1 Introduction↩︎

1.1 Motivation: global optimization under minimal regularity↩︎

Modern machine learning and scientific computing routinely give rise to optimization problems that are high-dimensional, highly non-convex, and geometrically intricate. In many of these settings, classical structural assumptions used to obtain convergence guarantees, such as convexity, smoothness, global Lipschitz bounds, or Polyak–Łojasiewicz-type conditions, are either false, unknown, or hard to justify from first principles. At the same time, much of the existing theory for popular methods such as (stochastic) gradient descent and its variants is inherently local: it characterizes convergence to stationary points under regularity assumptions on gradients and Hessians, and it is not designed to provide global progress guarantees for general objective functions. We have in mind very complicated loss functions, e.g., coming from deep neural networks, which are at best continuous and lack the convexity properties usually required to guarantee progress over iterations.

This motivates the question we study here: can one design a practically implementable stochastic procedure with a provable tendency toward global minima, under assumptions as weak as continuity? Of course, global optimization is hard in the worst case, even for simple model classes, and the present discussion is only meant to give an intuitive understanding of our approach. Rather than attempting to overcome worst-case hardness for all loss functions, our perspective is to obtain a guarantee whose rate depends on a measurable geometric property (the “gap parameter” (9 )) of the objective function, and to interpret this property as a form of tractability. Our results are meaningless when the gap parameter is zero, and most interesting when the gap parameter is sufficiently large. The latter is necessary in order to avoid numerous mathematical and complexity-theoretic difficulties that can arise, even in the context of smooth functions (where the set of critical points can be extremely complicated, e.g., a Cantor set [1]).

1.2 Random subspaces and Grassmannian randomness↩︎

A familiar strategy for high-dimensional optimization is to reduce dimension, for example through coordinate descent or block coordinate methods. Such approaches, however, are tied to a particular choice of coordinates. Ideally, one might instead desire the coordinate-independent viewpoint of considering quantities that are well-defined on a manifold. To make this approach feasible, we take the given coordinate system the optimization problem has been posed in, and create a problem on a manifold as follows. Specifically, we consider all rotations of the coordinate axes at once, and optimize on random linear subspaces that cross the coordinate grid at generic angles (not just ones aligned with any particular chosen coordinate system of \(\mathbb{R}^d\)). The collection of all \(k\)-dimensional subspaces of \(\mathbb{R}^d\) forms the Grassmannian \(\mathcal{G}_{k,d}\), a smooth manifold equipped with a canonical rotation-invariant probability measure. This geometry provides a natural specification for choosing subspaces at random, which is independent of the choice of coordinate axes.

1.3 A stochastic global optimization procedure↩︎

Our method proceeds by sequentially solving the restriction of the original problem to random \(k\)-dimensional subspaces (with \(k\ll d\)), using an arbitrary black-box optimization routine in these low-dimensional subspaces. Each successive subspace is sampled to contain the current iterate, so that the loss function values produced by the walk are non-increasing. The resulting algorithm (Algorithm 1) in Section 4 can be viewed as a random walk on \(\mathcal{G}_{k,d}\).

A key novelty is that our algorithm’s access to the continuous loss function is as a black box that lacks access to derivatives. Instead, it depends only on the distribution of restricted minima across random subspaces. This leads to a convergence estimate for continuous loss functions in terms of a single quantity — the gap parameter (Definition (9 )) — which plays a role analogous to the spectral gap controlling the mixing rate of random walks on graphs.

1.4 Blind-spot robustness to narrow pathological regions↩︎

Beyond convergence, the same perspective yields a useful robustness notion. If the loss function has an extremely narrow region where it dips sharply downward, a “spike” of very low values supported on a set of tiny measure, then a method based on random subspace sampling is unlikely to encounter this region. Section 5 formalizes this idea by analyzing a clipped objective function \(\ell_{\alpha'}:=\max(\ell,\alpha')\), showing that the algorithm effectively optimizes the loss function outside such small-measure blind spots. This robustness is subtle: it does not claim immunity to arbitrary perturbations, but it does explain why certain highly concentrated pathologies have limited impact on the algorithm’s output.

1.5 Organization of this paper↩︎

Section 2 discusses connections to existing stochastic and derivative-free optimization techniques. Section 3 contains the mathematical background on compact groups, Haar measure, and Grassmannians. In Section 4 we present the random-subspace walk and prove our main global convergence guarantee. Section 5 develops the blind-spot robustness viewpoint. We conclude in Section 6 with a discussion of implications and potential applications.

We wish to thank Henry Cohn, Michael Freedman, Ravi Kannan, Bhargav Narayanan, and Nikhil Srivastava for their helpful comments.

2 Related Work↩︎

Comparison to existing stochastic techniques. Given the current emphasis on applications of optimization in machine learning, we present a review of techniques focused around applications to computer science. However, optimization as a whole is of course a far broader topic that contains other methods such as basin-hopping that are decisive in other fields (see [2] for applications to chemistry and physics). Most of the existing literature focuses on stochastic gradient descent or its popular variants such as Adam [3] and AdaGrad [4]. In stochastic gradient descent one picks a random subset of the data, and computes the gradient of the loss function on this subset to update the parameters of the model. Convergence for this scheme can be shown under assumptions like strong convexity [5], the Polyak–Łojasiewicz condition [6], or convergence to stationary points for non-convex functions which satisfy an expected smoothness assumption [7]. These convergence results rely crucially on structural assumptions on the loss function and, by design, they emphasize local stationarity rather than global optimality.

Our approach is fundamentally different in that, rather than drawing randomness from subsampling the data, we select random linear subspaces of the Euclidean space in which the optimization problem is posed. This differs from coordinate descent in that random linear subspaces are typically not aligned with the coordinate grid; they cross it at generic angles. It also differs from random-direction search in that we randomize the subspace in which a subproblem is posed and solved, while treating the solver within that subspace as a black box. The only existing technique with superficial similarities is Dropout in deep learning [8], but Dropout typically considers subsets (not subspaces) of parameters, and is specialized to neural networks. By contrast, the set of all \(k\)-dimensional subspaces is a smooth manifold (hence uncountable), and its canonical probability measure comes from the rotation group.

Analysis via compact-group geometry. Analytically, our results rely on the fact that randomness is drawn from a homogeneous space (a quotient of a compact Lie group) equipped with Haar probability measure. This enables convergence guarantees for continuous objective functions in terms of global distributional properties of restricted minima, rather than derivative-based regularity conditions such as smoothness or Lipschitzness.

3 Mathematical ingredients↩︎

Geometric setup and invariant sampling↩︎

Grassmannians and “conditioned Grassmannians”. Fix integers \(d>k>1\). Let \(\mathcal{G}_{k,d}\) denote the Grassmannian of \(k\)-dimensional linear subspaces (“\(k\)-planes”) of \(\mathbb{R}^d\) (see [9]). For any nonzero vector \(x\in\mathbb{R}^d\), we let \[\label{eq:Gkmdmx} \mathcal{G}_{k,d,x} \;:=\;\{\eta\in\mathcal{G}_{k,d} : x\in \eta\}\tag{1}\] be the submanifold of \(k\)-planes that contain \(x\). We call these “conditioned Grassmannians”; they are related to generalized flag varieties and Schubert varieties. These compact homogeneous manifolds, equipped with rotation-invariant measures, provide the state spaces for our random-subspace walk.

Group viewpoint. Let \(G\) denote a Lie group and \(H\subset G\) a closed subgroup, with quotient \(G/H\). Our primary example is \(G=O(d)\) and \(H=O(k)\times O(d-k)\), embedded in the standard way so that \(H\) preserves the coordinate subspace \(V:=\mathrm{span}(e_1,\ldots,e_k)\subset\mathbb{R}^d\). Then the quotient \(G/H\) can be identified with \(\mathcal{G}_{k,d}\) via \(gH\mapsto gV\). This identification lets us translate probabilistic statements on \(\mathcal{G}_{k,d}\) into statements on the compact group \(O(d)\).

Invariant measures and sampling. Any locally compact Lie group admits a left-invariant Haar measure \(dg\), unique up to scaling [10]. After normalizing so that \(G\) has measure 1, \(dg\) becomes a probability measure when \(G\) is compact. Pushing \(dg\) forward under \(g\mapsto gV\) yields the canonical \(O(d)\)-invariant probability measure on \(\mathcal{G}_{k,d}\). Likewise, the “conditioned Grassmannians” \(\mathcal{G}_{k,d,x}\) (1 ) inherit unique probability measures invariant under the stabilizer subgroup of \(x\) (see e.g. [11] and [12] for quotient-measure constructions). Throughout, the measure of a measurable subset \(A\) under its ambient space’s Haar measure will be denoted by \(|A|\).

The objective function \(\phi\) and its restricted minima. Let \(\ell:\mathbb{R}^d\rightarrow\mathbb{R}\) be the continuous objective (loss) function we wish to minimize. We tacitly assume \(\ell\) is coercive, so that it attains minima on each \(\eta\in \mathcal{G}_{k,d}\). Our method queries \(\ell\) only through low-dimensional restrictions: given a \(k\)-plane \(\eta\), we assume access to a black box that returns (exactly or approximately) a minimizer of \(\ell\) restricted to \(\eta\). To study the distribution of these restricted optima, we define \[\label{eq:phidef} \phi(\eta) \;= \;\min_{y\in \eta}\ell(y).\tag{2}\] Note that \[\min_{y\in\mathbb{R}^d}\ell(y)\;=\;\min_{\eta\in\mathcal{G}_{k,d}}\phi(\eta),\] i.e., the global minimum value of \(\ell\) coincides with the global minimum of \(\phi\).

3.1 Probabilities for random values being near minima of functions on compact groups.↩︎

Algorithm 1 repeatedly samples random subspaces (elements of \(\mathcal{G}_{k,d}\) or \(\mathcal{G}_{k,d,x}\)) and keeps the best among \(T\) samples. To analyze such steps, we need a quantitative statement of the following form: if a bounded function on a compact homogeneous space is not almost constant under the invariant measure, then random sampling has a non-negligible chance of producing a value that is meaningfully below its maximum. When applied to \(\phi(\eta)=\min_{y\in\eta}\ell(y)\), this becomes a lower bound on the probability of drawing a subspace whose restricted optimum is closer to the global optimum than it is to the maximal (worst) restricted optimum.

The lemma below is a simple variance-to-level-set bound on a compact Lie group equipped with Haar probability measure. In its most transparent specialization, suppose \(f:G\to\mathbb{R}\) has been rescaled so that its range is contained in \([0,1]\). If the \(L^2\)-deviation \(\|f-\bar f\|_2\) is bounded away from zero, then \(f\) cannot concentrate near \(1\) everywhere, and a random draw \(g\sim dg\) has a positive chance of landing in a set where \(f(g)\le 1-\delta\) for an explicit \(\delta\). Lemma 1 records a slightly more flexible version that also allows an exceptional set \(U\) on which \(f\) dips below a threshold \(\alpha'\); this flexibility is what we later exploit in our blind-spot robustness analysis (Section 5).

Concretely, in the special case \(\alpha'=\alpha_\text{min}\) (so \(U=\varnothing\)) and after rescaling \(f\) so that \(\alpha_\text{min}=0\) and \(\alpha_\text{max}=1\), the parameter in Lemma 1 simplifies to \(\delta=\tfrac{1}{\sqrt{2}}\|f-\bar f\|_2\), and (?? ) becomes \[\mathbb{P}_{g\sim dg}\!\left[f(g)\le 1-\delta\right]\;\ge\;\delta^2.\] This provides an explicit way to read the lemma as a statement about the likelihood of sampling a value that is not “near the top” of the range.

Our presentation is motivated by (and can be derived from) Kazhdan’s Property (T) [13], which holds trivially for compact groups such as \(G=O(d)\).

3.1.0.1 How to read Lemma 1.

Think of \(G\) as a probability space and \(f(g)\) as a random variable when \(g\) is drawn from Haar measure. The parameters \(\alpha_\text{min}\) and \(\alpha_\text{max}\) describe the range of \(f\). The set \(U=\{g:f(g)<\alpha'\}\) isolates an exceptional region where \(f\) takes unusually small values. The conclusion (?? ) lower bounds the probability mass of points where \(f\) is not in the very top of its range above \(\alpha'\): with probability at least \(\delta^2\), a random sample satisfies \[f(g)\;\le\;\alpha_\text{max}-\delta(\alpha_\text{max}-\alpha').\] To connect with the comment above, in the special case \(\alpha'=\alpha_\text{min}\) (so \(U=\varnothing\)), this says that random sampling reaches values a positive fraction of the way from the worst value \(\alpha_\text{max}\) toward the best value \(\alpha_\text{min}\).

Proof. Augmenting (?? ), let \[\label{eq:msbds1} \aligned S \;& = \; \{g : f(g) \ge \alpha_\text{max}-\delta(\alpha_\text{max}-\alpha')\} \;\;\text{and}\\ T \;& = \;\{g : \alpha' \le f(g) <\alpha_\text{max}-\delta(\alpha_\text{max}-\alpha')\}, \endaligned\tag{3}\] so that \(|S|+|T|+|U|=1\). The key point is that \(f-\bar{f}\), having mean zero, is orthogonal to all constants, including \(\alpha_\text{max}-\bar{f}\). Thus \[\label{eq:msbds2} \aligned \|f-\bar{f}\|_2^2 \;& \le \; \|\alpha_\text{max}-\bar{f}\|_2^2 \, + \, \|f-\bar{f}\|_2^2 \;= \; \|\alpha_\text{max}-f\|_2^2 \\ & \le \;\delta^2(\alpha_\text{max}-\alpha')^2 |S| + (\alpha_\text{max}-\alpha')^2|T| + (\alpha_\text{max}-\alpha_\text{min})^2|U| \\ & \le \;(\alpha_\text{max}-\alpha')^2(|T|+\delta^2) \, + \, |U|(\alpha_\text{max}-\alpha_\text{min})^2\,, \endaligned\tag{4}\] where in the last step we have used \(|S|\le 1\). Were \(|T|<\delta^2\), then the right-hand side of (4 ) would be less than the left-hand side; hence \(|T|\ge \delta^2\), i.e., (?? ).  

Lemma 1 is stated in a fairly general form (in particular, allowing an exceptional set \(U\)). This generality becomes useful later in the blind-spot robustness analysis (Section 5), where we want to “clip away” tiny regions on which the loss function dips far below its typical values. For the main global-convergence analysis, however, we will invoke Lemma 1 primarily through two simplifications:

1. No exceptional set: \(\alpha'=\alpha_\text{min}\). As mentioned above, this forces \(U=\varnothing\) and (?? ) simplifies to \[\delta\;=\;\frac{1}{\sqrt{2}}\frac{\|f-\bar{f}\|_2}{\alpha_\text{max}-\alpha_\text{min}}\] (the constant \(\tfrac{1}{\sqrt{2}}\) can be improved).

2. Passing to homogeneous spaces. In our applications, the relevant functions live on homogeneous spaces \(G/H\) rather than on \(G\) itself. If \(f\) is right-invariant under a closed subgroup \(H\) of \(G\), then it descends to a function on the quotient manifold \(G/H\), and the same measure-theoretic conclusions hold with respect to the unique \(G\)-invariant probability measure on \(G/H\) (this is the push-forward measure mentioned above; see also [14]).

In the setting of this paper we take \(G=O(d)\) and \(H=O(k)\times O(d-k)\), so \(G/H\simeq \mathcal{G}_{k,d}\). We will also need the “conditioned Grassmannians” \(\mathcal{G}_{k,d,x}\) of \(k\)-planes constrained to contain a given nonzero vector \(x\) defined in (1 ). They play an important role in our analysis, because each iteration of Algorithm 1 samples \(\eta\) from \(\mathcal{G}_{k,d,x_i}\), the set of \(k\)-planes containing the current iterate \(x_i\). For fixed \(x\neq 0\), \(\mathcal{G}_{k,d,x}\) can be identified (non-canonically) with the lower-dimensional quotient manifold \(O(d-1)/(O(k-1)\times O(d-k))\), reflecting the freedom to choose a \((k-1)\)-plane inside the orthogonal complement of \(x\). In particular, both \(\mathcal{G}_{k,d}\) and \(\mathcal{G}_{k,d,x}\) are homogeneous spaces carrying unique invariant probability measures.

Applying Lemma 1 to the restriction of \(\phi\) to either \(S=\mathcal{G}_{k,d}\) or \(S=\mathcal{G}_{k,d,x}\) yields a constant \(\delta\) (defined below) such that a single uniform sample has probability at least \(\delta^2\) of producing a subspace \(\eta\) with \(\phi(\eta)\) bounded away from the largest value on \(S\). Taking the best among \(T\) independent samples amplifies this success probability to \(1-(1-\delta^2)^T\). We record this amplified sampling guarantee as the following corollary.

The fundamental nature of this Lemma should be compared with results like the Markov inequality or the Chebyshev inequality, which instead give a non-trivial lower bound on the probability of getting a value close to the mean via random sampling.

Though it is overkill to appeal to it for the analysis here, these sampling results were heavily motivated by Kazhdan’s Property (T), a very important and extensively studied notion used in analyzing random walks on certain noncompact Lie groups [15] that holds trivially for all compact Lie groups.

Generality of Lemma 1. Though our primary application is to the orthogonal group \(G=O(d)\), there are other compact groups (which also have Haar probability measures) of interest in the theory of optimization. For example, one may take any finite group or the \(n\)-dimensional torus \((\mathbb{Z}\backslash\mathbb{R})^n\), which is — issues of boundary identifications aside — nearly the \(n\)-dimensional hypercube.

4 Stochastic global optimization: algorithm, guarantees, and examples↩︎

In this section, we present our random-subspace walk, a stochastic global optimization procedure applicable to any continuous loss function, and we prove a global convergence guarantee for this general setting.

4.1 Random-subspace walk↩︎

Our objective is global minimization: we seek a point \(x^*\in\mathbb{R}^d\) satisfying \[x^*\in\arg\min_{x\in\mathbb{R}^d}\ell(x).\] Many popular methods in modern non-convex optimization are analyzed through local stationarity conditions (e.g., small gradients) and therefore do not directly track progress toward the global minimum value. In contrast, our random-subspace walk is built so that the analysis follows the gap \(\ell(x_i)-\alpha_\text{min}\) itself, yielding a global convergence guarantee under only continuity (cf. Theorem 1). Of course, when run for limited time the method may return a point near a non-global local minimum; our robustness discussion later in the paper shed light on which local minima can realistically attract the walk.

Assume that we are given access to a black box for solving (or at least approximating) the same optimization problem restricted to a smaller dimensional \(k\)-plane \(\eta\subset\mathbb{R}^d\), i.e., we can find an \(x^*_\eta\) such that \[x_\eta^*\in\arg\min\limits_{x\in\eta}\ell(x),\] or at least a good approximation to one. Our random walk attempts to use this black box repeatedly, over some sequence of \(k\)-planes \(\eta_1,\eta_2,\ldots\), to find a candidate for \(x^*\). Among various other possibilities, a natural random walk is given as follows: starting with some \(x_1\in\mathbb{R}^d\), sample a random \(k\)-plane \(\eta_1\in \mathcal{G}_{k,d,x_1}\) containing \(x_1\); at the \(i\)-th step, find some \(x_{i+1}\in \arg\min_{x\in \eta_i}\ell(x)\) (or an approximation to such a minimizer) and then sample a random \(k\)-plane \(\eta_{i+1}\in \mathcal{G}_{k,d,x_{i+1}}\); stop the algorithm after \(N\) iterations. We state one such variant of this more formally as Algorithm 1, with a concrete suggestion for the stopping rule’s number of steps \(N\).

This is a natural random walk from the computational complexity theory perspective. It leverages the ability to solve several lower-dimensional random problems when attempting to solve a higher-dimensional problem.

4.2 Analysis of convergence properties of Algorithm 1↩︎

We now study the convergence properties of Algorithm 1. Let \(\phi\) be as defined in (2 ) and let \(\alpha_{\text{max}}\) (resp., \(\alpha_\text{min}\)) denote its global maximum (resp., minimum) value. According to Corollary 1 applied to \(S=\mathcal{G}_{k,d}\), Step 3 succeeds in finding some \(\eta_1\in \mathcal{G}_{k,d}\) and an element \(x_1\in \eta_1\) satisfying \[\label{eq:analysisinitial} \ell(x_1) - \alpha_{\text{min}} \;\le \;(1-\delta) \left(\alpha_\text{max} - \alpha_\text{min} \right),\tag{5}\] with probability at least \(\delta_T=1-(1-\delta^2)^T\), where \[\label{eq:deltadef} \delta \;= \;\frac{1}{\sqrt{2}}\frac{\big\|\phi-\bar{\phi}\big\|_2}{\alpha_\text{max} -\alpha_\text{min}}\tag{6}\] and \(\bar{\phi}\) is the mean of \(\phi\). At each iterative step, we apply Corollary 1 to \(S=\mathcal{G}_{k,d,x_i}\), with \(\delta(x_i)\) as given in (?? ), and \(\delta(x_i)_T\) likewise denoting \(1-(1-\delta(x_i)^2)^T\). In this context, \(\phi\)’s minimum value \(m_S\) on \(S\) must equal the global minimum \(\alpha_\text{min}\), because any vector in \(\mathbb{R}^d\) (a global minimizer included) must lie inside some element of \(S\). At the same time, \(\phi\)’s maximum \(M_S\) on \(S\) cannot exceed \(\ell(x_i)\), because each \(\eta\in S\) already contains \(x_i\). Thus with probability at least \(\delta(x_i)_T\), the algorithm finds some \(\eta_{i+1}\in \mathcal{G}_{k,d,x_i}\) and some \(x_{i+1}\in \eta_{i+1}\) with \[\label{eq:analysisiter} \ell(x_{i+1}) - \alpha_{\text{min}} \;\le \;(1-\delta(x_i))\left(\ell(x_i)-\alpha_\text{min} \right).\tag{7}\] We deduce the inequality \[\label{eq:analysiscascade} \ell(x_{n+1}) - \alpha_{\text{min}} \;\le \;(1-\delta)\left( \prod_{i=1}^n(1-\delta(x_i))\right) (\alpha_\text{max}-\alpha_\text{min})\tag{8}\] by telescoping the inequalities (5 )-(7 ).

To simplify (8 ), we define “gap parameter” of the loss function \(\ell\) as \[\label{eq:gapdef} \Theta_\ell \;:= \; \inf_x \frac{\bigg\|\phi|_{\mathcal{G}_{k,d,x}}- \overline{\phi|_{\mathcal{G}_{k,d,x}}}\bigg\|_{2,\mathcal{G}_{k,d,x}}}{\max(\phi|_{\mathcal{G}_{k,d,x}}) - \min(\phi|_{\mathcal{G}_{k,d,x}})}\,,\tag{9}\] where the infimum is taken over all \(x\) for which the restriction of \(\phi(\eta):=\min_{x\in\eta}\ell(x)\) to \(\eta\in\mathcal{G}_{k,d,x}\) is nonconstant, i.e., \(x\) which are not already global minimizers of \(\ell\). Proving good lower bounds for \(\Theta_\ell\) (or even that it is positive) is beyond the scope of the present paper, and likely depends on specific information about the loss function \(\ell\). Therefore each \(\delta(x_i)\ge \frac{1}{\sqrt{2}}\Theta_\ell\) and the probability of success at each stage is at least \(\delta(x_i)_T\ge \Theta_{\ell,T}\), where \[\label{eq:deltabiggerTheta} \Theta_{\ell,T} \;= \; 1-(1-\Theta_\ell^2/2)^T.\tag{10}\] We conclude:

Remarks:

1. Unlike analysis of algorithms in the non-convex optimization literature that track consecutive iterates \(\ell(x_{n})-\ell(x_{n+1})\), and use local regularity properties of \(\ell\) to show they are small, our analysis is instead non-local in the sense that at each step we directly track progress towards the global minimum \(\alpha_\text{min}\), thus only indirectly obtaining the bound on iterates in (?? ) as a byproduct of (?? ). Furthermore, the search space for consecutive iterates \(\{x_i\}\) is in principal global, rather than simply nearby previous iterates.

2. The only regularity assumption made on the loss function \(\ell\) is continuity. In particular, we do not require any bounds on the Lipschitz constant of \(\ell\), as is the case in the analysis of convergence of algorithms such as gradient descent. This is because the relevant quantities such as the “gap parameter” \(\Theta_\ell\) from (9 ) are defined through integrals rather than derivatives.

3. The gap parameter \(\Theta_\ell\) is defined in terms of the ratio of two ways to measure the variation of a function: the \(L^2\)-distance from its mean, and the largest spread among its values. Both quantities are nonzero for nonconstant continuous functions, and each remains unchanged if the mean is subtracted from the function. We were very motivated by Kazhdan’s Property (T) because of its formulations in terms of harmonic analysis on spaces of mean-zero functions, and our results can be derived from Property (T) since the latter automatically holds for compact groups.

4. There are parallels between the gap parameter \(\Theta_\ell\) and the spectral gap of the laplacian on a graph. One of the very important and early applications of Kazhdan’s Property (T) was the first explicit construction of expander graphs in [16]. By virtue of their spectral gap (the difference between the largest two eigenvalues of the graph adjacency matrix), expander graphs have very good mixing properties, i.e., a random walk on one quickly becomes evenly distributed across the graph [13]. This can be understood in terms of how the adjacency matrix of an expander graph shrinks functions orthogonal to constants, which connects to the previous remark. The gap parameter dictates convergence for a similar reason (compare (?? ) with [17] and its proof). This common source of operating principles is the reason why we call \(\Theta_\ell\) the gap parameter of \(\ell\).

4.3 Experimental results↩︎

In order to test the ability of Algorithm 1 to solve optimization problems, we performed some experiments on J.J. Thomson’s classic problem of minimizing the Coulomb potential over all configurations of \(N\) points on the unit sphere in \(\mathbb{R}^{n+1}\) [18]. We selected this problem because it has been heavily studied over the last century, and is notorious for having a complicated landscape of myriad local minima [2].

We applied stereographic projection to convert to a minimization problem over \(\mathbb{R}^{Nn}\). Although we make no claims that Algorithm 1 has superior speed to known approaches, we did find that we could achieve known records on this problem [19] provided adequate CPU time — even with various choices of the “black-box” solver in \(k\)-dimensional subspaces.

The details of the implementation reported on here are as follows. We start from a random point in \(\mathbb{R}^n\) selected by independently sampling each coordinate from the normal distribution with mean zero and variance one. At step \(i\), we choose \(k-1\) vectors at random from the same Gaussian distribution on \(\mathbb{R}^n\), that together with \(x_i\) span a \(k\)-plane \(\eta_i\). Next, we perform the following variant of random descent within \(\eta_i\), starting from \(x_i\): for \(j=1,\ldots,m\) we compute a random vector of length \(2^{-j}\) in \(\mathbb{R}^k\), map it to a vector \(v\in\eta_i\) using the chosen spanning basis, and move by \(v\) if it lowers the value of the loss function. This results in a candidate minimizer \(x_{i+1}\), and the algorithm then repeats at step \(i+1\).

Algorithm 1 performed well in many situations, the following two being the most notable.

42 points in \(\mathbb{R}^7\): Using the values \(k=150\), \(m=20\), and \(T=1\), we found a configuration with Coulomb energy \(\approx227.410\), which ties the existing record in [19] that was announced in 2021, and is smaller than the previous record of \(\approx227.436\) set in [18]. This demonstrates that Algorithm 1 can be competitive in quality with existing approaches.

The optimal Coulomb energy for \(N=120\) points in \(\mathbb{R}^4\) is known be exactly 5395. In [18] the authors perform a fascinating study of how likely it is that their optimization method achieves various local minima. In particular, they found that their algorithm got stuck at different local, non-global minima in 6.8% of their 200,000 trials.

We performed the same experiment, but using Algorithm 1 with the parameter choices \(k=4\), \(m=50\), and \(T=1\), and letting the program run for exactly 24 hours of wall time on the Rutgers Amarel High Performance Cluster (approximately \(1.5\times 10^7\) iterative steps). In our 1,000 trials, only 1.9% got stuck at local minima other than the global minimum. Thus Algorithm 1 cut the rate of failure to achieve the global minimum by over a factor of three.

5 Blind-spot robustness↩︎

Theorem 1 shows that the iterations of Algorithm 1 converge towards the global minimum \(\alpha_\text{min}\) of the loss function \(\ell(\cdot)\), at a rate controlled by the gap parameter \(\Theta_\ell\) (cf. (?? )). However, in practice (given time constraints) the algorithm might yield an output near a different local minimum. One particular such scenario is when \(\ell(\cdot)\) spikes dramatically downward on a set of small measure, which would be a blind spot for random sampling. In this section we analyze this scenario, in which one is only concerned with minimizing the loss function away from such minuscule, invisible subsets.

Consider a value \(\alpha'\) between the extreme values \(\alpha_\text{min}\) and \(\alpha_\text{max}\) of the function \(\phi\) defined in (2 ), all three of which may be unknown. We define an auxiliary loss function \(\ell_{\alpha'}:=\max(\ell, \alpha')\), which is itself continuous (note that even if \(\ell\) is smooth, \(\ell_{\alpha'}\) may fail to inherit differentiability at the level set \(\ell=\alpha'\), but this is not relevant to the present analysis). By applying Theorem 1 to the loss function \(\ell_{\alpha'}\) instead of \(\ell\), we deduce the following estimates:

Theorem 2 specializes to Theorem 1 when \(\alpha'= \alpha_{\text{min}}:=\min(\ell)\), and is meaningless for \(\alpha\ge \alpha_{\text{max}}\). It is therefore important to understand how the quantities \(\Theta_{\ell_{\alpha'}}\) and \(\delta_{\alpha'}\), which are defined in terms of ratios of \(L^2\)-norms to \(L^\infty\)-norms (e.g., in (9 )), vary as \(\alpha'\) increases from \(\alpha_\text{min}\) to \(\alpha_\text{max}\). The denominator is of course nonincreasing in \(\alpha'\). At the same time, the numerators – being averages over many values – might not change much if the set \[\{\eta \in \mathcal{G}_{k,d} \mid \phi(\eta)<\alpha' \}\] is tiny. We conclude that when the loss function has sufficiently steep dips below \(\alpha'\), we can expect that the constants in Theorems 1 and Theorem 2 to be fairly similar. This can be quantified and extended using the full formulation of Lemma 1, which relates the effect of the parameter \(\alpha'\) and the size of the set (?? ) to \(\delta\) in a way very similar to its use in analyzing Algorithm 1.

When the clipped loss function \(\ell_{\alpha'}\) differs from \(\ell\) on only a small set, a random walk may have difficulty distinguishing them. Blindspot robustness is, in spirit, closely related to the set-hitting and set-avoidance problems that arise in the study of random walks on expander graphs, with applications in cryptography and random number generation. Let \(A\) be a measurable subset whose measure is proportion \(\mu\) of the full space’s measure. Given a random walk of length \(L\), the empirical occupation measure of \(A\) is defined as \(\frac{m}{L}\), where \(m\) denotes the number of times the walk visits the set \(A\). For symmetric random walks on expander graphs, it is known (see, e.g., [20]) that the occupation measure concentrates sharply around \(\mu\): specifically, Chernoff-type bounds hold for the deviation of \(\frac{m}{L}\) from \(\mu\). A symmetric walk can loop back to earlier steps, and converge to the uniform distribution. However, in many naturally arising applications the walk needs be asymmetric (e.g., the Pollard Rho walk for discrete logarithms [17]) and has no short cycles; adapting Chernoff bounds is non-trivial. PTW walks do not achieve uniform distribution since they converge to optima, and almost never loop, so a proof technique must be fundamentally different.

5.1 Potential application: adversarial data poisoning↩︎

One setting where blind-spot robustness may be relevant is adversarial data poisoning: an attacker modifies or injects a small fraction of the training data with the goal of steering learning toward a model with degraded (or targeted) test-time behavior. A large literature studies poisoning attacks and defenses in modern machine learning; see [21], [22] for surveys.

The clipped-loss viewpoint above suggests a complementary, geometry-driven mechanism. Many poisoning and backdoor constructions can be interpreted as creating a region of parameter space where the empirical loss function dips anomalously low because it overfits a tiny corrupted subset, while remaining comparatively higher elsewhere. If this “too-good-to-be-true” basin is highly concentrated (small measure under the random-subspace exploration distribution induced by \(\mathcal{G}_{k,d}\) and \(\mathcal{G}_{k,d,x}\)), then the walk is unlikely to encounter it, and Theorem 2 implies that optimizing the clipped objective function \(\ell_{\alpha'}\) behaves similarly to optimizing the original loss function outside that basin.

This should be interpreted as a mechanistic robustness guarantee rather than a worst-case security statement. Our results do not provide general immunity to poisoning: an adversary could in principle create a broad region of deceptively low loss, or otherwise alter the loss landscape on a non-negligible fraction of random subspaces, in which case the blind-spot assumption is violated. Understanding when a concrete poisoning model leads to a genuinely small-measure downward spike for the induced loss landscape is an interesting direction for future work, and likely requires combining our geometric viewpoint with robust-learning and robust-statistics techniques [23], [24].

Finally, note that this effect may become more pronounced for smaller values of \(k\): a perturbation that only “rewards” a very low-dimensional cone of directions is harder to encounter when the walk explores through random, low-dimensional \(k\)-planes.

6 Discussion↩︎

In this paper, we have presented a stochastic global optimization algorithm built from random walks on Grassmannians. The method repeatedly solves low-dimensional random restrictions of a given objective function and monotonically improves the loss function value. Our main convergence theorem provides a guarantee whose only regularity requirement on the objective function is continuity. The analysis highlights a single quantity — the gap parameter \(\Theta_\ell\) — as controlling the rate of progress, in a way reminiscent of how spectral gaps control the behavior of random walks on graphs. Understanding when \(\Theta_\ell\) is bounded away from zero, and how it behaves for concrete model classes, is an important direction for future work.

A second theme is robustness. Section 5 shows that the same analysis naturally supports a blind-spot viewpoint: if the loss function contains extremely narrow regions where it dips far below its typical values, then these regions may have limited influence on an algorithm that explores the landscape through random subspace sampling. This perspective does not replace classical notions of robustness (e.g., uniform stability or worst-case adversarial guarantees), but it captures a failure mode that is common in complex non-convex landscapes: “too-good-to-be-true” minima supported on vanishingly small regions.

Finally, apart from optimization and robustness, the extensive use of randomness in our walk suggests potential privacy-related properties; studying this connection is another interesting direction. Given the continued growth of large-scale, non-convex optimization problems across machine learning and beyond, we believe it is important to develop algorithms that have guarantees under minimal assumptions, alongside interpretations that reflect the geometry of modern objective function landscapes.

References↩︎