE]

∎

Proposition B.3**.**

Let $\Pr[\cdot]$ be any rank $r$ distribution over observation sequences of length $T$ and $\{B_{t}\}_{t\in[T]}$ be some basis of distribution $\Pr[\cdot]$. Then, we can build an estimate ${\rm diag}(\widehat{\Pr}[o|B_{t}])$ for every $o\in\mathcal{O}$ and $t\in[T]$, such that with probability $1-\delta$

[TABLE]

using $\widetilde{O}(|\mathcal{O}|Tn^{3}/\varepsilon^{2}\log(1/\delta))$ conditional samples.

Proof.

This follows from Hoeffding’s inequality (Proposition E.1). ∎

Using the approximations above, we are in position to present the approximation error for our estimate of operator $A_{o,t}$.

Lemma B.4** (Restatement of Lemma 4.4).**

Let $\Pr[\cdot]$ be any rank $r$ distribution over observation sequences of length $T$. Assume knowledge of a $\Delta$-robust basis $\{B_{t}\}_{t\in[T]}$ for distribution $\Pr[\cdot]$. Also, assume the coefficients under basis $\{B_{t}\}_{t\in[T]}$ are bounded i.e. $||\beta(h)||_{2}\leq c<1$ for all histories $h\in\mathcal{O}^{\leq T}$ and some constant $c\leq 1$. Then, we can build an estimate $\widehat{A}_{o,t}$ using

[TABLE]

many conditional samples such that with probability $1-\delta$, for any vector $v$

[TABLE]

where $\beta(B_{t+1})$ is a matrix with its columns given by coefficients $\beta(b_{i})$ for $B_{t+1}=\{b_{1},\ldots,b_{n}\}$ and $V_{t+1}^{\perp}$ is a matrix with its columns given by the right singular vectors corresponding to kernel of $\Pr[F_{t+1}|B_{t+1}]^{\top}D_{t+1}^{-1}\Pr[F_{t+1}|B_{t+1}]$. Moreover, the column vectors $\alpha(o,v)$ and $\alpha^{\perp}(o,v)$ are $\ell_{1}$ bounded i.e.

[TABLE]

Proof.

Using $P_{\widehat{V}_{t}}$, $\widehat{\beta}(B_{t}o)$ and ${\rm diag}(\widehat{\Pr}[o|B_{t}])$ from Propositions B.1, B.2 and B.3, define operator $\widehat{A}_{o,t}$, an approximation of $A_{o,t}$, as

[TABLE]

Let

[TABLE]

Also, note that $||\widehat{\beta}(b_{i}o)||_{2}\leq\sqrt{2}c$, $||{\rm diag}(\widehat{\Pr}[o|B_{t}])||_{2}\leq 2$ and $||P_{\widehat{V}_{t+1}}||_{2}\,,||P_{\widehat{V}_{t}}||_{2}\leq 1$. To prove our main claim, we will first show that for any unit vector $v$

[TABLE]

with $\|e(o,v)\|_{2},\|\alpha^{\perp}\|_{2}\leq 4\sqrt{2}c\alpha_{1}+\sqrt{2}c\alpha_{3}+\alpha_{2}$. To prove this, we will show that

[TABLE]

The first term is bounded by $2\sqrt{2}c\alpha_{1}$, second term by $2\sqrt{2}c\alpha_{1}$, similarly third term by $\sqrt{2}c\alpha_{3}$ and the last term by $\alpha_{2}$. Therefore, we get that

[TABLE]

which implies for any unit vector $v$

[TABLE]

with $\|e(o,v)\|_{2},\|\alpha^{\perp}(o,v)\|_{2}\leq\|\widehat{A}_{o,t}-A_{o,t}\|_{2}\leq 4\sqrt{2}c\alpha_{1}+\sqrt{2}c\alpha_{3}+\alpha_{2}$. To complete our proof, we will show that

[TABLE]

This is enough, as this gives,

[TABLE]

where $\alpha(o,v)=V_{t+1}e(o,v)$ as $||\alpha(o,v)||_{2}\leq||V_{t+1}e(o,v)||\leq||e(o,v)||$ as $V_{t+1}$ is a unit norm matrix.

We now prove our claim. First note that $\Pr[F|B_{t+1}]I=\Pr[F|B_{t+1}]$ and therefore by uniqueness of $\beta(B_{t+1})$,

[TABLE]

This whole process requires $O(n^{2}|\mathcal{O}|Tm_{\varepsilon})$ many conditional samples with $m_{\varepsilon}$ defined in Lemma B.12. Substituting in $\varepsilon=\Delta\varepsilon^{\prime}/(c\sqrt{r}n^{3/2})$ gives the result. ∎

B.3 Perturbation analysis: Error in coefficients

Our approach for learning the distribution $\Pr[\cdot]$ is to learn approximations of operators $\widehat{A}_{o,t}$ and use them to compute probabilities

[TABLE]

For clarity, let $A_{x_{1:t}}$ and $\widehat{A}_{x_{1:t}}$ represent the product of matrices $A_{x_{t},t-1}\ldots A_{x_{1},0}$ and $\widehat{A}_{x_{t},t-1}\ldots\widehat{A}_{x_{1},0}$ respectively. Similarly, we use $x_{1:t}$ to represent the sequence $(x_{1},\ldots,x_{t})$. In this section, we now present a technical lemma showing that the errors in our estimates of $\widehat{\Pr}[x_{1:T}]$ scales linearly with errors in our approximate operators $\widehat{A}_{o,t}$.

Proposition B.5**.**

Suppose we have estimates $\widehat{A}_{o,t}$ for every $t\in[T]$ and observation $o\in\mathcal{O}$ which satisfy the conditions Equations 24 and 25 in Lemma B.4. Then, for every sequence $(x_{1},\ldots,x_{t})$ with $t\in[T]$, the product $\widehat{A}_{x_{1:t}}$ satisfies:

[TABLE]

where $B_{x_{1:t}}\subset H_{t+1}$ i.e. a subset of histories of length $t+1$ (and of size possibly exponential in $t$). Moreover, the $\ell_{1}$ norm of the associated coefficients $\gamma_{x_{1:t}}$ and $\gamma^{\perp}_{x_{1:t}}$ grow moderately:

[TABLE]

Proof.

First, by our assumption, for any observation sequence $x=(x_{1},\ldots,x_{t-1})$

[TABLE]

with $\|\alpha(o,x)\|_{1},\|\alpha^{\perp}(o,x)\|_{1}\leq\varepsilon$ as $||\beta(x)||_{2}\leq 1$. Next, the following recursive relation holds

[TABLE]

We will bound the three terms separately for each $x_{t}$. We start by bounding the first term

[TABLE]

and here we can see that

[TABLE]

where we used $||\alpha^{\perp}(x_{t},x_{1:t-1})||_{1},||\alpha(x_{t},x_{1:t-1})||_{1}\leq\varepsilon||\beta(x_{1:t-1})||_{2}\leq\varepsilon$. This gives the first term in Equations 29 and 30 (with the $|\mathcal{O}|$ factor for sum over $x_{t}$). We bound the remaining terms by induction. Assume

[TABLE]

where $B_{x_{1:t-1}}$ is a set of observation sequences of length $t-1$. Let’s first bound the second term.

[TABLE]

We can bound the $\ell_{1}$ norm of the coefficients as

[TABLE]

where the second last step follows from $\sum_{x_{t}\in\mathcal{O}}\Pr[x_{t}|b_{i}]=1$. For clarity, let $\alpha(x_{t},B_{x_{1:t-1}})$ represent a matrix with its column given by $\alpha(x_{t},b)$ for $b\in B_{x_{1:t-1}}$. Similarly, define $\alpha^{\perp}(x_{t},B_{x_{1:t-1}})$, $\alpha(x_{t},V_{t-1}^{\perp})$ and $\alpha^{\perp}(x_{t},V_{t-1}^{\perp})$. We can then similarly bound the remaining term.

[TABLE]

Each of these terms can be bounded similarly. We show how to bound the first term:

[TABLE]

where we used $||\alpha(x_{t},b_{i})||_{1}\leq\varepsilon||\beta(b_{i})||_{2}\leq\varepsilon$. This gives the remaining terms in Equations 29 and 30 (with the $|\mathcal{O}|$ factor for sum over $x_{t}$). ∎

We next give a solution for the recursion from Proposition B.5. This is standard, but we give a proof for completeness.

Proposition B.6**.**

Consider the following recursions:

[TABLE]

Let $T\in\mathbb{Z}^{+}$. Then, the following holds for all $t\leq T$ and $\varepsilon\leq 1/12dT$:

[TABLE]

Proof.

We first claim: $g(t)\leq 6d\varepsilon$ for all $t\leq T$. We prove this by strong induction. This is true for $t=1$. Let’s assume $g(i)\leq 6\varepsilon$ for all $i\leq t-1$. Then, first we unroll the recursion for $f(t)$.

[TABLE]

where the last equation follows from observing the first few terms form a geometric series. Using our induction hypothesis, we get

[TABLE]

where we used that $(1+a)^{t}\leq 1+2at$ for $a<1/2t$ and $\varepsilon<1/12dT$. And therefore, we can bound $g(t)$ as

[TABLE]

where we used $g(t-1)\leq 6d\varepsilon$ and $\varepsilon\leq 1/12dT$. This proves that $g(t)\leq 6d\varepsilon$ for all $t<T$. Moreover, by arguments above, in that case $f(t)\leq 3dT\varepsilon$ for all $t\leq T$. ∎

Lemma B.7** (Restatement of Lemma 4.5).**

Let $\widehat{A}_{o,t}$ be the approximation of $A_{o,t}$ from Lemma B.4. Furthermore, for sequence $x_{1:T}$ of length $T$, define

[TABLE]

Then, for the values of $\varepsilon<(12|\mathcal{O}|T)^{-1}$, we get that the functions $\Pr$ and $\widehat{\Pr}$ are close in TV distance:

[TABLE]

Proof.

Recall $F_{T}=\{\varphi\}$. Then,

[TABLE]

The claim follows from Proposition B.5 and Proposition B.6. ∎

B.4 Proof of Theorem 2

See 2

Proof.

From Lemma 4.2, wp $1/2$, we built a $\Delta$-robust basis of size $n$ where

[TABLE]

using $nT$ samples from $\Pr[\cdot]$. From Lemma B.7, to get TV error $\varepsilon$ w.p. $1/2$, when given access to a $\Delta$-robust basis of size $n$, we need to learn the estimate operators $\widehat{A}_{o,t}$ using Lemma B.4 to accuracy

[TABLE]

Substituting this in Lemma B.4 gives the required sample complexity as

[TABLE]

Substituting the values of $\Delta,n$ and $c$ above, we need

[TABLE]

many queries to conditional sampling oracle. ∎

B.5 Properties of operator $A_{o}$

Let $\{B_{t}\}_{t\in[T]}$ be some basis of distribution $\Pr[\cdot]$ (as defined in Definition 3.1). In Proposition 3.2, we defined the operators $A_{o,t}$ under basis $\{B_{t}\}_{t\in[T]}$ and now prove some properties of this operator and the associated coefficients.

Consider the eigenvalue decomposition of the covariance matrix associated to $B_{t}$ where $D_{t}$ is a diagonal matrix of size $|F_{t}|\times|F_{t}|$ with entries $d_{t}(f):=\mathbb{E}_{b\in B_{t}}\Pr[f|b]$ on the diagonal:

[TABLE]

Here, $M_{t}$ is a set of all non-zero eigenvalues, $V_{t}$ is the eigenspace corresponding to non-zero eigenvalues and $V^{\perp}_{t}$ is the eigenspace corresponding to zero eigenvalues. Note that $d_{t}(\cdot)$ is a distribution.

Remark B.8**.**

We note a mild notational issue here. Since we invert the diagonal matrix $D_{t+1}$, we need to be careful about futures $f$ where $d_{t+1}(f)=0$. This is not an issue however as by definition of a basis, $d_{t+1}(f)=0$ implies $\Pr[f|h]=0$ for all histories $h\in H_{t+1}$. For now, we use $\Pr[F_{t}|B_{t}]^{\top}D_{t}^{-1}\Pr[F_{t}|B_{t}]$ to denote the sum

[TABLE]

where $\operatorname{dom}_{+}(t)$ is the set of all futures where $d_{t}(f)\neq 0$.

A basic property of $\operatorname{span}(V^{\perp}_{t})$ follows.

Proposition B.9**.**

$\operatorname{span}(V^{\perp}_{t})=\ker(\Pr[F_{t}|B_{t}]^{\top}D_{t}^{-1}\Pr[F_{t}|B_{t}])=\ker(\Pr[F_{t}|B_{t}])$**

Proof.

Define $d_{t}^{*}$ be the restriction of distribution $d_{t}$ over the set $\operatorname{dom}_{+}(t)$. Consider any vector $v\in\ker(\Pr[F_{t}|B_{t}]^{\top}D_{t}^{-1}\Pr[F_{t}|B_{t}])$. Then,

[TABLE]

This shows that $\ker(\Pr[F_{t}|B_{t}]^{\top}D_{t}^{-1}\Pr[F_{t}|B_{t}])\subset\ker(\Pr(F_{t}|B_{t}))$. The other direction follows because the definition of $\Pr[F_{t}|B_{t}]^{\top}D_{t}^{-1}\Pr[F_{t}|B_{t}]$ as sum above. This completes the proof. ∎

Recall we denote the coefficients associated to history $x$ of length $t$ under basis $B_{t}$ by $\beta(x)$ given by:

[TABLE]

By Proposition B.9, we can assume that $\beta(x)\in\operatorname{span}(V_{t})$ without loss of generality. We now show that the coefficients $\beta(x)$ satisfy some nice properties.

Proposition B.10**.**

Let $\beta(x)\in\operatorname{span}(V_{t})$ be coefficients associated to history $x$. Then, the following statements are true:

The coefficients $\beta(x)$ are uniquely defined in $\operatorname{span}(V_{t})$. Formally, Let $\beta^{\prime}(x)$ be any other coefficients which satisfy Equation 39. Then,

[TABLE]

where $P_{V_{t}}$ is the projection matrix onto subspace $V_{t}$. 2. 2.

The coefficients sum to one, even though some entries could be negative

[TABLE]

where ${\bf 1}^{\top}$ is all ones row vector.

Proof.

First, recall by definition, the coefficients satisfy Equation 39. The first claim follows from $\operatorname{span}(V_{t}^{\perp})=\ker(\Pr[F_{t}|B_{t}])$ (Proposition B.9). Finally, the last claim follows by multiplying both sides in Equation 39 by all ones row vector ${\bf 1}^{\top}$

[TABLE]

where the last equation follows by noting that ${\bf 1}^{\top}\Pr[F_{t}|x]$ for any probability vector $\Pr[F_{t}|x]$ is $1$ (recall $F_{t}$ is set of all futures of length exactly $T-t$). ∎

Even though, these are exponentially many coefficients, we next show existence of operators which can be used to construct these coefficients.

Proposition B.11**.**

Let $A_{o,t}$ be defined using basis $B_{t}=\{b_{1},\ldots,b_{n}\}$ and $B_{t+1}$ as:

[TABLE]

Then, it satisfies the following:

$span(A_{o,t})\subset span(V_{t+1})$** 2. 2.

$\ker(A_{o,t})\supset span(V^{\perp}_{t})$** 3. 3.

$A_{o,t}\beta(x)=\Pr[o|x]\beta(xo)$** 4. 4.

${\bf 1}^{\top}A_{o,t}\beta(x)=\Pr[o|x]$**

Proof.

The first two properties follow from the definition of $A_{o,t}$. Let $F_{t}$ be all observation sequences of length $T-t$. Similar to proof of Proposition 3.2, $A_{o,t}$ satisfies

[TABLE]

Since $oF_{t+1}$ is a subset of $F_{t}$, we get by multiplying $\beta(x)$ on both sides, we get

[TABLE]

By uniqueness of $\beta(xo)$ (Item 1) and that $\operatorname{span}(A_{o,t})\subset\operatorname{span}(V_{t+1})$ (Item 1), we get that

[TABLE]

The last one follows from multiplying both sides above by all ones row vector and then using ${\bf 1}^{\top}\beta(x)=1$ (Item 2).

[TABLE]

∎

B.6 Estimating covariance matrix in Frobenius norm

In this section, we would show how to estimate the following objects:

[TABLE]

which we need for estimating the operator $A_{o,t}$. We ignore $t$ subscript when clear from context.

Lemma B.12** (Restatement of Lemma 4.3).**

Let $\{B_{t}\}_{t\in[T]}$ be a basis of distribution $\Pr[\cdot]$. Let $c$ be some upper bound on the coefficients under basis $B_{t}$ and $n$ be the size of basis $B_{t}$. Define $s(b^{*},x)$ as the following sum where $b^{*}\in B_{t}$ and $x$ is a history of length $t$:

[TABLE]

Then we can learn estimate $\widehat{s}(b^{*},x)$ such that with probability $1-\delta$,

[TABLE]

using at most

[TABLE]

conditional samples.

Proof.

We start by writing $s(b^{*},x)$ in terms of expectation under $\Pr[\cdot|x]$:

[TABLE]

With this, we define (un-normalized) probability functions $\widehat{\Pr}[\cdot|b]$ for $b\in B_{t}$ which set probability to [math] if $f$ is part of a set $F_{b}$ (to be defined) which will depend only on history $b$ i.e.

[TABLE]

We define $\widehat{d}$ as mixture distribution of $\widehat{\Pr}[\cdot|b]$ for $b\in B_{t}$ i.e. $\widehat{d}(f)=\mathbb{E}_{b\in B_{t}}[\widehat{\Pr}[f|b]]$. An important aspect of our definitions is that for any future $f$,

[TABLE]

are upper bounded. Now, suppose $\max_{b}\Pr[F_{b}|b]\leq p$. Then,

[TABLE]

where the last step follows from $\max_{b}\Pr[F_{b}|b]\leq p$. Now, we can sample $m=(1/2c^{2}n^{3}p^{2})\log(2/\delta)$ random futures from $\Pr[\cdot|x]$ and call this set $S$. Then, again by Equation 40 and Hoeffding’s inequality (Proposition E.1), we get that

[TABLE]

Together, with $(1/2c^{2}n^{3}p^{2})\log(2/\delta)$ conditional samples, we get the following guarantee with probability $1-\delta$,

[TABLE]

To estimate $\widehat{\Pr}[f|b^{*}]$ and $\widehat{d}(f)$, we need to identify the case when $f$ is “irregular”.

Definition B.13** ($\alpha$-regular future).**

We define a future $f$ to be $\alpha$-regular for history $b$ if for all $\tau\in[t]$

[TABLE]

Otherwise, $f$ is $\alpha$-irregular for history $b$.

To do this, define empirical estimates $\widetilde{\Pr}[f_{\tau}|bf_{1:\tau-1}]$ for every future $f$ and basis $b\in B_{t}$ using $\widetilde{O}(n|S|T/\alpha^{2}\log(1/\delta))$ many samples. Then, we perform the following test $A(f,b)$ for each future $f$ and basis history $b$ using these estimates:

Definition B.14** (Test $A(f,b)$).**

Test $A(f,b)$ passes if the empirical estimate $\widetilde{\Pr}[f_{\tau}|bf_{1:\tau-1}]>2\alpha$ for all $\tau\in[t]$ and fails otherwise.

Note that with probability $1-\delta$, (i) if test $A(f,b)$ passes for future $f$ and history $b$, then $f$ is $\alpha$-regular for $b$, and (ii) if test $A(f,b)$ fails for future $f$ and history $b$, then $f$ is $3\alpha$-irregular for $b$. In the rest of the proof, we condition on the event that this relationship between test $A(f,b)$ and irregular futures holds for all futures $f\in S$ and $b\in B_{t}$. We set $F_{b}$ to be the set of futures $f$ which are $3\alpha$-regular for basis $b$ and removing the ones where test $A(f,b)$ passes. By Proposition B.17,

[TABLE]

Now, we define estimates $\widetilde{\Pr}[f|b]$ for each future $f\in S$ and basis history $b\in B_{t}$ by first running test $A(f,b)$ on future $f$ and history $b$. If test $A(f,b)$ fails we set $\widetilde{\Pr}[f|b]=0$. Note that otherwise $\widetilde{\Pr}[f|b]$ to be the estimate from Proposition B.16 i.e. with probability $1-\delta$,

[TABLE]

This requires $\widetilde{O}(n|S|T^{2}/(\gamma\alpha)^{2}\log(1/\delta))$ many samples. Moreover, because $\widehat{\Pr}[f|b^{*}]=\Pr[f|b^{*}]$ for futures where tests passes, we can estimate the probability ratios with additive error:

[TABLE]

where the second inequality holds for $\gamma<1/2$ and

[TABLE]

where the second inequality holds for $\gamma<1/2$. This means

[TABLE]

Combining Equations 40, 41 and 42, we can build estimate $[\widehat{q}(x)]_{i}$ for $[q(x)]_{i}$ such that with probability $1-O(\delta)$

[TABLE]

using

[TABLE]

many conditional samples. ∎

Corollary B.15**.**

We can learn approximations $\widehat{q}(bo)$ and $\widehat{\Sigma}_{B_{t}}$ for all $b\in B_{t}$, observations $o$ and time $t\in[T]$ such that with probability $1-\delta$,

[TABLE]

using at most $\widetilde{O}(n^{2}|\mathcal{O}|Tm_{\varepsilon})$ conditional samples.

Proof.

Each entry of $\Sigma_{B_{t}}$ and $q(bo)$ is given by sums of the form $s(b*,x)$ where $b^{*}$ is a basis history and $x$ is arbitrary history of length $t$. Therefore, we can estimate each of them using estimates given by Lemma B.12:

[TABLE]

The result similarly holds for $q(bo)$. There are $O(n^{2})$ entries in each matrix with $T$ many $\Sigma_{B_{t}}$ matrices and $O(n)$ entries in each vector with $n|\mathcal{O}|T$ many $q(bo)$ vectors. ∎

Proposition B.16**.**

Consider a future $f_{1:t}$ of length $t$ and history $x$. Fix $\gamma>0$. Suppose $\Pr[f_{\tau}|xf_{1:\tau-1}]>\alpha$ for each $\tau\in[t]$. Then, we can build estimate $\widehat{\Pr}[f_{1:t}|x]$ such that with probability $1-\delta$

[TABLE]

using at most $O(t^{2}/(\gamma\alpha)^{2}\log(t/\delta))$ conditional samples.

Proof.

By Hoeffding’s inequality, using $m=16t^{2}\log(t/\delta)/(\gamma\alpha)^{2}$ samples, we have, with probability greater than $1-\delta$, that for all $\tau\in[t]$,

[TABLE]

where the last step uses our assumption above. For an upper bound, we have,

[TABLE]

where the last step follows with $(1+a)^{t}\leq 1+2at$ for $a<1/2t$. Similarly, for a lower bound we have:

[TABLE]

where the last step follows with $(1+a)^{t}\geq 1+at$ for $a\geq-1$. ∎

Proposition B.17**.**

Define a future $f$ to be $\alpha$-irregular for history $b\in B_{t}$ if there exists some $\tau\in[t]$ ($\tau$ can depend on $b$) such that

[TABLE]

Let $F_{b}$ be the set of futures $f$ where $f$ is $\alpha$-irregular for history $b$. Then,

[TABLE]

Proof.

Let future $f$ be of length $T$. We first partition the set $F_{b}$ into $T$ sets: $F_{b,1},\ldots,F_{b,T}$ based on the first time irregular is observed i.e. $f\in F_{b,t}\iff t=\min_{\tau}\Pr[f_{\tau}|bf_{1:\tau-1}]<\alpha$. Now,

[TABLE]

Summing over all $T$ of these sets gives the claim. ∎

Appendix C Examples

In this section, we show that our parity with noise and all previously known positive results: full rank HMMs from [MR05, HKZ12] and overcomplete HMMs from [Sha+17] can be learned by our algorithm using conditional sampling oracle. Note that we will use the alternate form of fidelity which is more amenable to analysis given by

[TABLE]

C.1 HMMs induce low rank distributions

Before we show that Hidden Markov Models induce low rank distributions, we clarify a subtle issue in our definition of rank.

Remark C.1**.**

If $\Pr[h]=0$ for some history $h$, then we can not use Bayes rule $\Pr[f|h]\Pr[h]=\Pr[hf]$ to define the conditional distribution $\Pr[\cdot|h]$. Moreover, in this case, any consistent setting of $\Pr[\cdot|h]$ i.e. which satisfies $\Pr[of|h]=\Pr[f|ho]\Pr[o|h]$ will induce the same joint distribution $\Pr[\cdot]$, and might have possibly different $\operatorname{rank}(\Pr[\mathcal{O}^{\leq T-t}|\mathcal{O}^{t}])$. For Hidden Markov Models, if $\Pr[h]=0$ for some history $h\in\mathcal{O}^{t}$, this can be resolved by assuming $\Pr[\textbf{s}_{t+1}=s|h]=1/S$ for $s\in\mathcal{S}$. We will show below that for this choice, rank of HMM is at most the number of hidden states.

Proposition C.2**.**

The distribution induced by HMM with $S$ hidden states has rank $\leq S$.

Proof.

We will show that $\operatorname{rank}(\Pr[\mathcal{O}^{\leq T-t}|\mathcal{O}^{t}])\leq S$ under the choice that if $\Pr[h]=0$ for some history $h\in\mathcal{O}^{t}$, then $\Pr[\textbf{s}_{t+1}=s|h]=1/S$ for $s\in\mathcal{S}$. Define $\mathcal{S}^{+}\subset\mathcal{S}$ be hidden states $s$ for which $\Pr[\textbf{s}_{t+1}=s]>0$. Define $\Pr[\mathcal{O}^{\leq T-t}|\mathcal{S}^{+}]$ to be a matrix whose $((x_{t+\tau},\ldots,x_{t+1}),s)$ entry is given by probability $\Pr[x_{t+\tau},\ldots,x_{t+1}|\textbf{s}_{t+1}=s]$. Similarly define $\Pr[\mathcal{S}^{+}|H_{t}]$ to be a matrix whose $(s,(x_{t},\ldots,x_{1}))$ entry is given by probability $\Pr[\textbf{s}_{t+1}=s|x_{t},\ldots,x_{1}]$. Then, $\Pr[\mathcal{O}^{\leq T-t}|H_{t}]$ can be decomposed into $\Pr[\mathcal{O}^{\leq T-t}|H_{t}]=\Pr[\mathcal{O}^{\leq T-t}|\mathcal{S}^{+}]\Pr[\mathcal{S}^{+}|H_{t}]$, and therefore $\operatorname{rank}(\Pr[\mathcal{O}^{\leq T-t}|H_{t}])\leq S$. ∎

C.2 Parity with noise

We first formally define the distribution induced by parity with noise which has been extensively studied in the computational learning theory [Blu+94a].

Definition C.3** (Parity with noise).**

Let $(x_{1},\ldots,x_{T-1})$ be a vector in $\{0,1\}^{T-1}$, $S$ a subset of $[T-1]$ and $0<\alpha<1/2$. The parity of $(x_{1},\ldots,x_{T-1})$ on $S$ is the boolean function $\phi(x_{1},\ldots,x_{T-1})$ which outputs [math] if the number of ones in the sub-vector $(x_{i})_{i\in S}$ is even and [math] otherwise. Then the distribution induced by HMM for parity with noise is such that the first $T-1$ bits are uniform over $\{0,1\}^{T-1}$ and the last bit is $\phi(x_{1},\ldots,x_{T-1})$ with probability $1-\alpha$ and $1-\phi(x_{1},\ldots,x_{T-1})$ otherwise.

We now show that Parity with noise HMM satisfies the conditions of Theorem 2. Note that for $\alpha=1/2$, each bit becomes a random bit. And then as can be seen in the proof below, the fidelity is $1$ (since the second eigenvalue goes to [math]).

Proposition C.4**.**

The distribution induced by Parity with noise HMM has rank $\leq 2T$ and fidelity $(1-2\alpha)^{2}/2$ under a basis of size $\leq 2$ for every sequence length $t\in T$.

Proof.

The claim about rank follows from noting that $\Pr[\cdot|x]=\Pr[\cdot|y]$ is same if both $x$ and $y$ have the same length $t$; and the subvectors $(x_{i})_{i\in S\cap[t]}$ and $(y_{i})_{i\in S\cap[t]}$ have the same number of ones modulo $2$.

We only need to show that the distribution has large fidelity. The proof is same for all $t\in[T]$, so we prove this for a particular $t$. For a fixed $t$, $\Pr[F_{t}|x]$ only depends on the parity of the secret subset $(x_{i})_{S\cap[t]}$, so there are only two options for $\Pr[F_{t}|x]$. Let those be $v_{1}$ and $v_{2}$. We choose the basis $B_{t}$ to be any histories with probability vector $v_{1}$ and $v_{2}$. Note that if the last bit in the future $f$ matches the parity, then the corresponding probability entry is $(1-\alpha)/2^{T-t-1}$, otherwise it is $\alpha/2^{T-t-1}$. Also, we have $v_{1}(f)=(1-\alpha)/2^{T-t-1}$ if and only if $v_{2}(f)=\alpha/2^{T-t-1}$. Therefore, $d_{t}(f)=1/2^{T-t}$ for every future $f\in F_{t}$.

Let $V$ be the matrix with columns given by $v_{1}$ and $v_{2}$. Our goal is to show that following matrix has large non-zero eigenvalues

[TABLE]

Since $D_{t}^{-1/2}VV^{\top}D_{t}^{-1/2}$ has same eigenvalues as $V^{\top}D_{t}^{-1}V$, we compute its eigenvalues. The diagonal entries of $V^{\top}D_{t}^{-1}V$ are $\alpha^{2}+(1-\alpha)^{2}$ and the off-diagonal entries are $2\alpha(1-\alpha)$. Therefore, the eigenvalues of $V^{\top}D_{t}^{-1}V$ are $1$ and $(1-2\alpha)^{2}$. This gives us a lower bound on $\Delta\geq(1-2\alpha)^{2}/2$. ∎

C.3 Full rank HMMs

We first define full rank HMMs studied in [MR05, HKZ12]. Recall the definition of HMMs (Definition 1.1).

Definition C.5** (Full rank HMMs).**

We say an HMM is full rank, if its emission matrix $\mathbb{O}$ and state transition matrix $\mathbb{T}$ have full column rank.

We next show the distribution induced by full rank HMM can be learned by our algorithm in Theorem 2. Let $P_{2,1}$ be an $O\times O$ matrix with $(i,j)$th entry $\Pr[o_{2}=i,o_{1}=j]$. Note that the previous result [HKZ12] in this setting depend on smallest eigenvalues of $P_{2,1}$.

Proposition C.6**.**

The distribution induced by full rank HMM has rank $S$ and fidelity $\sigma_{\min}(P_{2,1})^{2}$ under a basis of size $\leq O$ for every sequence length $t\in T$.

Proof.

We note that in the full rank case, we can simplify our algorithm considerably. By our assumptions, $\operatorname{rank}(\Pr[\mathcal{O}|\mathcal{O}])=S$. And therefore instead of picking all distributions, we can replace $\Pr[F_{t}|x]$ by one-step probabilities $\Pr[\mathcal{O}|x]$. We can do this because $\Pr[\mathcal{O}|x]=\Pr[\mathcal{O}|\mathcal{O}]\beta(x)$ implies $\Pr[F_{t}|x]=\Pr[F_{t}|\mathcal{O}]\beta(x)$ as $\operatorname{rank}(\Pr[F_{t}|H_{t}])=\operatorname{rank}(\Pr[F_{t}|\mathcal{O}])=\operatorname{rank}(\Pr[\mathcal{O}|\mathcal{O}])=S$ and $\mathcal{O}\subset F_{t}$ where $H_{t}$ and $F_{t}$ is the set of all observation sequences of length $t\geq 1$ and $\leq T-t$ respectively. This also means we do not need a different basis for each $t\in[T]$. And only need to show that the following matrix has large eigenvalues:

[TABLE]

where $D$ is a diagonal matrix with $d(o^{\prime})=\mathbb{E}_{o\in\mathcal{O}}[\Pr[o^{\prime}|o]]$. Since, eigenvalues of $D^{-1/2}$ are $\geq 1$, we are interested in eigenvalues of

[TABLE]

where $U$ is a diagonal matrix of size $|\mathcal{O}|\times|\mathcal{O}|$ with its diagonal entries given by $\Pr[o]$. Then, by definition, $P_{2,1}=\Pr[\mathcal{O}|\mathcal{O}]U$. Using this we can lower bound $\Delta$ by $\sigma_{\min}(P_{2,1})^{2}$.

∎

C.4 Overcomplete HMMs

We first define the class of overcomplete HMMs which can be learned using techniques in [Sha+17]. In [Sha+17], the authors were concerned with the stationary distribution induced by HMMs. To define their assumptions, let $S$ be the number of hidden states and $\tau=O(\lceil\log_{|\mathcal{O}|}S\rceil)$. Moreover, let $H_{\tau}$ be the set of histories of length $\tau$ of the form $x_{-\tau},\ldots,x_{-1}$, $F_{\tau}$ be the set of histories of length $\tau$ of the form $x_{0},\ldots,x_{\tau}$, and $\mathcal{S}=\{1,\ldots,S\}$ be the set of hidden states. Note that by our setting of $\tau$, $H_{\tau}$, $F_{\tau}$ and $\mathcal{S}$ are all size $O(S)$. Define $\Pr[F_{\tau}|\mathcal{S}]$ to be a matrix of size $|F_{\tau}|\times S$ whose $((x_{0},\ldots,x_{\tau}),s)$ entry is given by probability $\Pr[x_{0},\ldots,x_{\tau}|s_{0}=s]$. Similarly define $\Pr[H_{\tau}|\mathcal{S}]$ as the equivalent matrix for time-reversed Markov chain whose $((x_{-\tau},\ldots,x_{-1}),s)$ entry is given by probability $\Pr[x_{-\tau},\ldots,x_{-1}|s_{0}=s]$.

[Sha+17] showed efficient algorithms for HMMs under assumptions which imply (a) $\Pr[F_{\tau}|\mathcal{S}]$ and $\Pr[H_{\tau}|\mathcal{S}]$ matrices are rank $S$, (b) the condition number of $\Pr[F_{\tau}|\mathcal{S}]$ and $\Pr[H_{\tau}|\mathcal{S}]$ is $\operatorname{poly}(S)$ and (c) every hidden state has stationary probability at least $1/\operatorname{poly}(S)$.

We next show the distribution induced by these HMMs can be learned by our algorithm in Theorem 2. We will not require the uniqueness of columns of $\Pr[\mathcal{O},\mathcal{S}]$.

Proposition C.7**.**

The distribution induced by HMMs defined above has rank $S$ and fidelity $(\operatorname{poly}(S))^{-1}$ under a basis of size $O(S)$ for every sequence length $t\in T$.

Proof.

Just like the full rank case (Proposition C.6), we can simplify our algorithm considerably. We choose $B$ to be the set $H_{\tau}$. Let $F$ be the set of all observation sequences of length $T-\tau$ for some $T>2\tau$. Now, we can replace $\Pr[F|x]$ by probabilities $\Pr[F_{\tau}|x]$ in our algorithm. We can do this because $\Pr[F_{\tau}|x]=\Pr[F_{\tau}|B]\beta(x)$ implies $\Pr[F|x]=\Pr[F|B]\beta(x)$ as $\operatorname{rank}(\Pr[F|B])=\operatorname{rank}(\Pr[F_{\tau}|B])=S$ and $F_{\tau}\subset F$. Second, we do not need a different basis for each $t\in[T]$ as $\Pr[F_{\tau}|x]$ lives in the span of $\Pr[F_{\tau}|B]$ for every history $x$ as $\operatorname{rank}(\Pr[F_{\tau}|B])=\operatorname{rank}(\Pr[F_{\tau}|H])=S$ and $B\subset H$ where $H$ is the set of all histories of length $\leq T$. This means we only need to show that the following matrix has large eigenvalues:

[TABLE]

where $D$ is a diagonal matrix with entries $d(f):=\mathbb{E}_{b\in B}\Pr[f|b]$ on the diagonal. Since, eigenvalues of $D^{-1/2}$ are $\geq 1$, we are interested in eigenvalues of

[TABLE]

where $K$ is a diagonal matrix of size $|B|\times|B|$ with diagonal entries given by $k(b)=\Pr[b]$. Define $\Pr[F_{\tau}H_{\tau}]$ to be a matrix of size $|F_{\tau}|\times|H_{\tau}|$ whose $((x_{0},\ldots,x_{\tau}),(x_{-\tau},\ldots,x_{-1}))$ entry is given by probability $\Pr[x_{-\tau},\ldots,x_{-1},x_{0},\ldots,x_{\tau}]$. Then, by definition, $\Pr[F_{\tau}|B]K=\Pr[F_{\tau}H_{\tau}]$. Using this, each entry of $K<1$ and every hidden state has stationary probability at least $1/\operatorname{poly}(S)$, we can lower bound $\Delta$ by $\sigma_{\min}(\Pr[F_{\tau}H_{\tau}])^{2}=1/\operatorname{poly}(S)$.

∎

Appendix D General algorithm for finding approximate basis

In this section, we learn an approximate version of basis for probability vectors. Throughout this section, we ignore $t$ subscript in $F_{t}$ and $B_{t}$ when clear from context. We define an approximate basis, which allows us to ignore histories which have very low probability under the distribution $\Pr[\cdot]$:

Definition D.1** (Approximate Basis).**

Fix $0<\varepsilon<1$. For a distribution $\Pr[\cdot]$ over observation sequences of length $T$, we say a subset of observations sequences $B$ forms an $\varepsilon$-basis for $\Pr[\cdot]$ at length $t\in[T]$, if for every observation sequence $x=(x_{1},\ldots,x_{t})$, there exists coefficients $\beta(x)$ with $\ell_{2}$ norm $||\beta(x)||_{2}\leq 1$ such that:

[TABLE]

We first define regular distributions.

Definition D.2** (Regular distribution).**

We say a distribution $\Pr[\cdot]$ is $\alpha$-regular if $\min\Pr[o|x]\geq\alpha$ where the minimum is over all histories $x$ and observations $o$ where $\Pr[o|x]\neq 0$.

We now present the main result in this section: how to build an approximate basis for a regular low rank distribution.

Theorem 3**.**

Let $\Pr[\cdot]$ be an $\alpha$-regular distribution over observation sequences of length $T$ with rank $r$. Fix $0<\varepsilon<\alpha/Tr$ and $0<\delta<1$. Then, in $\operatorname{poly}(r,T,1/\varepsilon,1/\alpha,\log(1/\delta))$ time, with probability $1-\delta$, we can find an $\varepsilon$-basis of size at most $O(r^{2}T^{3}\log(1/\alpha\varepsilon))$ using conditional sampling oracle.

We believe the regularity assumption on the distribution can be removed using the ideas from Section B.6 but leave it as future work.

D.1 Learning coefficients

Towards this goal, we first show how to check given an observation sequence $x$ and set of observation sequences $B$ if there exists $\beta(x)$ such that

[TABLE]

Instead of directly working with the $\ell_{1}$ loss, we first define an $\ell_{2}$ loss which we can use as its proxy.

Definition D.3**.**

For set of observation sequences $B=\{b_{1},\ldots,b_{h}\}$, observation sequence $x$, column vector $\beta\in\mathbb{R}^{|B|}$, we define the $\ell_{2}$ approximation error as:

[TABLE]

where $d$ is the mixture distribution for $B$:

[TABLE]

When clear from context, we drop the $B,x$ superscript.

We will use our ability to simulate relative probabilities for regular distributions (Proposition B.16) to build our guess for approximate basis.

Proposition D.4**.**

Let $\Pr[\cdot]$ be an $\alpha$-regular distribution, $x$ be an observation sequence of length $t\in[T]$ and $B$ be any set of observation sequences of length $t$. Suppose $f_{1},\ldots f_{m}$ are i.i.d. samples from $d$. Then, using $\operatorname{poly}(T,1/\varepsilon,1/\alpha,\log(1/\delta))$ many conditional samples, we can have estimates $\widehat{\Pr}[f_{i}|b]$ for all $i\in[m]$ and $b\in B\cup\{x\}$ such that with probability $1-\delta$,

[TABLE]

where the estimated $\ell_{2}$ error function $\widehat{L}_{B,x}$ is defined as

[TABLE]

and $\widehat{d}(f_{i})$ is the mixture distribution defined with the estimated probabilities.

Proof.

For notational convenience, we will drop the $B,x$ superscript and simply write $L(\cdot)$ in the proof. Using $\operatorname{poly}(1/\gamma,1/\alpha,T,\log(1/\delta))$ conditional samples (Proposition B.16), with probability $1-\delta$, we can have estimates $\widehat{\Pr}[f_{i}|b]$ such that, for all $i\in[m]$ and $b\in B\cup\{x\}$,

[TABLE]

Define:

[TABLE]

We have that:

[TABLE]

We will handle the two terms above separately.

To bound $|L(\beta)-\overline{L}(\beta)|$, let us first show that for any observation sequence $f$:

[TABLE]

To see this, observe that:

[TABLE]

which implies:

[TABLE]

Now using that the square loss is a $2$-smooth function and a standard uniform convergence argument, we have that

[TABLE]

holds with probability greater than $1-\delta$.

For the second term, define

[TABLE]

Let us first show that, for all $i\in[m]$,

[TABLE]

We have that:

[TABLE]

The claim would follow provided we have that, for all $i\in[m]$ and all $b\in B\cup\{x\}$,

[TABLE]

To see this, using that $\gamma\leq 1/2$, we have the following upper bound that:

[TABLE]

and the lower bound:

[TABLE]

This completes the proof of Equation 43.

Now consider any $\beta$ such that $\widehat{L}(\beta)\leq\widehat{L}(0)$. Also, it is straightforward that $\widehat{L}(0)\leq 4$. This implies that:

[TABLE]

where the first step follows from $a^{2}-b^{2}=2b(a-b)+(a-b)^{2}$ and the second step from Cauchy–Schwarz inequality. Combining the bounds for the first and second terms completes the proof. ∎

D.2 Algorithm

We now ready to present our algorithm. The user furnishes $\varepsilon$, the accuracy with which approximate basis is to be learned; and $\delta$, a confidence parameter. The parameter $n$ and $H$ depend on the input.

By Hoeffding’s inequality, it is clear that if this algorithm ends then we have found an approximate basis. We know show that with high probability, this algorithm ends in small number of rounds.

Proposition D.5**.**

Let $\Pr[\cdot]$ be an $\alpha$-regular distribution over observation sequences of length $T$ with rank $r$. Fix $0\leq\varepsilon\leq\alpha^{2}/T^{2}r^{2}$. Let $C=\sqrt{2Tr\log(1/\alpha\varepsilon)}$ and let $H$ be any natural number provided $H\geq 8rT^{2}\log(1/\varepsilon\alpha)$. Consider any sequence of observation sequences $b_{1},b_{2},\ldots,b_{H}$. Let $B_{h}=\{b_{1},\ldots,b_{h}\}$. Then, there exists $h\leq H$ such that:

[TABLE]

Proof.

For $\beta\in\mathbb{R}^{h}$, define:

[TABLE]

and $d^{(h)}$ to be the mixture distribution corresponding to $B_{h}$. Define:

[TABLE]

It will be helpful to overload notation and view $P_{j}=\Pr[F|b_{j}]$ as a vector of length $|F|$ and $P_{1:h}=\Pr[F|B_{h}]$ as a matrix of size $|F|\times h$, whose columns are $P_{1},\ldots P_{h}$. We overload notation analogously for the vector $\overline{P}_{j}$ and the matrix $\overline{P}_{1:h}$. Also, let $D^{(h)}$ be a diagonal matrix of size $|F|\times|F|$, whose diagonal entries are $d^{(h)}$, where we drop the $h$ superscript when clear from context. With our notation, we have that $\overline{P}_{h+1}=D^{-1/2}P_{h+1}$ and $\overline{P}_{1:h}=D^{-1/2}P_{1:h}$. We will write $P$ and $\overline{P}$ in lieu of $P_{1:h}$ and $\overline{P}_{1:h}$, when clear from context.

We have:

[TABLE]

where we have used our matrix notation. Let us consider the following ridge regression estimator:

[TABLE]

For $p_{\textrm{min}}=\alpha^{T}$, define $\Sigma_{h}$ to be the $|F|\times|F|$ sized matrix, as follows

[TABLE]

We will now show that:

[TABLE]

Define:

[TABLE]

One can verify that:

[TABLE]

and that:

[TABLE]

Using this, we have:

[TABLE]

By our assumption on $p_{\textrm{min}}$, we have that $d^{(h)}(f)\geq p_{\textrm{min}}$, which implies $D\succeq\frac{1}{p_{\textrm{min}}}I$. Using this, we have

[TABLE]

where we have used the definition of $\Sigma_{h}$ in the last step. This proves the claim in Equation 44.

This implies:

[TABLE]

Using that $\|P_{h+1}\|^{2}\leq 1$ (since $P_{h+1}$ is a probability distribution), Lemma D.6 implies:

[TABLE]

which implies there exists an $h\leq H$ such that:

[TABLE]

For $H\geq 4r\log(1+H/(p_{\textrm{min}}\lambda))$, exponentiating leads to:

[TABLE]

where the last step follows due to our choice of $H$. This shows that there exists an $h\leq H$ such that:

[TABLE]

Choosing $\lambda=\varepsilon^{2}$, setting $H=8rT^{2}\log(1/\varepsilon\alpha)$ suffices to satisfy our assumptions. This implies we get the minimum loss is achieved at $\beta$ whose norm is bounded by

[TABLE]

and therefore for $\varepsilon<\alpha^{2}/T^{2}r^{2}$, we get

[TABLE]

∎

The following is a variant of the Elliptical Potential Lemma, from the analysis of linear bandits [DHK08].

Lemma D.6** (Elliptical potential).**

Consider a sequence of vectors $\{x_{1},\dots,x_{T}\}$ where, for all $i\in[T]$, each $x_{i}\in\mathcal{V}$, where $\mathcal{V}$ is a $d$-dimensional subspace of a Hilbert space, and $\|x_{i}\|\leq B$. Let $\lambda\in\mathbb{R}^{+}$. Denote $\Sigma_{t}=\Sigma_{0}+\sum_{i=1}^{t}x_{i}x_{i}^{\top}$. We have that:

[TABLE]

We now finish the proof of Theorem 3 by adding the missing details. We first show that $\ell_{1}$ loss is upper bounded by our $\ell_{2}$ proxy loss.

Proposition D.7**.**

Let $x$ be an observation sequence of length $t\in[T]$ and $B=\{b_{1},\ldots,b_{h}\}$ be any set of observation sequences of length $t$. We have that:

[TABLE]

Proof.

We have that:

[TABLE]

where the last step uses Jensen’s inequality. ∎

Proof of Theorem 3.

We choose $C=\sqrt{2Tr\log(16/\alpha\varepsilon^{2})}$, $H=8rT^{2}\log(16/\varepsilon^{2}\alpha)$ and $n=O(1/\varepsilon^{2}\log(H/\delta))$. Let $\beta_{h}(x)$ be the coefficients such that

[TABLE]

From Proposition D.4, using $\operatorname{poly}(r,T,1/\varepsilon,1/\alpha,\log(1/\delta))$ many conditionally samples, we can get with probability $1-\delta/2$, for all $h\in[H]$ and observation sequence $x$ in our random sample

[TABLE]

In our algorithm, when we find a counterexample, it means for some observation sequence $x$ in the random sample:

[TABLE]

This means, by Equation 45, for that observation sequence $x$,

[TABLE]

However, by our choice of $C$ and $H$, by Proposition D.5 this can not happen $H$ times and therefore, our algorithm should end in at most $H$ rounds.

We will now show that the overall error of our basis is small. When our algorithm ends, then we should have for all observation sequence $x$ in our random sample:

[TABLE]

This means by Equation 45, we should have for all observation sequence $x$ in our random sample:

[TABLE]

By Proposition D.7, this implies for all observation sequence $x$ in our random sample:

[TABLE]

Therefore, for our choice of $n$, by Hoeffding inequality, we get with probability $1-\delta/2$,

[TABLE]

Since, for all histories $x$

[TABLE]

we get that

[TABLE]

Let $B^{\prime}$ be the set where we repeat $C$ times every $b\in B$, then we get that

[TABLE]

This completes the proof. ∎

Appendix E Helper propositions

Proposition E.1** (Hoeffding’s inequality).**

Let $X_{1},X_{2},\ldots,X_{n}$ be independent random variables such that $a\leq X_{i}\leq b$ almost surely. Consider the sum of these random variables,

[TABLE]

Then for all $t>0$,

[TABLE]

Here $E[S_{n}]$ is the expected value of $S_{n}$.

In our work, we care about how far are the projection operators onto top eigenspace for two symmetric matrices which are close to each other. This follows as a corollary of Davis-Kahan theorem.

Proposition E.2** (Davis-Kahan theorem).**

Let $\Sigma$ and $\widehat{\Sigma}$ be real symmetric matrices with the eigenvalue decomposition: $V\Lambda_{0}V^{\top}+V^{\perp}\Lambda_{1}V^{\perp\top}$ and $\widehat{V}\widehat{\Lambda}_{0}\widehat{V}^{\top}+\widehat{V}^{\perp}\widehat{\Lambda}_{1}\widehat{V}^{\perp\top}$. If the eigenvalues of $\Lambda_{0}$ are contained in an interval $[a,b]$, and the eigenvalues of $\widehat{\Lambda}_{1}$ are excluded from the interval $[a-\gamma,b+\gamma]$ for some $\gamma>0$, then

[TABLE]

Corollary E.3** ([Pen20]).**

Let $\Sigma$, $\widehat{\Sigma}$, $V$, $\widehat{V}$ and $\gamma$ be as defined above. Assume $V,\widehat{V}\in\mathbb{R}^{n\times r}$. Then,

[TABLE]